\n| \n >>>\n<\/td>\n | \n def plot_bernstein_polynomials():
\n """
\n \u7ed8\u5236\u4f2f\u6069\u65af\u5766\u591a\u9879\u5f0f
\n """
\n n = 4 # \u9636\u6570
\n t = np.linspace(0, 1, 200)<\/p>\n # \u521b\u5efa\u56fe\u5f62
\n fig, axes = plt.subplots(2, 3, figsize=(12, 8))
\n axes = axes.flatten()<\/p>\n for i in range(n+1):
\n # \u8ba1\u7b97\u57fa\u51fd\u6570\u503c
\n B = bernstein_basis_numeric(n, i, t)<\/p>\n # \u7ed8\u5236\u56fe\u5f62
\n ax = axes[i]
\n ax.plot(t, B, 'b-', linewidth=2.5, label=f'$B_{{{i},{n}}}(t)$')
\n ax.fill_between(t, 0, B, alpha=0.2)<\/p>\n # \u8bbe\u7f6e\u56fe\u5f62\u5c5e\u6027
\n ax.set_xlabel('t', fontsize=12)
\n ax.set_ylabel('\u503c', fontsize=12)
\n ax.set_title(f'$B_{{{i},{n}}}(t)$', fontsize=14)
\n ax.grid(True, alpha=0.3)
\n ax.set_xlim(0, 1)
\n ax.set_ylim(0, 1)<\/p>\n # \u9a8c\u8bc1\u5355\u4f4d\u5206\u5272\u6027\u8d28
\n ax_sum = axes[5]
\n sum_B = np.sum([bernstein_basis_numeric(n, i, t) for i in range(n+1)], axis=0)
\n ax_sum.plot(t, sum_B, 'r-', linewidth=2.5, label='\u548c')
\n ax_sum.axhline(y=1, color='k', linestyle='–', alpha=0.5)
\n ax_sum.set_xlabel('t', fontsize=12)
\n ax_sum.set_ylabel(r'$\\\\sum B_{i,n}(t)$', fontsize=12)
\n ax_sum.set_title('\u5355\u4f4d\u5206\u5272\u6027\u8d28', fontsize=14)
\n ax_sum.grid(True, alpha=0.3)
\n ax_sum.set_xlim(0, 1)
\n ax_sum.set_ylim(0.99, 1.01)<\/p>\n fig.suptitle(f'{n}\u6b21\u4f2f\u6069\u65af\u5766\u57fa\u51fd\u6570', fontsize=16, fontweight='bold')
\n fig.tight_layout()
\n return fig<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n \u00a0\n<\/p>\n 1.4 \u7ed3\u679c\u8f93\u51fa<\/h5>\n\n\n\n| \n >>>\n<\/td>\n | \n # \u6267\u884c\u7ed8\u56fe
\nfig_bernstein = plot_bernstein_polynomials()
\nfig_bernstein<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n ![\"\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/02\/20260222072608-699aaf906c644.png\") <\/p>\n
\u56fe1：4\u6b21\u4f2f\u6069\u65af\u5766\u57fa\u51fd\u6570\u3002\u6bcf\u4e2a\u57fa\u51fd\u6570\u5728\u00a0[0,1]\u00a0\u533a\u95f4\u4e0a\u975e\u8d1f，\u4e14\u6240\u6709\u57fa\u51fd\u6570\u7684\u548c\u4e3a1，\u6ee1\u8db3\u5355\u4f4d\u5206\u5272\u6027\u8d28\u3002<\/p>\n 2. \u6807\u51c6（\u591a\u9879\u5f0f）\u8d1d\u585e\u5c14\u66f2\u7ebf<\/h4>\n2.1 \u6570\u5b66\u5b9a\u4e49<\/h5>\n\u7ed9\u5b9a\u00a0n+1\u00a0\u4e2a\u63a7\u5236\u70b9\u00a0P0,P1,\u2026,Pn\u2208\u211dd（\u901a\u5e38\u00a0d=2\u00a0\u6216\u00a03），n\u00a0\u6b21\u6807\u51c6\u8d1d\u585e\u5c14\u66f2\u7ebf\u5b9a\u4e49\u4e3a：<\/p>\n C(t)=\u2211i=0nBi,n(t)\u2062Pi=\u2211i=0nti\u2062(1-t)n-i\u2062Pi<\/p>\n \u5176\u4e2d\u00a0t\u2208[0,1]\u3002<\/p>\n 2.2 \u516c\u5f0f\u63a8\u5bfc<\/h5>\n<\/p>\n2.2.1 \u4ece\u7ebf\u6027\u63d2\u503c\u63a8\u5bfc（\u5fb7\u5361\u65af\u7279\u91cc\u5965\u7b97\u6cd5）<\/h6>\n<\/p>\n\u5fb7\u5361\u65af\u7279\u91cc\u5965\u7b97\u6cd5\u63d0\u4f9b\u4e86\u4e00\u79cd\u9012\u5f52\u8ba1\u7b97\u8d1d\u585e\u5c14\u66f2\u7ebf\u70b9\u7684\u65b9\u6cd5：<\/p>\n<\/p>\n \n\u5bf9\u4e8e\u00a0t\u2208[0,1]，\u5728\u7ebf\u6bb5\u00a0Pi\u2062Pi+1\u00a0\u4e0a\u53d6\u70b9\u00a0Pi1(t)=(1-t)\u2062Pi+t\u2062Pi+1<\/p>\n<\/li>\n \n\u5728\u7ebf\u6bb5\u00a0Pi1\u2062Pi+11\u00a0\u4e0a\u53d6\u70b9\u00a0Pi2(t)=(1-t)\u2062Pi1+t\u2062Pi+11<\/p>\n<\/li>\n \n\u91cd\u590d\u6b64\u8fc7\u7a0b\u00a0n\u00a0\u6b21，\u5f97\u5230\u00a0P0n(t)，\u5373\u66f2\u7ebf\u4e0a\u7684\u70b9\u00a0C(t)<\/p>\n<\/li>\n<\/p>\n 2.2.2 \u4ece\u4e8c\u9879\u5f0f\u5b9a\u7406\u63a8\u5bfc<\/h6>\n<\/p>\n\u5229\u7528\u4e8c\u9879\u5f0f\u5b9a\u7406\u00a0(a+b)n=\u2211i=0nai\u2062bn-i，\u4ee4\u00a0a=t，b=1-t，\u5f97\u5230：<\/p>\n 1=(t+(1-t))n=\u2211i=0nti\u2062(1-t)n-i<\/p>\n \u6bcf\u4e2a\u63a7\u5236\u70b9\u4e58\u4ee5\u76f8\u5e94\u7684\u4f2f\u6069\u65af\u5766\u57fa\u51fd\u6570，\u5f97\u5230\u66f2\u7ebf\u65b9\u7a0b\u3002<\/p>\n 2.3 \u6570\u5b66\u6027\u8d28<\/h5>\n<\/p>\n2.3.1 \u7aef\u70b9\u6027\u8d28<\/h6>\n<\/p>\n<\/p>\n\n\u7aef\u70b9\u63d2\u503c：<\/p>\n<\/li>\n C(0)=P0,C(1)=Pn<\/p>\n \u66f2\u7ebf\u59cb\u7ec8\u901a\u8fc7\u7b2c\u4e00\u4e2a\u548c\u6700\u540e\u4e00\u4e2a\u63a7\u5236\u70b9\u3002<\/p>\n<\/p>\n \n\u7aef\u70b9\u5207\u7ebf：<\/p>\n<\/li>\n C'(0)=n\u2062(P1-P0),C'(1)=n\u2062(Pn-Pn-1)<\/p>\n \u66f2\u7ebf\u5728\u7aef\u70b9\u7684\u5207\u7ebf\u65b9\u5411\u7531\u76f8\u90bb\u63a7\u5236\u70b9\u51b3\u5b9a\u3002<\/p>\n<\/p>\n \n\u7aef\u70b9\u66f2\u7387：<\/p>\n<\/li>\n C''(0)=n\u2062(n-1)\u2062(P2-2\u2062P1+P0)<\/p>\n \u66f2\u7ebf\u5728\u7aef\u70b9\u7684\u66f2\u7387\u7531\u524d\u4e09\u4e2a\u63a7\u5236\u70b9\u51b3\u5b9a\u3002<\/p>\n<\/p>\n 2.3.2 \u51f8\u5305\u6027\u8d28<\/h6>\n<\/p>\n\u51f8\u5305\u662f\u5305\u542b\u6240\u6709\u70b9\u7684\u6700\u5c0f\u51f8\u591a\u8fb9\u5f62\u3002\u5bf9\u4e8e\u4efb\u610f\u00a0t\u2208[0,1]，C(t)\u00a0\u4f4d\u4e8e\u63a7\u5236\u70b9\u7684\u51f8\u5305\u5185\u3002<\/p>\n \u8bc1\u660e：\u7531\u4e8e\u4f2f\u6069\u65af\u5766\u57fa\u51fd\u6570\u975e\u8d1f\u4e14\u548c\u4e3a1，C(t)\u00a0\u662f\u63a7\u5236\u70b9\u7684\u51f8\u7ec4\u5408\u3002<\/p>\n<\/p>\n 2.3.3 \u4eff\u5c04\u4e0d\u53d8\u6027<\/h6>\n<\/p>\n\u5bf9\u4e8e\u4efb\u4f55\u4eff\u5c04\u53d8\u6362\u00a0T：<\/p>\n T(\u2211i=0nBi,n(t)\u2062Pi)=\u2211i=0nBi,n(t)\u2062T(Pi)<\/p>\n<\/p>\n 2.3.4 \u53d8\u5dee\u7f29\u51cf\u6027<\/h6>\n<\/p>\n\u4efb\u4f55\u76f4\u7ebf\u4e0e\u8d1d\u585e\u5c14\u66f2\u7ebf\u7684\u4ea4\u70b9\u6570\u4e0d\u8d85\u8fc7\u8be5\u76f4\u7ebf\u4e0e\u63a7\u5236\u591a\u8fb9\u5f62\u7684\u4ea4\u70b9\u6570\u3002<\/p>\n 2.4 \u4ee3\u7801\u5b9e\u73b0<\/h5>\n\n\n\n| \n >>>\n<\/td>\n | \n def compute_bezier_curve(control_points, t_values=None):
\n """
\n \u8ba1\u7b97\u6807\u51c6\u8d1d\u585e\u5c14\u66f2\u7ebf<\/p>\n \u53c2\u6570:
\n control_points: \u63a7\u5236\u70b9\u6570\u7ec4，\u5f62\u72b6\u4e3a (n+1, 2) \u6216 (n+1, 3)
\n t_values: \u53c2\u6570\u503c\u6570\u7ec4，\u9ed8\u8ba4\u4e3a [0, 1] \u533a\u95f4\u4e0a\u7684100\u4e2a\u70b9<\/p>\n \u8fd4\u56de:
\n \u66f2\u7ebf\u70b9\u6570\u7ec4
\n """
\n control_points = np.array(control_points)
\n n = len(control_points) – 1<\/p>\n if t_values is None:
\n t_values = np.linspace(0, 1, 100)<\/p>\n curve_points = np.zeros((len(t_values), control_points.shape[1]))<\/p>\n for i, t in enumerate(t_values):
\n for j in range(n+1):
\n basis = bernstein_basis_numeric(n, j, t)
\n curve_points[i] += basis * control_points[j]<\/p>\n return curve_points<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n \n\n\n| \n >>>\n<\/td>\n | \n def de_casteljau_algorithm(control_points, t):
\n """
\n \u5fb7\u5361\u65af\u7279\u91cc\u5965\u7b97\u6cd5<\/p>\n \u53c2\u6570:
\n control_points: \u63a7\u5236\u70b9\u6570\u7ec4
\n t: \u53c2\u6570\u503c<\/p>\n \u8fd4\u56de:
\n \u66f2\u7ebf\u5728\u53c2\u6570t\u5904\u7684\u70b9
\n """
\n points = control_points.copy()
\n n = len(points) – 1<\/p>\n for r in range(1, n+1):
\n for i in range(n-r+1):
\n points[i] = (1-t) * points[i] + t * points[i+1]<\/p>\n return points[0]<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n \n\n\n| \n >>>\n<\/td>\n | \n def plot_bezier_curve_example():
\n """
\n \u7ed8\u5236\u6807\u51c6\u8d1d\u585e\u5c14\u66f2\u7ebf\u793a\u4f8b
\n """
\n # \u5b9a\u4e49\u63a7\u5236\u70b9
\n control_points_list = [
\n np.array([[0, 0], [1, 2], [3, 0]]), # \u4e8c\u6b21\u8d1d\u585e\u5c14\u66f2\u7ebf
\n np.array([[0, 0], [1, 3], [2, -1], [4, 2]]), # \u4e09\u6b21\u8d1d\u585e\u5c14\u66f2\u7ebf
\n np.array([[0, 0], [1, 2], [2, -1], [3, 3], [5, 0]]) # \u56db\u6b21\u8d1d\u585e\u5c14\u66f2\u7ebf
\n ]<\/p>\n titles = ['\u4e8c\u6b21\u8d1d\u585e\u5c14\u66f2\u7ebf (n=2)', '\u4e09\u6b21\u8d1d\u585e\u5c14\u66f2\u7ebf (n=3)', '\u56db\u6b21\u8d1d\u585e\u5c14\u66f2\u7ebf (n=4)']<\/p>\n # \u521b\u5efa\u56fe\u5f62
\n fig, axes = plt.subplots(1, 3, figsize=(15, 5))<\/p>\n for idx, (control_points, title) in enumerate(zip(control_points_list, titles)):
\n ax = axes[idx]<\/p>\n # \u8ba1\u7b97\u66f2\u7ebf\u70b9
\n curve_points = compute_bezier_curve(control_points)<\/p>\n # \u7ed8\u5236\u63a7\u5236\u591a\u8fb9\u5f62
\n ax.plot(control_points[:, 0], control_points[:, 1],
\n 'ro-', linewidth=1.5, markersize=8, label='\u63a7\u5236\u591a\u8fb9\u5f62')<\/p>\n # \u7ed8\u5236\u63a7\u5236\u70b9\u6807\u7b7e
\n for i, point in enumerate(control_points):
\n ax.text(point[0]+0.05, point[1]+0.05, f'$P_{i}$',
\n fontsize=12, fontweight='bold')<\/p>\n # \u7ed8\u5236\u8d1d\u585e\u5c14\u66f2\u7ebf
\n ax.plot(curve_points[:, 0], curve_points[:, 1],
\n 'b-', linewidth=2.5, label='\u8d1d\u585e\u5c14\u66f2\u7ebf')<\/p>\n # \u4f7f\u7528\u5fb7\u5361\u65af\u7279\u91cc\u5965\u7b97\u6cd5\u8ba1\u7b97\u66f2\u7ebf\u4e0a\u7684\u70b9
\n t_values = [0.25, 0.5, 0.75]
\n for t in t_values:
\n point = de_casteljau_algorithm(control_points, t)
\n ax.scatter(point[0], point[1], s=80, c='green',
\n edgecolors='black', linewidth=2, zorder=5)
\n ax.text(point[0]+0.05, point[1]+0.05, f't={t}',
\n fontsize=10, fontweight='bold')<\/p>\n ax.set_title(title, fontsize=14, fontweight='bold')
\n ax.set_xlabel('x', fontsize=12)
\n ax.set_ylabel('y', fontsize=12)
\n ax.grid(True, alpha=0.3)
\n ax.legend(loc='best')
\n ax.axis('equal')<\/p>\n fig.suptitle('\u6807\u51c6\u8d1d\u585e\u5c14\u66f2\u7ebf\u793a\u4f8b', fontsize=16, fontweight='bold')
\n fig.tight_layout()
\n return fig<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n \u00a0\n<\/p>\n 2.5 \u7ed3\u679c\u8f93\u51fa<\/h5>\n\n\n\n| \n >>>\n<\/td>\n | \n # \u6267\u884c\u7ed8\u56fe
\nfig_bezier = plot_bezier_curve_example()
\nfig_bezier<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n ![\"\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/02\/20260222072608-699aaf908f532.png\") <\/p>\n<\/p>\n
537 msec<\/p>\n<\/p>\n \u56fe2：\u4e0d\u540c\u9636\u6570\u7684\u6807\u51c6\u8d1d\u585e\u5c14\u66f2\u7ebf\u3002\u4ece\u5de6\u5230\u53f3：\u4e8c\u6b21\u3001\u4e09\u6b21\u3001\u56db\u6b21\u8d1d\u585e\u5c14\u66f2\u7ebf\u3002\u66f2\u7ebf\u59cb\u7ec8\u4f4d\u4e8e\u63a7\u5236\u591a\u8fb9\u5f62\u5f62\u6210\u7684\u51f8\u5305\u5185\u3002<\/p>\n 3. \u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf<\/h4>\n3.1 \u6570\u5b66\u5b9a\u4e49<\/h5>\n\u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf\u662f\u6807\u51c6\u8d1d\u585e\u5c14\u66f2\u7ebf\u7684\u63a8\u5e7f，\u6bcf\u4e2a\u63a7\u5236\u70b9\u00a0Pi\u00a0\u5173\u8054\u4e00\u4e2a\u6743\u91cd\u00a0wi>0\u3002n\u00a0\u6b21\u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf\u5b9a\u4e49\u4e3a：<\/p>\n R(t)=\u2211i=0nwi\u2062Bi,n(t)\u2062Pi\u2211i=0nwi\u2062Bi,n(t)=\u2211i=0nRi,n(t)\u2062Pi<\/p>\n \u5176\u4e2d\u6709\u7406\u4f2f\u6069\u65af\u5766\u57fa\u51fd\u6570：<\/p>\n Ri,n(t)=wi\u2062Bi,n(t)\u2211j=0nwj\u2062Bj,n(t)<\/p>\n 3.2 \u516c\u5f0f\u63a8\u5bfc<\/h5>\n\u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf\u53ef\u4ee5\u901a\u8fc7\u9f50\u6b21\u5750\u6807\u63a8\u5bfc\u3002\u5728\u9f50\u6b21\u5750\u6807\u4e2d，\u6bcf\u4e2a\u63a7\u5236\u70b9\u8868\u793a\u4e3a\u00a0P\u223ci=(wi\u2062xi,wi\u2062yi,wi\u2062zi,wi)\u3002\u66f2\u7ebf\u5728\u9f50\u6b21\u5750\u6807\u4e0b\u4e3a：<\/p>\n R\u223c(t)=\u2211i=0nBi,n(t)\u2062P\u223ci<\/p>\n \u901a\u8fc7\u900f\u89c6\u9664\u6cd5\u5f97\u5230\u6b27\u51e0\u91cc\u5f97\u5750\u6807：<\/p>\n R(t)=(x\u223c(t)w\u223c(t),y\u223c(t)w\u223c(t),z\u223c(t)w\u223c(t))<\/p>\n \u5f53\u6240\u6709\u6743\u91cd\u76f8\u7b49\u65f6，\u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf\u9000\u5316\u4e3a\u6807\u51c6\u8d1d\u585e\u5c14\u66f2\u7ebf\u3002<\/p>\n 3.3 \u6570\u5b66\u6027\u8d28<\/h5>\n<\/p>\n\n\u6295\u5f71\u4e0d\u53d8\u6027：\u5728\u6295\u5f71\u53d8\u6362\u4e0b\u4fdd\u6301\u4e0d\u53d8<\/p>\n<\/li>\n \n\u7aef\u70b9\u63d2\u503c：R(0)=P0，R(1)=Pn（\u5047\u8bbe\u00a0w0,wn>0）<\/p>\n<\/li>\n \n\u51f8\u5305\u6027：\u66f2\u7ebf\u4f4d\u4e8e\u63a7\u5236\u70b9\u7684\u51f8\u5305\u5185<\/p>\n<\/li>\n \n\u6743\u91cd\u5f71\u54cd：wi\u00a0\u589e\u5927\u65f6，\u66f2\u7ebf\u88ab\u62c9\u5411\u63a7\u5236\u70b9\u00a0Pi<\/p>\n<\/li>\n \n\u7cbe\u786e\u8868\u793a\u5706\u9525\u66f2\u7ebf：\u53ef\u4ee5\u7cbe\u786e\u8868\u793a\u5706\u3001\u692d\u5706\u3001\u629b\u7269\u7ebf\u3001\u53cc\u66f2\u7ebf<\/p>\n<\/li>\n 3.4 \u4ee3\u7801\u5b9e\u73b0<\/h5>\n\n\n\n| \n >>>\n<\/td>\n | \n def compute_rational_bezier_curve(control_points, weights, t_values=None):
\n """
\n \u8ba1\u7b97\u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf<\/p>\n \u53c2\u6570:
\n control_points: \u63a7\u5236\u70b9\u6570\u7ec4
\n weights: \u6743\u91cd\u6570\u7ec4
\n t_values: \u53c2\u6570\u503c\u6570\u7ec4<\/p>\n \u8fd4\u56de:
\n \u66f2\u7ebf\u70b9\u6570\u7ec4
\n """
\n control_points = np.array(control_points)
\n weights = np.array(weights)
\n n = len(control_points) – 1<\/p>\n if t_values is None:
\n t_values = np.linspace(0, 1, 100)<\/p>\n curve_points = np.zeros((len(t_values), control_points.shape[1]))<\/p>\n for i, t in enumerate(t_values):
\n numerator = np.zeros(control_points.shape[1])
\n denominator = 0.0<\/p>\n for j in range(n+1):
\n basis = bernstein_basis_numeric(n, j, t)
\n weighted_basis = weights[j] * basis
\n numerator += weighted_basis * control_points[j]
\n denominator += weighted_basis<\/p>\n if denominator != 0:
\n curve_points[i] = numerator \/ denominator<\/p>\n return curve_points<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n \n\n\n| \n >>>\n<\/td>\n | \n def plot_rational_bezier_comparison():
\n """
\n \u6bd4\u8f83\u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf\u548c\u6807\u51c6\u8d1d\u585e\u5c14\u66f2\u7ebf
\n """
\n # \u5b9a\u4e49\u63a7\u5236\u70b9
\n control_points = np.array([
\n [0, 0],
\n [1, 2],
\n [2, -1],
\n [3, 1],
\n [4, 0]
\n ])<\/p>\n # \u5b9a\u4e49\u6743\u91cd\u914d\u7f6e
\n weight_configs = [
\n {'weights': np.ones(5), 'label': '\u6807\u51c6\u8d1d\u585e\u5c14\u66f2\u7ebf (\u6240\u6709\u6743\u91cd=1)', 'color': 'b'},
\n {'weights': np.array([1, 3, 1, 1, 1]), 'label': '\u6709\u7406\u66f2\u7ebf (P\u2081\u6743\u91cd=3)', 'color': 'r'},
\n {'weights': np.array([1, 0.2, 1, 1, 1]), 'label': '\u6709\u7406\u66f2\u7ebf (P\u2081\u6743\u91cd=0.2)', 'color': 'g'},
\n {'weights': np.array([1, 1, 5, 1, 1]), 'label': '\u6709\u7406\u66f2\u7ebf (P\u2082\u6743\u91cd=5)', 'color': 'm'}
\n ]<\/p>\n # \u521b\u5efa\u56fe\u5f62
\n fig, axes = plt.subplots(2, 2, figsize=(12, 10))
\n axes = axes.flatten()<\/p>\n for idx, config in enumerate(weight_configs):
\n ax = axes[idx]
\n weights = config['weights']<\/p>\n # \u8ba1\u7b97\u66f2\u7ebf
\n if np.all(weights == 1):
\n curve = compute_bezier_curve(control_points)
\n else:
\n curve = compute_rational_bezier_curve(control_points, weights)<\/p>\n # \u7ed8\u5236\u63a7\u5236\u591a\u8fb9\u5f62
\n ax.plot(control_points[:, 0], control_points[:, 1],
\n 'ko-', linewidth=1, markersize=6, alpha=0.5)<\/p>\n # \u7ed8\u5236\u63a7\u5236\u70b9（\u5927\u5c0f\u8868\u793a\u6743\u91cd）
\n for i, (point, weight) in enumerate(zip(control_points, weights)):
\n ax.scatter(point[0], point[1], s=weight*80,
\n alpha=0.7, edgecolors='black', linewidth=1)
\n ax.text(point[0]+0.05, point[1]+0.05, f'w={weight}',
\n fontsize=10, fontweight='bold')<\/p>\n # \u7ed8\u5236\u66f2\u7ebf
\n ax.plot(curve[:, 0], curve[:, 1],
\n config['color'] + '-', linewidth=2.5, label=config['label'])<\/p>\n ax.set_title(config['label'], fontsize=12, fontweight='bold')
\n ax.set_xlabel('x', fontsize=12)
\n ax.set_ylabel('y', fontsize=12)
\n ax.grid(True, alpha=0.3)
\n ax.legend(loc='best')
\n ax.axis('equal')<\/p>\n fig.suptitle('\u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf\u4e0e\u6807\u51c6\u8d1d\u585e\u5c14\u66f2\u7ebf\u6bd4\u8f83', fontsize=16, fontweight='bold')
\n fig.tight_layout()
\n return fig<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n \n\n\n| \n >>>\n<\/td>\n | \n def plot_rational_bezier_circle():
\n """
\n \u4f7f\u7528\u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf\u7ed8\u5236\u5706\u5f27
\n """
\n # \u4f7f\u7528\u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf\u8868\u793a90\u5ea6\u5706\u5f27
\n theta = np.pi \/ 4 # 45\u5ea6
\n r = 1.0 # \u534a\u5f84
\n # \u63a7\u5236\u70b9\u5750\u6807
\n P0 = np.array([r, 0])
\n P1 = np.array([r, r * np.tan(theta\/2)])
\n P2 = np.array([r * np.cos(theta), r * np.sin(theta)])
\n # \u6743\u91cd\u8ba1\u7b97（\u7528\u4e8e\u7cbe\u786e\u8868\u793a\u5706\u5f27）
\n w0 = 1.0
\n w1 = np.cos(theta\/2)
\n w2 = 1.0
\n control_points = np.array([P0, P1, P2])
\n weights = np.array([w0, w1, w2])
\n # \u8ba1\u7b97\u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf
\n t_values = np.linspace(0, 1, 100)
\n curve = compute_rational_bezier_curve(control_points, weights, t_values)
\n # \u8ba1\u7b97\u6807\u51c6\u8d1d\u585e\u5c14\u66f2\u7ebf（\u7528\u4e8e\u6bd4\u8f83）
\n std_curve = compute_bezier_curve(control_points, t_values)
\n # \u7cbe\u786e\u5706\u5f27
\n arc_theta = np.linspace(0, theta, 100)
\n exact_arc = np.column_stack([r * np.cos(arc_theta), r * np.sin(arc_theta)])
\n # \u521b\u5efa\u56fe\u5f62
\n fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 5))
\n # \u5de6\u56fe：\u66f2\u7ebf\u6bd4\u8f83
\n ax1.plot(exact_arc[:, 0], exact_arc[:, 1], 'g-',
\n linewidth=3, alpha=0.7, label='\u7cbe\u786e\u5706\u5f27')
\n ax1.plot(curve[:, 0], curve[:, 1], 'b-',
\n linewidth=2, label='\u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf')
\n ax1.plot(std_curve[:, 0], std_curve[:, 1], 'r–',
\n linewidth=2, alpha=0.7, label='\u6807\u51c6\u8d1d\u585e\u5c14\u66f2\u7ebf')
\n # \u7ed8\u5236\u63a7\u5236\u591a\u8fb9\u5f62
\n ax1.plot(control_points[:, 0], control_points[:, 1],
\n 'ko-', linewidth=1, markersize=8, alpha=0.5)
\n # \u6807\u8bb0\u63a7\u5236\u70b9
\n for i, (point, weight) in enumerate(zip(control_points, weights)):
\n ax1.scatter(point[0], point[1], s=weight*100,
\n alpha=0.7, edgecolors='black', linewidth=1)
\n ax1.text(point[0]+0.05, point[1]+0.05, f'P{i}\\\\nw={weight:.3f}',
\n fontsize=10, fontweight='bold')
\n ax1.set_xlabel('x', fontsize=12)
\n ax1.set_ylabel('y', fontsize=12)
\n ax1.set_title('\u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf\u8868\u793a\u5706\u5f27', fontsize=14, fontweight='bold')
\n ax1.grid(True, alpha=0.3)
\n ax1.legend(loc='best')
\n ax1.axis('equal')
\n ax1.set_xlim(-0.2, 1.2)
\n ax1.set_ylim(-0.2, 1.2)
\n # \u53f3\u56fe：\u8bef\u5dee\u5206\u6790
\n # \u8ba1\u7b97\u8bef\u5dee
\n error_rational = np.linalg.norm(curve – exact_arc, axis=1)
\n error_standard = np.linalg.norm(std_curve – exact_arc, axis=1)
\n ax2.plot(t_values, error_rational, 'b-', linewidth=2,
\n label='\u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf\u8bef\u5dee')
\n ax2.plot(t_values, error_standard, 'r–', linewidth=2,
\n label='\u6807\u51c6\u8d1d\u585e\u5c14\u66f2\u7ebf\u8bef\u5dee')
\n ax2.set_xlabel('\u53c2\u6570 t', fontsize=12)
\n ax2.set_ylabel('\u8bef\u5dee', fontsize=12)
\n ax2.set_title('\u4e0e\u7cbe\u786e\u5706\u5f27\u7684\u8bef\u5dee\u6bd4\u8f83', fontsize=14, fontweight='bold')
\n ax2.grid(True, alpha=0.3)
\n ax2.legend(loc='best')
\n fig.suptitle('\u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf\u8868\u793a\u5706\u9525\u66f2\u7ebf', fontsize=16, fontweight='bold')
\n fig.tight_layout()
\n return fig<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n \u00a0\n<\/p>\n 3.5 \u7ed3\u679c\u8f93\u51fa<\/h5>\n\n\n\n| \n >>>\n<\/td>\n | \n # \u6267\u884c\u7ed8\u56fe
\nfig_rational_comparison = plot_rational_bezier_comparison()
\nfig_rational_comparison<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n ![\"\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/02\/20260222072608-699aaf90ae506.png\") <\/p>\n<\/p>\n
681 msec<\/p>\n<\/p>\n \u56fe3：\u4e0d\u540c\u6743\u91cd\u914d\u7f6e\u7684\u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf\u3002\u6743\u91cd\u8d8a\u5927，\u66f2\u7ebf\u8d8a\u63a5\u8fd1\u5bf9\u5e94\u63a7\u5236\u70b9\u3002<\/p>\n \n\n\n| \n >>>\n<\/td>\n | \n # \u6267\u884c\u7ed8\u56fe
\nfig_rational_circle = plot_rational_bezier_circle()
\nfig_rational_circle<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n ![\"\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/02\/20260222072608-699aaf90ca404.png\") <\/p>\n<\/p>\n
299 msec<\/p>\n<\/p>\n \u56fe4：\u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf\u53ef\u4ee5\u7cbe\u786e\u8868\u793a\u5706\u5f27，\u800c\u6807\u51c6\u8d1d\u585e\u5c14\u66f2\u7ebf\u53ea\u80fd\u8fd1\u4f3c\u8868\u793a\u3002<\/p>\n 4. \u8fde\u7eed\u6027\u5206\u6790<\/h4>\n4.1 \u6570\u5b66\u5b9a\u4e49<\/h5>\n<\/p>\n4.1.1 \u53c2\u6570\u8fde\u7eed\u6027 (Ck\u00a0\u8fde\u7eed)<\/h6>\n<\/p>\nC\u8fde\u7eed\u6027\u5173\u6ce8\u7684\u662f\u53c2\u6570\u57df\u4e0a\u7684\u5bfc\u6570\u662f\u5426\u8fde\u7eed，\u5373\u66f2\u7ebf\u5728\u53c2\u6570\u7a7a\u95f4\u4e2d\u7684\u53d8\u5316\u662f\u5426\u5e73\u6ed1\u3002<\/p>\n \n- \n
C\u8fde\u7eed\u6027\u662f\u4e25\u683c\u7684\u6570\u5b66\u5b9a\u4e49，\u8981\u6c42\u53c2\u6570\u5316\u7684\u5bfc\u6570\u5b8c\u5168\u5339\u914d\u3002<\/p>\n<\/li>\n - \n
\u9002\u7528\u4e8e\u9700\u8981\u4e25\u683c\u63a7\u5236\u53c2\u6570\u5316\u884c\u4e3a\u7684\u573a\u666f（\u5982\u52a8\u753b\u8def\u5f84\u89c4\u5212\u3001\u673a\u5668\u4eba\u8f68\u8ff9\u8bbe\u8ba1\u7b49）\u3002<\/p>\n<\/li>\n<\/ul>\n \u5bf9\u4e8e\u4e24\u6761\u8d1d\u585e\u5c14\u66f2\u7ebf：<\/p>\n \n- \n
\u66f2\u7ebf1：C1(t)=\u2211i=0nBi,n(t)\u2062Pi，t\u2208[0,1]<\/p>\n<\/li>\n - \n
\u66f2\u7ebf2：C2(t)=\u2211i=0mBi,m(t)\u2062Qi，t\u2208[0,1]<\/p>\n<\/li>\n<\/ul>\n - \n
C0\u00a0\u8fde\u7eed（\u4f4d\u7f6e\u8fde\u7eed）：<\/p>\n<\/li>\n C1(1)=C2(0)\u21d2Pn=Q0<\/p>\n<\/p>\n - \n
C1\u00a0\u8fde\u7eed（\u5207\u7ebf\u8fde\u7eed）：<\/p>\n \n- \n
Pn=Q0（\u4f4d\u7f6e\u8fde\u7eed）<\/p>\n<\/li>\n - \n
C1'(1)=C2'(0)\u21d2n\u2062(Pn-Pn-1)=m\u2062(Q1-Q0)<\/p>\n<\/li>\n<\/ul>\n<\/li>\n - \n
C2\u00a0\u8fde\u7eed（\u66f2\u7387\u8fde\u7eed）：<\/p>\n \n- \n
C1\u00a0\u8fde\u7eed\u6761\u4ef6<\/p>\n<\/li>\n - \n
C1''(1)=C2''(0)<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/p>\n 4.1.2 \u51e0\u4f55\u8fde\u7eed\u6027 (Gk\u00a0\u8fde\u7eed)<\/h6>\n<\/p>\nG\u8fde\u7eed\u6027\u5173\u6ce8\u7684\u662f\u51e0\u4f55\u5f62\u72b6\u4e0a\u7684\u5149\u6ed1\u7a0b\u5ea6，\u4e0d\u8981\u6c42\u53c2\u6570\u5316\u5bfc\u6570\u5b8c\u5168\u4e00\u81f4，\u53ea\u9700\u51e0\u4f55\u7279\u6027（\u5982\u5207\u7ebf\u65b9\u5411\u3001\u66f2\u7387）\u8fde\u7eed\u5373\u53ef\u3002<\/p>\n \n- \n
G\u8fde\u7eed\u6027\u662f\u51e0\u4f55\u610f\u4e49\u4e0a\u7684\u5b9a\u4e49，\u66f4\u6ce8\u91cd\u89c6\u89c9\u4e0a\u7684\u5e73\u6ed1\u6548\u679c\u3002<\/p>\n<\/li>\n - \n
\u9002\u7528\u4e8e\u5bf9\u53c2\u6570\u5316\u4e0d\u654f\u611f\u7684\u573a\u666f（\u5982\u5de5\u4e1a\u8bbe\u8ba1\u3001\u5b57\u4f53\u8bbe\u8ba1\u7b49）\u3002<\/p>\n<\/li>\n<\/ul>\n - \n
G1\u00a0\u8fde\u7eed（\u5207\u7ebf\u65b9\u5411\u8fde\u7eed）：<\/p>\n \n- \n
C1(1)=C2(0)<\/p>\n<\/li>\n - \n
C1'(1)\u00a0\u4e0e\u00a0C2'(0)\u00a0\u65b9\u5411\u76f8\u540c（\u5927\u5c0f\u53ef\u4e0d\u540c）<\/p>\n<\/li>\n<\/ul>\n<\/li>\n - \n
G2\u00a0\u8fde\u7eed（\u66f2\u7387\u8fde\u7eed）：<\/p>\n \n- \n
G1\u00a0\u8fde\u7eed<\/p>\n<\/li>\n - \n
\u66f2\u7387\u5728\u8fde\u63a5\u70b9\u5904\u8fde\u7eed<\/p>\n<\/li>\n<\/ul>\n<\/li>\n 4.2 \u516c\u5f0f\u63a8\u5bfc<\/h5>\n<\/p>\n4.2.1 \u5bfc\u6570\u516c\u5f0f<\/h6>\n<\/p>\n\u8d1d\u585e\u5c14\u66f2\u7ebf\u7684\u5bfc\u6570：<\/p>\n \u4e00\u9636\u5bfc\u6570：<\/p>\n C'(t)=n\u2062\u2211i=0n-1Bi,n-1(t)\u2062(Pi+1-Pi)<\/p>\n \u4e8c\u9636\u5bfc\u6570：<\/p>\n C''(t)=n\u2062(n-1)\u2062\u2211i=0n-2Bi,n-2(t)\u2062(Pi+2-2\u2062Pi+1+Pi)<\/p>\n<\/p>\n 4.2.2 \u66f2\u7387\u516c\u5f0f<\/h6>\n<\/p>\n\u5bf9\u4e8e\u5e73\u9762\u66f2\u7ebf\u00a0C(t)=(x(t),y(t))，\u66f2\u7387\u4e3a：<\/p>\n \u03ba(t)=|x'(t)\u2062y''(t)-y'(t)\u2062x''(t)|[x'(t)2+y'(t)2]3\/2<\/p>\n 4.3 \u4ee3\u7801\u5b9e\u73b0<\/h5>\n\n\n\n| \n >>>\n<\/td>\n | \n def compute_bezier_derivative(control_points, t, order=1):
\n """
\n \u8ba1\u7b97\u8d1d\u585e\u5c14\u66f2\u7ebf\u7684\u5bfc\u6570<\/p>\n \u53c2\u6570:
\n control_points: \u63a7\u5236\u70b9\u6570\u7ec4
\n t: \u53c2\u6570\u503c
\n order: \u5bfc\u6570\u9636\u6570 (0: \u4f4d\u7f6e, 1: \u4e00\u9636\u5bfc\u6570, 2: \u4e8c\u9636\u5bfc\u6570)<\/p>\n \u8fd4\u56de:
\n \u6307\u5b9a\u9636\u6570\u7684\u5bfc\u6570
\n """
\n control_points = np.array(control_points)
\n n = len(control_points) – 1<\/p>\n if order == 0:
\n # \u4f4d\u7f6e
\n return de_casteljau_algorithm(control_points, t)<\/p>\n if n < order:
\n # \u9636\u6570\u4e0d\u591f
\n return np.zeros_like(control_points[0])<\/p>\n # \u8ba1\u7b97\u5bfc\u6570\u63a7\u5236\u70b9
\n points = control_points.copy()
\n for r in range(order):
\n m = n – r
\n new_points = np.zeros((m, control_points.shape[1]))
\n for i in range(m):
\n # \u5bfc\u6570\u516c\u5f0f: C^{(r)}(t) \u7684\u63a7\u5236\u70b9\u662f\u539f\u59cb\u63a7\u5236\u70b9\u7684r\u9636\u5dee\u5206
\n new_points[i] = m * (points[i+1] – points[i])
\n points = new_points<\/p>\n # \u4f7f\u7528\u5fb7\u5361\u65af\u7279\u91cc\u5965\u7b97\u6cd5\u8ba1\u7b97\u5bfc\u6570\u503c
\n return de_casteljau_algorithm(points, t)<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n 52 msec<\/p>\n<\/p>\n \n\n\n| \n >>>\n<\/td>\n | \n def compute_curvature(control_points, t):
\n """
\n \u8ba1\u7b97\u8d1d\u585e\u5c14\u66f2\u7ebf\u7684\u66f2\u7387<\/p>\n \u53c2\u6570:
\n control_points: \u63a7\u5236\u70b9\u6570\u7ec4 (\u4e8c\u7ef4)
\n t: \u53c2\u6570\u503c<\/p>\n \u8fd4\u56de:
\n \u66f2\u7387\u503c
\n """
\n # \u8ba1\u7b97\u4e00\u9636\u548c\u4e8c\u9636\u5bfc\u6570
\n deriv1 = compute_bezier_derivative(control_points, t, 1)
\n deriv2 = compute_bezier_derivative(control_points, t, 2)<\/p>\n x1, y1 = deriv1[0], deriv1[1]
\n x2, y2 = deriv2[0], deriv2[1]<\/p>\n # \u8ba1\u7b97\u66f2\u7387
\n numerator = abs(x1*y2 – y1*x2)
\n denominator = (x1**2 + y1**2)**1.5<\/p>\n return numerator \/ denominator if denominator != 0 else 0<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n 36 msec<\/p>\n<\/p>\n \n\n\n| \n >>>\n<\/td>\n | \n def analyze_continuity(curve1_points, curve2_points):
\n """
\n \u5206\u6790\u4e24\u6761\u8d1d\u585e\u5c14\u66f2\u7ebf\u7684\u8fde\u7eed\u6027
\n \u53c2\u6570:
\n curve1_points: \u7b2c\u4e00\u6761\u66f2\u7ebf\u7684\u63a7\u5236\u70b9
\n curve2_points: \u7b2c\u4e8c\u6761\u66f2\u7ebf\u7684\u63a7\u5236\u70b9
\n """
\n curve1_points = np.array(curve1_points);curve2_points = np.array(curve2_points)
\n # \u8ba1\u7b97\u8fde\u63a5\u70b9\u4f4d\u7f6e
\n C1_end = compute_bezier_derivative(curve1_points, 1.0, 0)
\n C2_start = compute_bezier_derivative(curve2_points, 0.0, 0)
\n # \u8ba1\u7b97\u5bfc\u6570
\n C1_end_deriv1 = compute_bezier_derivative(curve1_points, 1.0, 1)
\n C2_start_deriv1 = compute_bezier_derivative(curve2_points, 0.0, 1)
\n C1_end_deriv2 = compute_bezier_derivative(curve1_points, 1.0, 2)
\n C2_start_deriv2 = compute_bezier_derivative(curve2_points, 0.0, 2)
\n # \u8ba1\u7b97\u66f2\u7387
\n curvature1_end = compute_curvature(curve1_points, 1.0)
\n curvature2_start = compute_curvature(curve2_points, 0.0)
\n # \u5224\u65ad\u8fde\u7eed\u6027
\n C0_continuous = np.allclose(C1_end, C2_start, rtol=1e-10, atol=1e-10)
\n # G1\u8fde\u7eed：\u5207\u7ebf\u65b9\u5411\u76f8\u540c
\n if np.linalg.norm(C1_end_deriv1) > 1e-10 and np.linalg.norm(C2_start_deriv1) > 1e-10:
\n unit_deriv1 = C1_end_deriv1 \/ np.linalg.norm(C1_end_deriv1)
\n unit_deriv2 = C2_start_deriv1 \/ np.linalg.norm(C2_start_deriv1)
\n G1_continuous = np.dot(unit_deriv1, unit_deriv2) > 0.999
\n else:
\n G1_continuous = False
\n # C1\u8fde\u7eed：\u5bfc\u6570\u76f8\u7b49
\n C1_continuous = np.allclose(C1_end_deriv1, C2_start_deriv1, rtol=1e-10, atol=1e-10)
\n # G2\u8fde\u7eed：\u66f2\u7387\u8fde\u7eed
\n G2_continuous = np.allclose(curvature1_end, curvature2_start, rtol=1e-5, atol=1e-5)
\n # C2\u8fde\u7eed：\u4e8c\u9636\u5bfc\u6570\u76f8\u7b49
\n C2_continuous = np.allclose(C1_end_deriv2, C2_start_deriv2, rtol=1e-10, atol=1e-10)
\n # \u521b\u5efa\u56fe\u5f62
\n fig, axes = plt.subplots(2, 2, figsize=(12, 10))
\n # \u5de6\u4e0a：\u66f2\u7ebf\u62fc\u63a5
\n ax = axes[0, 0]
\n # \u8ba1\u7b97\u66f2\u7ebf
\n curve1 = compute_bezier_curve(curve1_points);curve2 = compute_bezier_curve(curve2_points)
\n # \u7ed8\u5236\u63a7\u5236\u591a\u8fb9\u5f62
\n ax.plot(curve1_points[:, 0], curve1_points[:, 1], 'ro-', linewidth=1.5, markersize=8, alpha=0.7, label='\u66f2\u7ebf1\u63a7\u5236\u591a\u8fb9\u5f62')
\n ax.plot(curve2_points[:, 0], curve2_points[:, 1], 'bo-', linewidth=1.5, markersize=8, alpha=0.7, label='\u66f2\u7ebf2\u63a7\u5236\u591a\u8fb9\u5f62')
\n # \u7ed8\u5236\u66f2\u7ebf
\n ax.plot(curve1[:, 0], curve1[:, 1], 'r-', linewidth=2.5, label='\u66f2\u7ebf1')
\n ax.plot(curve2[:, 0], curve2[:, 1], 'b-', linewidth=2.5, label='\u66f2\u7ebf2')
\n # \u6807\u8bb0\u8fde\u63a5\u70b9
\n joint_point = C1_end
\n ax.scatter(joint_point[0], joint_point[1], s=150, color='green', edgecolors='black', linewidth=2, zorder=5, label='\u8fde\u63a5\u70b9')
\n # \u7ed8\u5236\u5207\u7ebf
\n scale = 0.3
\n ax.arrow(joint_point[0], joint_point[1], C1_end_deriv1[0]*scale, C1_end_deriv1[1]*scale, head_width=0.05, head_length=0.1, fc='red', ec='red', linewidth=2, label='\u66f2\u7ebf1\u7ec8\u70b9\u5207\u7ebf')
\n ax.arrow(joint_point[0], joint_point[1], C2_start_deriv1[0]*scale, C2_start_deriv1[1]*scale, head_width=0.05, head_length=0.1, fc='blue', ec='blue', linewidth=2, label='\u66f2\u7ebf2\u8d77\u70b9\u5207\u7ebf')
\n ax.set_title('\u66f2\u7ebf\u62fc\u63a5', fontsize=14, fontweight='bold')
\n ax.set_xlabel('x', fontsize=12);ax.set_ylabel('y', fontsize=12)
\n ax.grid(True, alpha=0.3);ax.legend(loc='best');ax.axis('equal')
\n # \u53f3\u4e0a：\u66f2\u7387\u5206\u6790
\n ax = axes[0, 1];t_values = np.linspace(0, 1, 100)
\n curvature1 = [compute_curvature(curve1_points, t) for t in t_values]
\n curvature2 = [compute_curvature(curve2_points, t) for t in t_values]
\n ax.plot(t_values, curvature1, 'r-', linewidth=2, label='\u66f2\u7ebf1\u66f2\u7387')
\n ax.plot(t_values, curvature2, 'b-', linewidth=2, label='\u66f2\u7ebf2\u66f2\u7387')
\n # \u6807\u8bb0\u8fde\u63a5\u70b9\u5904\u7684\u66f2\u7387
\n ax.scatter([1.0], [curvature1_end], s=100, c='red', edgecolors='black', linewidth=2, zorder=5)
\n ax.scatter([0.0], [curvature2_start], s=100, c='blue', edgecolors='black', linewidth=2, zorder=5)
\n ax.set_title('\u66f2\u7387\u5206\u6790', fontsize=14, fontweight='bold')
\n ax.set_xlabel('\u53c2\u6570 t', fontsize=12)
\n ax.set_ylabel(r'\u66f2\u7387 $\\\\kappa$', fontsize=12)
\n ax.grid(True, alpha=0.3)
\n ax.legend(loc='best')
\n # \u5de6\u4e0b：\u5bfc\u6570\u5206\u6790
\n ax = axes[1, 0]
\n # \u8ba1\u7b97\u5bfc\u6570\u5927\u5c0f
\n deriv1_mag = [np.linalg.norm(compute_bezier_derivative(curve1_points, t, 1)) for t in t_values]
\n deriv2_mag = [np.linalg.norm(compute_bezier_derivative(curve2_points, t, 1)) for t in t_values]
\n ax.plot(t_values, deriv1_mag, 'r-', linewidth=2, label='\u66f2\u7ebf1\u5bfc\u6570\u5927\u5c0f')
\n ax.plot(t_values, deriv2_mag, 'b-', linewidth=2, label='\u66f2\u7ebf2\u5bfc\u6570\u5927\u5c0f')
\n ax.set_title('\u4e00\u9636\u5bfc\u6570\u5927\u5c0f', fontsize=14, fontweight='bold')
\n ax.set_xlabel('\u53c2\u6570 t', fontsize=12)
\n ax.set_ylabel('$|C\\\\'(t)|$', fontsize=12)
\n ax.grid(True, alpha=0.3)
\n ax.legend(loc='best')
\n # \u53f3\u4e0b：\u8fde\u7eed\u6027\u7ed3\u679c
\n ax = axes[1, 1]
\n ax.axis('off')
\n # \u663e\u793a\u8fde\u7eed\u6027\u7ed3\u679c
\n result_text = "\u8fde\u7eed\u6027\u5206\u6790\u7ed3\u679c:\\\\n\\\\n"
\n result_text += f"\u4f4d\u7f6e\u8fde\u7eed (C0): {'\u6ee1\u8db3' if C0_continuous else '\u4e0d\u6ee1\u8db3'}\\\\n"
\n result_text += f"\u5207\u7ebf\u65b9\u5411\u8fde\u7eed (G1): {'\u6ee1\u8db3' if G1_continuous else '\u4e0d\u6ee1\u8db3'}\\\\n"
\n result_text += f"\u5207\u7ebf\u8fde\u7eed (C1): {'\u6ee1\u8db3' if C1_continuous else '\u4e0d\u6ee1\u8db3'}\\\\n"
\n result_text += f"\u66f2\u7387\u8fde\u7eed (G2): {'\u6ee1\u8db3' if G2_continuous else '\u4e0d\u6ee1\u8db3'}\\\\n"
\n result_text += f"\u4e8c\u9636\u5bfc\u6570\u8fde\u7eed (C2): {'\u6ee1\u8db3' if C2_continuous else '\u4e0d\u6ee1\u8db3'}\\\\n\\\\n"
\n result_text += "\u8be6\u7ec6\u6570\u503c:\\\\n"
\n result_text += f"\u66f2\u7ebf1\u7ec8\u70b9\u4f4d\u7f6e: {C1_end}\\\\n"
\n result_text += f"\u66f2\u7ebf2\u8d77\u70b9\u4f4d\u7f6e: {C2_start}\\\\n\\\\n"
\n result_text += f"\u66f2\u7ebf1\u7ec8\u70b9\u5207\u7ebf: {C1_end_deriv1}\\\\n"
\n result_text += f"\u66f2\u7ebf2\u8d77\u70b9\u5207\u7ebf: {C2_start_deriv1}\\\\n\\\\n"
\n result_text += f"\u66f2\u7ebf1\u7ec8\u70b9\u66f2\u7387: {curvature1_end:.6f}\\\\n"
\n result_text += f"\u66f2\u7ebf2\u8d77\u70b9\u66f2\u7387: {curvature2_start:.6f}\\\\n\\\\n"
\n result_text += f"\u66f2\u7ebf1\u7ec8\u70b9\u4e8c\u9636\u5bfc\u6570: {C1_end_deriv2}\\\\n"
\n result_text += f"\u66f2\u7ebf2\u8d77\u70b9\u4e8c\u9636\u5bfc\u6570: {C2_start_deriv2}"
\n ax.text(0.1, 0.5, result_text, fontsize=11, verticalalignment='center', bbox=dict(boxstyle='round', facecolor='wheat', alpha=0.8))
\n fig.suptitle('\u8d1d\u585e\u5c14\u66f2\u7ebf\u8fde\u7eed\u6027\u5206\u6790', fontsize=16, fontweight='bold')
\n fig.tight_layout()
\n return fig<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n \n\n\n| \n >>>\n<\/td>\n | \n def plot_continuity_examples():
\n """
\n \u7ed8\u5236\u8fde\u7eed\u6027\u793a\u4f8b
\n """
\n # \u793a\u4f8b1：C0\u8fde\u7eed
\n curve1_c0 = np.array([[0, 0], [1, 2], [2, 1]])
\n curve2_c0 = np.array([[2, 1], [3, 0], [4, 2]])<\/p>\n # \u793a\u4f8b2：G1\u8fde\u7eed
\n curve1_g1 = np.array([[0, 0], [1, 2], [2, 1]])
\n curve2_g1 = np.array([[2, 1], [3, 0.5], [4, 2]]) # P2, Q0, Q1\u5171\u7ebf<\/p>\n # \u793a\u4f8b3：C1\u8fde\u7eed
\n curve1_c1 = np.array([[0, 0], [1, 2], [2, 1]])
\n # \u6ee1\u8db3 C1'(1) = C2'(0)
\n # n=2, m=2, \u9700\u8981 P2-P1 = Q1-Q0
\n Q0 = np.array([2, 1]) # = P2
\n Q1 = Q0 + (curve1_c1[2] – curve1_c1[1])
\n Q2 = np.array([4, 1])
\n curve2_c1 = np.array([Q0, Q1, Q2])<\/p>\n # \u793a\u4f8b4：C2\u8fde\u7eed
\n curve1_c2 = np.array([[0, 0], [1, 1], [2, 2], [3, 1]])
\n # \u6ee1\u8db3 C1''(1) = C2''(0)
\n # \u8fd9\u91cc\u7b80\u5316\u5904\u7406，\u4f7f\u7528\u76f8\u540c\u63a7\u5236\u70b9
\n curve2_c2 = np.array([[3, 1], [4, 0], [5, 1], [6, 2]])<\/p>\n # \u521b\u5efa\u56fe\u5f62
\n fig, axes = plt.subplots(2, 2, figsize=(12, 10))<\/p>\n examples = [
\n (curve1_c0, curve2_c0, "C0\u8fde\u7eed（\u4ec5\u4f4d\u7f6e\u8fde\u7eed）", axes[0, 0]),
\n (curve1_g1, curve2_g1, "G1\u8fde\u7eed（\u5207\u7ebf\u65b9\u5411\u8fde\u7eed）", axes[0, 1]),
\n (curve1_c1, curve2_c1, "C1\u8fde\u7eed（\u5207\u7ebf\u8fde\u7eed）", axes[1, 0]),
\n (curve1_c2, curve2_c2, "C2\u8fde\u7eed（\u66f2\u7387\u8fde\u7eed）", axes[1, 1])
\n ]<\/p>\n for curve1, curve2, title, ax in examples:
\n # \u8ba1\u7b97\u66f2\u7ebf
\n curve1_pts = compute_bezier_curve(curve1)
\n curve2_pts = compute_bezier_curve(curve2)<\/p>\n # \u7ed8\u5236\u63a7\u5236\u591a\u8fb9\u5f62
\n ax.plot(curve1[:, 0], curve1[:, 1], 'ro-',
\n linewidth=1.5, markersize=6, alpha=0.7)
\n ax.plot(curve2[:, 0], curve2[:, 1], 'bo-',
\n linewidth=1.5, markersize=6, alpha=0.7)<\/p>\n # \u7ed8\u5236\u66f2\u7ebf
\n ax.plot(curve1_pts[:, 0], curve1_pts[:, 1], 'r-',
\n linewidth=2.5, alpha=0.8)
\n ax.plot(curve2_pts[:, 0], curve2_pts[:, 1], 'b-',
\n linewidth=2.5, alpha=0.8)<\/p>\n # \u6807\u8bb0\u8fde\u63a5\u70b9
\n joint_point = curve1[-1]
\n ax.scatter(joint_point[0], joint_point[1], s=100,
\n color='green', edgecolors='black', linewidth=2)<\/p>\n ax.set_title(title, fontsize=12, fontweight='bold')
\n ax.set_xlabel('x', fontsize=10)
\n ax.set_ylabel('y', fontsize=10)
\n ax.grid(True, alpha=0.3)
\n ax.axis('equal')<\/p>\n fig.suptitle('\u8d1d\u585e\u5c14\u66f2\u7ebf\u8fde\u7eed\u6027\u793a\u4f8b', fontsize=16, fontweight='bold')
\n fig.tight_layout()
\n return fig<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n 4.4 \u7ed3\u679c\u8f93\u51fa<\/h5>\n\n\n\n| \n >>>\n<\/td>\n | \n # \u6267\u884c\u8fde\u7eed\u6027\u5206\u6790
\nfig_continuity = analyze_continuity(
\n np.array([[0, 0], [1, 2], [2, 1]]),
\n np.array([[2, 1], [3, 0.5], [4, 2]])
\n)<\/p>\n fig_examples = plot_continuity_examples()
\nfig_continuity<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n ![\"\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/02\/20260222072608-699aaf90debd7.png\") <\/p>\n<\/p>\n
651 msec<\/p>\n<\/p>\n \u56fe5：\u8d1d\u585e\u5c14\u66f2\u7ebf\u8fde\u7eed\u6027\u5206\u6790\u3002\u663e\u793a\u4e86\u66f2\u7ebf\u62fc\u63a5\u3001\u66f2\u7387\u53d8\u5316\u3001\u5bfc\u6570\u5927\u5c0f\u548c\u8fde\u7eed\u6027\u5224\u65ad\u7ed3\u679c\u3002<\/p>\n \n\n\n| \n >>>\n<\/td>\n | \n fig_examples = plot_continuity_examples()
\nfig_examples<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n ![\"\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/02\/20260222072609-699aaf9106c0f.png\") <\/p>\n<\/p>\n
557 msec<\/p>\n<\/p>\n \u56fe6：\u4e0d\u540c\u8fde\u7eed\u6027\u7ea7\u522b\u7684\u8d1d\u585e\u5c14\u66f2\u7ebf\u62fc\u63a5\u793a\u4f8b\u3002<\/p>\n 5. \u66f2\u7ebf\u6027\u8d28\u5206\u6790<\/h4>\n5.1 \u51f8\u5305\u6027\u8bc1\u660e<\/h5>\n<\/p>\n5.1.1 \u6570\u5b66\u8bc1\u660e<\/h6>\n<\/p>\n\u51f8\u5305\u6027\u8bc1\u660e\u57fa\u4e8e\u4f2f\u6069\u65af\u5766\u57fa\u51fd\u6570\u7684\u6027\u8d28：<\/p>\n<\/p>\n - \n
\u975e\u8d1f\u6027：Bi,n(t)\u22650，\u5bf9\u00a0t\u2208[0,1]<\/p>\n<\/li>\n - \n
\u5355\u4f4d\u5206\u5272：\u2211i=0nBi,n(t)=1<\/p>\n<\/li>\n \u56e0\u6b64，\u5bf9\u4e8e\u4efb\u610f\u00a0t\u2208[0,1]：<\/p>\n C(t)=\u2211i=0nBi,n(t)\u2062Pi<\/p>\n \u662f\u63a7\u5236\u70b9\u00a0Pi\u00a0\u7684\u51f8\u7ec4\u5408，\u6545\u00a0C(t)\u00a0\u4f4d\u4e8e\u63a7\u5236\u70b9\u7684\u51f8\u5305\u5185\u3002<\/p>\n 5.2 \u53d8\u5dee\u7f29\u51cf\u6027<\/h5>\n<\/p>\n5.2.1 \u6570\u5b66\u5b9a\u4e49<\/h6>\n<\/p>\n\u5bf9\u4e8e\u4efb\u4f55\u76f4\u7ebf\u00a0L，\u8bbe\u00a0NC\u00a0\u4e3a\u00a0L\u00a0\u4e0e\u8d1d\u585e\u5c14\u66f2\u7ebf\u00a0C\u00a0\u7684\u4ea4\u70b9\u6570，NP\u00a0\u4e3a\u00a0L\u00a0\u4e0e\u63a7\u5236\u591a\u8fb9\u5f62\u00a0P\u00a0\u7684\u4ea4\u70b9\u6570，\u5219：<\/p>\n NC\u2264NP<\/p>\n 5.3 \u4ee3\u7801\u5b9e\u73b0<\/h5>\n\n\n\n| \n >>>\n<\/td>\n | \n def line_intersection(p1, p2, line_slope, line_intercept):
\n """\u8ba1\u7b97\u7ebf\u6bb5\u4e0e\u76f4\u7ebf\u7684\u4ea4\u70b9"""
\n x1, y1 = p1
\n x2, y2 = p2
\n # \u7ebf\u6bb5\u65b9\u7a0b\u53c2\u6570\u5f62\u5f0f
\n dx = x2 – x1
\n dy = y2 – y1
\n if abs(dx) < 1e-10:
\n # \u5782\u76f4\u7ebf\u6bb5
\n x = x1
\n y = line_slope * x + line_intercept
\n if min(y1, y2) <= y <= max(y1, y2):
\n return [(x, y)]
\n return []
\n # \u53c2\u6570t\u6ee1\u8db3 y1 + t*dy = line_slope*(x1 + t*dx) + line_intercept，\u89e3\u51fat
\n t = (line_slope*x1 + line_intercept – y1) \/ (dy – line_slope*dx)
\n if 0 <= t <= 1:
\n x = x1 + t*dx
\n y = y1 + t*dy
\n return [(x, y)]
\n return []<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n 28 msec<\/p>\n<\/p>\n \n\n\n| \n >>>\n<\/td>\n | \n def point_in_hull(point, hull_points):
\n """\u68c0\u67e5\u70b9\u662f\u5426\u5728\u51f8\u5305\u5185（\u5c04\u7ebf\u6cd5）"""
\n x, y = point
\n n = len(hull_points) – 1
\n inside = False
\n p1x, p1y = hull_points[0]
\n for i in range(1, n+1):
\n p2x, p2y = hull_points[i % n]
\n if y > min(p1y, p2y):
\n if y <= max(p1y, p2y):
\n if x <= max(p1x, p2x):
\n if p1y != p2y:
\n xinters = (y-p1y)*(p2x-p1x)\/(p2y-p1y)+p1x
\n if p1x == p2x or x <= xinters:
\n inside = not inside
\n p1x, p1y = p2x, p2y
\n return inside<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n 28 msec<\/p>\n<\/p>\n \n\n\n| \n >>>\n<\/td>\n | \n def verify_convex_hull(control_points):
\n """
\n \u9a8c\u8bc1\u8d1d\u585e\u5c14\u66f2\u7ebf\u7684\u51f8\u5305\u6027\u8d28
\n """
\n control_points = np.array(control_points)
\n # \u8ba1\u7b97\u51f8\u5305
\n from scipy.spatial import ConvexHull
\n hull = ConvexHull(control_points)
\n # \u8ba1\u7b97\u8d1d\u585e\u5c14\u66f2\u7ebf
\n t_values = np.linspace(0, 1, 200)
\n curve_points = compute_bezier_curve(control_points, t_values)
\n # \u521b\u5efa\u56fe\u5f62
\n fig, axes = plt.subplots(1, 2, figsize=(12, 5))
\n # \u5de6\u56fe：\u51f8\u5305\u6f14\u793a
\n ax = axes[0]
\n # \u7ed8\u5236\u63a7\u5236\u591a\u8fb9\u5f62
\n ax.plot(control_points[:, 0], control_points[:, 1], 'ro-', linewidth=1.5, markersize=8, alpha=0.7, label='\u63a7\u5236\u591a\u8fb9\u5f62')
\n # \u7ed8\u5236\u51f8\u5305
\n hull_points = control_points[hull.vertices]
\n hull_points = np.vstack([hull_points, hull_points[0]]) # \u95ed\u5408
\n ax.plot(hull_points[:, 0], hull_points[:, 1], 'g–', linewidth=2, alpha=0.7, label='\u51f8\u5305')
\n ax.fill(hull_points[:, 0], hull_points[:, 1], 'green', alpha=0.2)
\n # \u7ed8\u5236\u8d1d\u585e\u5c14\u66f2\u7ebf
\n ax.plot(curve_points[:, 0], curve_points[:, 1], 'b-', linewidth=2.5, label='\u8d1d\u585e\u5c14\u66f2\u7ebf')
\n # \u968f\u673a\u91c7\u6837\u66f2\u7ebf\u4e0a\u7684\u70b9
\n np.random.seed(42)
\n sample_t = np.random.rand(50)
\n sample_points = np.array([de_casteljau_algorithm(control_points, t)
\n for t in sample_t])
\n ax.scatter(sample_points[:, 0], sample_points[:, 1], s=30, c='orange', alpha=0.6, label='\u66f2\u7ebf\u91c7\u6837\u70b9')
\n ax.set_title('\u51f8\u5305\u6027\u8d28\u6f14\u793a', fontsize=14, fontweight='bold')
\n ax.set_xlabel('x', fontsize=12)
\n ax.set_ylabel('y', fontsize=12)
\n ax.grid(True, alpha=0.3)
\n ax.legend(loc='best')
\n ax.axis('equal')
\n # \u53f3\u56fe：\u53d8\u5dee\u7f29\u51cf\u6027\u6f14\u793a
\n ax = axes[1]
\n # \u7ed8\u5236\u63a7\u5236\u591a\u8fb9\u5f62
\n ax.plot(control_points[:, 0], control_points[:, 1], 'ro-', linewidth=1.5, markersize=8, alpha=0.7, label='\u63a7\u5236\u591a\u8fb9\u5f62')
\n # \u7ed8\u5236\u8d1d\u585e\u5c14\u66f2\u7ebf
\n ax.plot(curve_points[:, 0], curve_points[:, 1], 'b-', linewidth=2.5, label='\u8d1d\u585e\u5c14\u66f2\u7ebf')
\n # \u7ed8\u5236\u6d4b\u8bd5\u76f4\u7ebf
\n x_min, x_max = control_points[:, 0].min(), control_points[:, 0].max()
\n y_min, y_max = control_points[:, 1].min(), control_points[:, 1].max()
\n # \u968f\u673a\u751f\u6210\u76f4\u7ebf
\n np.random.seed(42)
\n line_slope = np.random.uniform(-2, 2)
\n line_intercept = np.random.uniform(y_min, y_max)
\n line_x = np.linspace(x_min – 1, x_max + 1, 100)
\n line_y = line_slope * line_x + line_intercept
\n ax.plot(line_x, line_y, 'g–', linewidth=2, alpha=0.7, label='\u6d4b\u8bd5\u76f4\u7ebf')
\n # \u8ba1\u7b97\u63a7\u5236\u591a\u8fb9\u5f62\u4e0e\u76f4\u7ebf\u7684\u4ea4\u70b9（\u7b80\u5316\u5904\u7406）
\n control_intersections = []
\n for i in range(len(control_points)-1):
\n intersections = line_intersection(
\n control_points[i], control_points[i+1],
\n line_slope, line_intercept
\n )
\n control_intersections.extend(intersections)
\n # \u4e0e\u8d1d\u585e\u5c14\u66f2\u7ebf\u7684\u8fd1\u4f3c\u4ea4\u70b9
\n curve_intersections = []
\n for i in range(len(curve_points)-1):
\n intersections = line_intersection(
\n curve_points[i], curve_points[i+1],
\n line_slope, line_intercept
\n )
\n curve_intersections.extend(intersections)
\n # \u7ed8\u5236\u4ea4\u70b9
\n if control_intersections:
\n control_intersections = np.array(control_intersections)
\n ax.scatter(control_intersections[:, 0], control_intersections[:, 1], s=100, c='red', edgecolors='black', linewidth=2, label=f'\u63a7\u5236\u591a\u8fb9\u5f62\u4ea4\u70b9 ({len(control_intersections)}\u4e2a)')
\n if curve_intersections:
\n curve_intersections = np.array(curve_intersections)
\n ax.scatter(curve_intersections[:, 0], curve_intersections[:, 1], s=100, c='blue', marker='s', edgecolors='black', linewidth=2,label=f'\u8d1d\u585e\u5c14\u66f2\u7ebf\u4ea4\u70b9 ({len(curve_intersections)}\u4e2a)')
\n ax.set_title('\u53d8\u5dee\u7f29\u51cf\u6027\u6f14\u793a', fontsize=14, fontweight='bold')
\n ax.set_xlabel('x', fontsize=12)
\n ax.set_ylabel('y', fontsize=12)
\n ax.grid(True, alpha=0.3)
\n ax.legend(loc='best')
\n ax.axis('equal')
\n fig.suptitle('\u8d1d\u585e\u5c14\u66f2\u7ebf\u51e0\u4f55\u6027\u8d28\u9a8c\u8bc1', fontsize=16, fontweight='bold')
\n fig.tight_layout()
\n # \u8f93\u51fa\u9a8c\u8bc1\u7ed3\u679c
\n print("\u51f8\u5305\u6027\u9a8c\u8bc1:")
\n print(f" \u63a7\u5236\u70b9\u6570\u91cf: {len(control_points)}")
\n print(f" \u51f8\u5305\u9876\u70b9\u6570\u91cf: {len(hull.vertices)}")
\n print(f" \u66f2\u7ebf\u91c7\u6837\u70b9\u6570\u91cf: {len(sample_points)}")
\n # \u68c0\u67e5\u91c7\u6837\u70b9\u662f\u5426\u5728\u51f8\u5305\u5185
\n all_inside = all(point_in_hull(point, hull_points) for point in sample_points)
\n print(f" \u6240\u6709\u91c7\u6837\u70b9\u90fd\u5728\u51f8\u5305\u5185: {'\u662f' if all_inside else '\u5426'}")
\n print("\\\\n\u53d8\u5dee\u7f29\u51cf\u6027\u9a8c\u8bc1:")
\n print(f" \u76f4\u7ebf\u65b9\u7a0b: y = {line_slope:.2f}x + {line_intercept:.2f}")
\n print(f" \u4e0e\u63a7\u5236\u591a\u8fb9\u5f62\u4ea4\u70b9\u6570: {len(control_intersections)}")
\n print(f" \u4e0e\u8d1d\u585e\u5c14\u66f2\u7ebf\u4ea4\u70b9\u6570: {len(curve_intersections)}")
\n print(f" \u4ea4\u70b9\u6570\u51cf\u5c11: {'\u662f' if len(curve_intersections) <= len(control_intersections) else '\u5426'}")
\n return fig<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n 92 msec<\/p>\n<\/p>\n \n\n\n| \n >>>\n<\/td>\n | \n def analyze_endpoint_properties(control_points):
\n """
\n \u5206\u6790\u8d1d\u585e\u5c14\u66f2\u7ebf\u7684\u7aef\u70b9\u6027\u8d28
\n """
\n control_points = np.array(control_points)
\n n = len(control_points) – 1
\n # \u8ba1\u7b97\u7aef\u70b9\u5904\u7684\u503c
\n start_point = compute_bezier_derivative(control_points, 0.0, 0)
\n end_point = compute_bezier_derivative(control_points, 1.0, 0)
\n # \u8ba1\u7b97\u7aef\u70b9\u5bfc\u6570
\n start_deriv = compute_bezier_derivative(control_points, 0.0, 1)
\n end_deriv = compute_bezier_derivative(control_points, 1.0, 1)
\n # \u8ba1\u7b97\u7aef\u70b9\u4e8c\u9636\u5bfc\u6570
\n start_second_deriv = compute_bezier_derivative(control_points, 0.0, 2)
\n # \u7406\u8bba\u503c
\n theoretical_start_deriv = n * (control_points[1] – control_points[0])
\n theoretical_end_deriv = n * (control_points[-1] – control_points[-2])
\n if n >= 2:
\n theoretical_start_second_deriv = n * (n-1) * (control_points[2] – 2*control_points[1] + control_points[0])
\n else:
\n theoretical_start_second_deriv = np.zeros_like(control_points[0])
\n # \u521b\u5efa\u56fe\u5f62
\n fig, axes = plt.subplots(1, 2, figsize=(12, 5))
\n # \u5de6\u56fe：\u66f2\u7ebf\u548c\u7aef\u70b9
\n ax = axes[0]
\n # \u8ba1\u7b97\u66f2\u7ebf
\n curve_points = compute_bezier_curve(control_points)
\n # \u7ed8\u5236\u63a7\u5236\u591a\u8fb9\u5f62
\n ax.plot(control_points[:, 0], control_points[:, 1], 'ro-', linewidth=1.5, markersize=8, alpha=0.7, label='\u63a7\u5236\u591a\u8fb9\u5f62')
\n # \u7ed8\u5236\u63a7\u5236\u70b9\u6807\u7b7e
\n for i, point in enumerate(control_points):
\n ax.text(point[0]+0.05, point[1]+0.05, f'P{i}', fontsize=12, fontweight='bold')
\n # \u7ed8\u5236\u8d1d\u585e\u5c14\u66f2\u7ebf
\n ax.plot(curve_points[:, 0], curve_points[:, 1], 'b-', linewidth=2.5, label='\u8d1d\u585e\u5c14\u66f2\u7ebf')
\n # \u6807\u8bb0\u8d77\u70b9\u548c\u7ec8\u70b9
\n ax.scatter([start_point[0], end_point[0]], [start_point[1], end_point[1]], s=150, c=['green', 'purple'], edgecolors='black', linewidth=2, zorder=5, label=['\u8d77\u70b9', '\u7ec8\u70b9'])
\n # \u7ed8\u5236\u7aef\u70b9\u5207\u7ebf
\n scale = 0.5
\n ax.arrow(start_point[0], start_point[1], start_deriv[0]*scale\/n, start_deriv[1]*scale\/n, head_width=0.08, head_length=0.12, fc='green', ec='green', linewidth=2, label='\u8d77\u70b9\u5207\u7ebf')
\n ax.arrow(end_point[0], end_point[1], end_deriv[0]*scale\/n, end_deriv[1]*scale\/n, head_width=0.08, head_length=0.12, fc='purple', ec='purple', linewidth=2, label='\u7ec8\u70b9\u5207\u7ebf')
\n ax.set_title('\u8d1d\u585e\u5c14\u66f2\u7ebf\u7aef\u70b9\u6027\u8d28', fontsize=14, fontweight='bold')
\n ax.set_xlabel('x', fontsize=12)
\n ax.set_ylabel('y', fontsize=12)
\n ax.grid(True, alpha=0.3)
\n ax.legend(loc='best')
\n ax.axis('equal')
\n # \u53f3\u56fe：\u6027\u8d28\u9a8c\u8bc1\u7ed3\u679c
\n ax = axes[1]
\n ax.axis('off')
\n # \u663e\u793a\u9a8c\u8bc1\u7ed3\u679c
\n result_text = "\u7aef\u70b9\u6027\u8d28\u9a8c\u8bc1:\\\\n\\\\n"
\n result_text += f"\u66f2\u7ebf\u6b21\u6570: n = {n}\\\\n\\\\n"
\n result_text += "1. \u7aef\u70b9\u63d2\u503c\u6027\u8d28:\\\\n"
\n result_text += f" C(0) = [{start_point[0]:.4f}, {start_point[1]:.4f}]\\\\n"
\n result_text += f" P0 = [{control_points[0,0]:.4f}, {control_points[0,1]:.4f}]\\\\n"
\n result_text += f" \u76f8\u7b49: {'\u662f' if np.allclose(start_point, control_points[0]) else '\u5426'}\\\\n\\\\n"
\n result_text += f" C(1) = [{end_point[0]:.4f}, {end_point[1]:.4f}]\\\\n"
\n result_text += f" P\u2099 = [{control_points[-1,0]:.4f}, {control_points[-1,1]:.4f}]\\\\n"
\n result_text += f" \u76f8\u7b49: {'\u662f' if np.allclose(end_point, control_points[-1]) else '\u5426'}\\\\n\\\\n"
\n result_text += "2. \u7aef\u70b9\u5207\u7ebf\u6027\u8d28:\\\\n"
\n result_text += f" C'(0) = [{start_deriv[0]:.4f}, {start_deriv[1]:.4f}]\\\\n"
\n result_text += f" n(P\u2081-P0) = [{theoretical_start_deriv[0]:.4f}, {theoretical_start_deriv[1]:.4f}]\\\\n"
\n result_text += f" \u76f8\u7b49: {'\u662f' if np.allclose(start_deriv, theoretical_start_deriv) else '\u5426'}\\\\n\\\\n"
\n result_text += f" C'(1) = [{end_deriv[0]:.4f}, {end_deriv[1]:.4f}]\\\\n"
\n result_text += f" n(P\u2099-P\u2099\u208b\u2081) = [{theoretical_end_deriv[0]:.4f}, {theoretical_end_deriv[1]:.4f}]\\\\n"
\n result_text += f" \u76f8\u7b49: {'\u662f' if np.allclose(end_deriv, theoretical_end_deriv) else '\u5426'}\\\\n\\\\n"
\n if n >= 2:
\n result_text += "3. \u7aef\u70b9\u4e8c\u9636\u5bfc\u6570:\\\\n"
\n result_text += f" C''(0) = [{start_second_deriv[0]:.4f}, {start_second_deriv[1]:.4f}]\\\\n"
\n result_text += f" n(n-1)(P\u2082-2P\u2081+P0) = [{theoretical_start_second_deriv[0]:.4f}, {theoretical_start_second_deriv[1]:.4f}]\\\\n"
\n result_text += f" \u76f8\u7b49: {'\u662f' if np.allclose(start_second_deriv, theoretical_start_second_deriv) else '\u5426'}\\\\n"
\n ax.text(0.1, 0.5, result_text, fontsize=11, verticalalignment='center', bbox=dict(boxstyle='round', facecolor='wheat', alpha=0.8))
\n fig.suptitle('\u8d1d\u585e\u5c14\u66f2\u7ebf\u7aef\u70b9\u6027\u8d28\u9a8c\u8bc1', fontsize=16, fontweight='bold')
\n fig.tight_layout()
\n return fig<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n 5.4 \u7ed3\u679c\u8f93\u51fa<\/h5>\n\n\n\n| \n >>>\n<\/td>\n | \n # \u6267\u884c\u6027\u8d28\u9a8c\u8bc1
\ncontrol_points = np.array([[0, 0], [1, 3], [3, -1], [4, 2]])
\nfig_convex = verify_convex_hull(control_points)
\nfig_convex<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n ![\"\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/02\/20260222072609-699aaf9123f01.png\") <\/p>\n<\/p>\n<\/p>\n
474 msec<\/p>\n<\/p>\n \u56fe7：\u8d1d\u585e\u5c14\u66f2\u7ebf\u7684\u51f8\u5305\u6027\u548c\u53d8\u5dee\u7f29\u51cf\u6027\u9a8c\u8bc1\u3002\u5de6\u56fe\u663e\u793a\u66f2\u7ebf\u4f4d\u4e8e\u63a7\u5236\u70b9\u7684\u51f8\u5305\u5185，\u53f3\u56fe\u9a8c\u8bc1\u53d8\u5dee\u7f29\u51cf\u6027\u3002<\/p>\n \n\n\n| \n >>>\n<\/td>\n | \n fig_endpoints = analyze_endpoint_properties(control_points)
\nfig_endpoints<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n ![\"\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/02\/20260222072609-699aaf9142c45.png\") <\/p>\n<\/p>\n
222 msec<\/p>\n<\/p>\n \u56fe8：\u8d1d\u585e\u5c14\u66f2\u7ebf\u7aef\u70b9\u6027\u8d28\u9a8c\u8bc1\u3002\u663e\u793a\u4e86\u7aef\u70b9\u63d2\u503c\u3001\u7aef\u70b9\u5207\u7ebf\u548c\u7aef\u70b9\u66f2\u7387\u6027\u8d28\u3002<\/p>\n 6. \u5e94\u7528\u6848\u4f8b<\/h4>\n6.1 \u56fe\u5f62\u8bbe\u8ba1<\/h5>\n\u4f7f\u7528\u6807\u51c6\u8d1d\u585e\u5c14\u66f2\u7ebf，\u8ba1\u7b97\u7b80\u5355<\/p>\n \n\n\n| \n >>>\n<\/td>\n | \n def bezier_application_graphics_design():
\n """
\n \u8d1d\u585e\u5c14\u66f2\u7ebf\u5e94\u7528\u5b9e\u4f8b：\u56fe\u5f62\u8bbe\u8ba1
\n \u4f7f\u7528\u6807\u51c6\u8d1d\u585e\u5c14\u66f2\u7ebf\u521b\u5efa\u7b80\u5355\u56fe\u5f62
\n """
\n # \u521b\u5efa\u56fe\u5f62
\n fig, axes = plt.subplots(2, 2, figsize=(12, 10))<\/p>\n # \u5de6\u4e0a：\u7b80\u5355\u5f62\u72b6 – \u5fc3\u5f62
\n ax = axes[0, 0]
\n heart_points = [
\n np.array([[0, 0], [0.7, 1.0], [1, 0]]),
\n np.array([[1, 0], [1.0, -0.5], [0.5, -1]]),
\n np.array([[0.5, -1], [0, -1.5], [-0.5, -1]]),
\n np.array([[-0.5, -1], [-1, -0.5], [-1, 0]]),
\n np.array([[-1, 0], [-0.7, 1.0], [0, 0]])
\n ]<\/p>\n for points in heart_points:
\n curve = compute_bezier_curve(points)
\n ax.plot(curve[:, 0], curve[:, 1], 'r-', linewidth=2)
\n ax.plot(points[:, 0], points[:, 1], 'ko', markersize=4, alpha=0.5)<\/p>\n ax.set_xlim(-1.5, 1.5)
\n ax.set_ylim(-1.5, 1.5)
\n ax.set_title('\u5fc3\u5f62\u8bbe\u8ba1 (\u6807\u51c6\u8d1d\u585e\u5c14\u66f2\u7ebf)', fontsize=12, fontweight='bold')
\n ax.grid(True, alpha=0.3)
\n ax.axis('equal')<\/p>\n # \u53f3\u4e0a：Logo\u8bbe\u8ba1 – \u6ce2\u6d6a
\n ax = axes[0, 1]
\n wave_points = [
\n np.array([[0, 0], [0.5, 0.8], [1, 0]]),
\n np.array([[1, 0], [1.5, -0.8], [2, 0]]),
\n np.array([[2, 0], [2.5, 0.8], [3, 0]])
\n ]<\/p>\n for points in wave_points:
\n curve = compute_bezier_curve(points)
\n ax.plot(curve[:, 0], curve[:, 1], 'b-', linewidth=3)
\n ax.plot(points[:, 0], points[:, 1], 'go', markersize=4, alpha=0.5)<\/p>\n ax.set_xlim(-0.2, 3.2)
\n ax.set_ylim(-1, 1)
\n ax.set_title('\u6ce2\u6d6aLogo\u8bbe\u8ba1', fontsize=12, fontweight='bold')
\n ax.grid(True, alpha=0.3)
\n ax.axis('equal')<\/p>\n # \u5de6\u4e0b：\u88c5\u9970\u56fe\u6848 – \u82b1\u74e3
\n ax = axes[1, 0]
\n petal_points = [
\n np.array([[0, 0], [0.7, 0.5], [1, 0]]),
\n np.array([[1, 0], [0.7, -0.5], [0, 0]])
\n ]<\/p>\n # \u65cb\u8f6c\u521b\u5efa\u591a\u4e2a\u82b1\u74e3
\n angles = np.linspace(0, 2*np.pi, 6, endpoint=False)
\n for angle in angles:
\n rotation_matrix = np.array([
\n [np.cos(angle), -np.sin(angle)],
\n [np.sin(angle), np.cos(angle)]
\n ])<\/p>\n for points in petal_points:
\n rotated_points = points @ rotation_matrix.T
\n curve = compute_bezier_curve(rotated_points)
\n ax.plot(curve[:, 0], curve[:, 1], 'm-', linewidth=2, alpha=0.8)<\/p>\n ax.set_xlim(-1.2, 1.2)
\n ax.set_ylim(-1.2, 1.2)
\n ax.set_title('\u82b1\u74e3\u88c5\u9970\u56fe\u6848', fontsize=12, fontweight='bold')
\n ax.grid(True, alpha=0.3)
\n ax.axis('equal')<\/p>\n # \u53f3\u4e0b：\u56fe\u6807\u8bbe\u8ba1 – \u4e91\u6735
\n ax = axes[1, 1]
\n cloud_points = [
\n np.array([[0, 0], [0.3, 0.5], [0.7, 0.5], [1, 0]]),
\n np.array([[0.3, 0], [0.5, 0.3], [0.7, 0]]),
\n np.array([[0.7, 0], [0.9, 0.3], [1.1, 0]]),
\n np.array([[0.9, 0], [1.1, 0.2], [1.3, 0]])
\n ]<\/p>\n for points in cloud_points:
\n curve = compute_bezier_curve(points)
\n ax.plot(curve[:, 0], curve[:, 1], 'c-', linewidth=2)
\n ax.plot(points[:, 0], points[:, 1], 'bo', markersize=3, alpha=0.5)<\/p>\n ax.fill_between([-0.2, 1.5], -0.2, -0.2, color='lightblue', alpha=0.3)
\n ax.set_xlim(-0.2, 1.5)
\n ax.set_ylim(-0.2, 0.7)
\n ax.set_title('\u4e91\u6735\u56fe\u6807\u8bbe\u8ba1', fontsize=12, fontweight='bold')
\n ax.grid(True, alpha=0.3)
\n ax.axis('equal')<\/p>\n fig.suptitle('\u56fe\u5f62\u8bbe\u8ba1\u5e94\u7528：\u6807\u51c6\u8d1d\u585e\u5c14\u66f2\u7ebf', fontsize=16, fontweight='bold')
\n fig.tight_layout()
\n return fig<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n \n\n\n| \n >>>\n<\/td>\n | \n # \u6267\u884c\u56fe\u5f62\u8bbe\u8ba1\u6848\u4f8b
\nfig_graphics = bezier_application_graphics_design()
\nfig_graphics<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n ![\"\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/02\/20260222072609-699aaf9158116.png\") <\/p>\n<\/p>\n
6.2 \u5b57\u4f53\u8bbe\u8ba1<\/h5>\n\u4f7f\u7528\u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf\u7cbe\u786e\u8868\u793a\u5706\u5f27<\/p>\n \n\n\n| \n >>>\n<\/td>\n | \n def bezier_application_font_design():
\n """
\n \u8d1d\u585e\u5c14\u66f2\u7ebf\u5e94\u7528\u5b9e\u4f8b：\u5b57\u4f53\u8bbe\u8ba1
\n \u4f7f\u7528\u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf\u7cbe\u786e\u8868\u793a\u5706\u5f27\u548c\u590d\u6742\u66f2\u7ebf
\n """
\n # \u521b\u5efa\u56fe\u5f62
\n fig, axes = plt.subplots(2, 3, figsize=(15, 10))
\n # \u5de6\u4e0a：\u5b57\u6bcd'S'\u8bbe\u8ba1
\n ax = axes[0, 0]
\n # \u4f7f\u7528\u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf\u7cbe\u786e\u8868\u793a\u5706\u5f27\u90e8\u5206
\n # \u4e0a\u534a\u5706\u90e8\u5206
\n theta = np.pi \/ 4
\n r = 0.5
\n P0 = np.array([r * np.cos(theta), r * np.sin(theta)])
\n P1 = np.array([r * np.tan(theta\/2), r])
\n P2 = np.array([0, r])
\n P3 = np.array([-r * np.tan(theta\/2), r])
\n P4 = np.array([-r * np.cos(theta), r * np.sin(theta)])
\n control_points1 = np.array([P0, P1, P2, P3, P4])
\n weights1 = np.array([1, np.cos(theta\/2),1, np.cos(theta\/2), 1])
\n curve1 = compute_rational_bezier_curve(control_points1, weights1)
\n # \u7ed8\u5236\u4e0a\u534a\u5706\u66f2\u7ebf\u53ca\u5176\u63a7\u5236\u70b9
\n ax.plot(curve1[:, 0], curve1[:, 1], 'b-', linewidth=3, label="\u4e0a\u534a\u5706")
\n ax.plot(control_points1[:, 0], control_points1[:, 1], 'ro-', linewidth=1, markersize=6, alpha=0.7, label="\u4e0a\u534a\u5706\u63a7\u5236\u70b9")
\n # \u4e0b\u534a\u5706\u90e8\u5206
\n P5 = np.array([r * np.cos(theta), -r * np.sin(theta)])
\n P6 = np.array([r * np.tan(theta\/2), -r])
\n P7 = np.array([0, -r])
\n P8 = np.array([-r * np.tan(theta\/2), -r])
\n P9 = np.array([-r * np.cos(theta), -r * np.sin(theta)])
\n control_points3 = np.array([P5, P6, P7, P8, P9])
\n curve3 = compute_bezier_curve(control_points3)
\n # \u7ed8\u5236\u4e0b\u534a\u5706\u66f2\u7ebf\u53ca\u5176\u63a7\u5236\u70b9
\n ax.plot(curve3[:, 0], curve3[:, 1], 'm-', linewidth=3, label="\u4e0b\u534a\u5706")
\n ax.plot(control_points3[:, 0], control_points3[:, 1], 'mo-', linewidth=1, markersize=6, alpha=0.7, label="\u4e0b\u534a\u5706\u63a7\u5236\u70b9")
\n # \u4e2d\u95f4\u90e8\u5206
\n control_points2 = np.array([P4, [-r*np.cos(theta), 0], [r*np.cos(theta), 0], P5])
\n curve2 = compute_bezier_curve(control_points2)
\n # \u7ed8\u5236\u4e2d\u95f4\u66f2\u7ebf\u53ca\u5176\u63a7\u5236\u70b9
\n ax.plot(curve2[:, 0], curve2[:, 1], 'g-', linewidth=3, label="\u4e2d\u95f4\u8fc7\u6e21")
\n ax.plot(control_points2[:, 0], control_points2[:, 1], 'go-', linewidth=1, markersize=6, alpha=0.7, label="\u4e2d\u95f4\u63a7\u5236\u70b9")
\n ax.set_xlim(-0.1, 1)
\n ax.set_ylim(-0.6, 0.6)
\n ax.set_title('\u5b57\u6bcd"S"\u8bbe\u8ba1 (\u542b\u5706\u5f27)', fontsize=12, fontweight='bold')
\n ax.grid(True, alpha=0.3)
\n ax.axis('equal')
\n # \u4e2d\u4e0a：\u5b57\u6bcd'A'\u8bbe\u8ba1
\n ax = axes[0, 1]
\n # \u5de6\u659c\u7ebf
\n control_points_a1 = np.array([[0, 0], [0.2, 0.8], [0.4, 1]])
\n curve_a1 = compute_bezier_curve(control_points_a1)
\n # \u53f3\u659c\u7ebf
\n control_points_a2 = np.array([[0.4, 1], [0.6, 0.8], [0.8, 0]])
\n curve_a2 = compute_bezier_curve(control_points_a2)
\n # \u6a2a\u7ebf
\n control_points_a3 = np.array([[0.1, 0.5], [0.4, 0.55], [0.7, 0.5]])
\n curve_a3 = compute_bezier_curve(control_points_a3)
\n ax.plot(curve_a1[:, 0], curve_a1[:, 1], 'r-', linewidth=3)
\n ax.plot(curve_a2[:, 0], curve_a2[:, 1], 'r-', linewidth=3)
\n ax.plot(curve_a3[:, 0], curve_a3[:, 1], 'r-', linewidth=3)
\n ax.set_xlim(-0.1, 0.9)
\n ax.set_ylim(-0.1, 1.1)
\n ax.set_title('\u5b57\u6bcd"A"\u8bbe\u8ba1', fontsize=12, fontweight='bold')
\n ax.grid(True, alpha=0.3)
\n ax.axis('equal')
\n # \u53f3\u4e0a：\u7cbe\u786e\u5706\u5f27\u6bd4\u8f83
\n ax = axes[0, 2]
\n # \u4f7f\u7528\u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf\u7cbe\u786e\u8868\u793a90\u5ea6\u5706\u5f27
\n theta = np.pi\/2
\n r = 0.5
\n center = np.array([0, 0])
\n # \u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf\u8868\u793a
\n w = np.cos(theta\/4) # \u6743\u91cd
\n # \u7b2c\u4e00\u4e2a1\/4\u5706
\n P0 = center + np.array([r, 0])
\n P1 = center + np.array([r, r * np.tan(theta\/4)])
\n P2 = center + np.array([r*np.cos(theta\/2), r*np.sin(theta\/2)])
\n control_rational = np.array([P0, P1, P2])
\n weights = np.array([1, w, 1])
\n curve_rational = compute_rational_bezier_curve(control_rational, weights)
\n # \u6807\u51c6\u8d1d\u585e\u5c14\u66f2\u7ebf\u8fd1\u4f3c
\n control_standard = np.array([P0, P1, P2])
\n curve_standard = compute_bezier_curve(control_standard)
\n # \u7cbe\u786e\u5706\u5f27
\n angles = np.linspace(0, theta\/2, 50)
\n exact_arc = np.column_stack([r*np.cos(angles), r*np.sin(angles)])
\n ax.plot(exact_arc[:, 0], exact_arc[:, 1], 'g-', linewidth=2, label='\u7cbe\u786e\u5706\u5f27')
\n ax.plot(curve_rational[:, 0], curve_rational[:, 1], 'b–', linewidth=2, label='\u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf')
\n ax.plot(curve_standard[:, 0], curve_standard[:, 1], 'r:', linewidth=2, label='\u6807\u51c6\u8d1d\u585e\u5c14\u66f2\u7ebf')
\n ax.set_xlim(-0.1, 0.6)
\n ax.set_ylim(-0.1, 0.6)
\n ax.set_title('\u5706\u5f27\u7cbe\u786e\u8868\u793a\u6bd4\u8f83', fontsize=12, fontweight='bold')
\n ax.legend(fontsize=9)
\n ax.grid(True, alpha=0.3)
\n ax.axis('equal')
\n # \u5de6\u4e0b：\u4e2d\u6587\u5b57\u4f53\u8bbe\u8ba1\u793a\u4f8b
\n ax = axes[1, 0]
\n # "\u4e2d"\u5b57\u8bbe\u8ba1
\n # \u7ad6\u7ebf
\n control_zhong1 = np.array([[0.3, 0], [0.3, 0.3], [0.3, 0.6]])
\n curve_zhong1 = compute_bezier_curve(control_zhong1)
\n # \u6a2a\u7ebf
\n control_zhong2 = np.array([[0, 0.2], [0.3, 0.2], [0.6, 0.2]])
\n curve_zhong2 = compute_bezier_curve(control_zhong2)
\n # \u6a2a\u7ebf2
\n control_zhong3 = np.array([[0, 0.4], [0.3, 0.4], [0.6, 0.4]])
\n curve_zhong3 = compute_bezier_curve(control_zhong3)
\n # \u7ad6\u7ebf4
\n control_zhong4 = np.array([[0.0, 0.2], [0.0, 0.3], [0.0, 0.4]])
\n curve_zhong4 = compute_bezier_curve(control_zhong4)
\n # \u7ad6\u7ebf5
\n control_zhong5 = np.array([[0.6, 0.2], [0.6, 0.3], [0.6, 0.4]])
\n curve_zhong5 = compute_bezier_curve(control_zhong5)
\n ax.plot(curve_zhong1[:, 0], curve_zhong1[:, 1], 'k-', linewidth=3)
\n ax.plot(curve_zhong2[:, 0], curve_zhong2[:, 1], 'k-', linewidth=3)
\n ax.plot(curve_zhong3[:, 0], curve_zhong3[:, 1], 'k-', linewidth=3)
\n ax.plot(curve_zhong4[:, 0], curve_zhong4[:, 1], 'k-', linewidth=3)
\n ax.plot(curve_zhong5[:, 0], curve_zhong5[:, 1], 'k-', linewidth=3)
\n ax.set_xlim(-0.1, 0.7)
\n ax.set_ylim(-0.1, 1.1)
\n ax.set_title('\u4e2d\u6587\u5b57\u4f53："\u4e2d"\u5b57\u8bbe\u8ba1', fontsize=12, fontweight='bold')
\n ax.grid(True, alpha=0.3)
\n ax.axis('equal')
\n # \u4e2d\u4e0b：\u5b57\u4f53\u5e73\u6ed1\u5ea6\u5bf9\u6bd4
\n ax = axes[1, 1]
\n # \u7c97\u7cd9\u7684\u5b57\u6bcd'O'
\n rough_points = [
\n np.array([[0, 0], [0.2, 0], [0.5, 0.2], [0.8, 0], [1, 0]]),
\n np.array([[1, 0], [1, 0.3], [1, 0.6], [1, 1]]),
\n np.array([[1, 1], [0.8, 1], [0.5, 0.8], [0.2, 1], [0, 1]]),
\n np.array([[0, 1], [0, 0.6], [0, 0.3], [0, 0]])
\n ]
\n for points in rough_points:
\n curve = compute_bezier_curve(points)
\n ax.plot(curve[:, 0], curve[:, 1], 'r-', linewidth=2, alpha=0.5, label='\u7c97\u7cd9\u8bbe\u8ba1' if points is rough_points[0] else "")
\n # \u5e73\u6ed1\u7684\u5b57\u6bcd'O'
\n smooth_points = [
\n np.array([[0.2, 0], [0.3, -0.1], [0.7, -0.1], [0.8, 0]]),
\n np.array([[0.8, 0], [1, 0.2], [1, 0.8], [0.8, 1]]),
\n np.array([[0.8, 1], [0.7, 1.1], [0.3, 1.1], [0.2, 1]]),
\n np.array([[0.2, 1], [0, 0.8], [0, 0.2], [0.2, 0]])
\n ]
\n for points in smooth_points:
\n curve = compute_bezier_curve(points)
\n ax.plot(curve[:, 0], curve[:, 1], 'b-', linewidth=3, label='\u5e73\u6ed1\u8bbe\u8ba1' if points is smooth_points[0] else "")
\n ax.set_xlim(-0.2, 1.2)
\n ax.set_ylim(-0.2, 1.2)
\n ax.set_title('\u5b57\u4f53\u5e73\u6ed1\u5ea6\u5bf9\u6bd4', fontsize=12, fontweight='bold')
\n ax.legend(fontsize=9)
\n ax.grid(True, alpha=0.3)
\n ax.axis('equal')
\n # \u53f3\u4e0b：\u8fde\u7b14\u5b57\u4f53\u793a\u4f8b
\n ax = axes[1, 2]
\n # \u8fde\u7b14\u5b57\u90e8\u5206\u8bbe\u8ba1
\n control_love = [
\n np.array([[0, 0.8], [0.2, 1], [0.4, 0.8]]),
\n np.array([[0.4, 0.8], [0.6, 0.6], [0.8, 0.8]]),
\n np.array([[0.8, 0.8], [1, 1], [1.2, 0.8]]),
\n np.array([[0.6, 0.6], [0.6, 0.3], [0.6, 0]]),
\n np.array([[0.6, 0], [0.4, -0.2], [0.2, 0]])
\n ]
\n colors = ['r', 'g', 'b', 'm', 'c']
\n labels = ['\u4e0a\u90e8\u66f2\u7ebf', '\u4e2d\u95f4\u8fde\u63a5', '\u53f3\u90e8\u66f2\u7ebf', '\u7ad6\u7b14', '\u5e95\u90e8\u56de\u7b14']
\n for i, points in enumerate(control_love):
\n curve = compute_bezier_curve(points)
\n ax.plot(curve[:, 0], curve[:, 1], f'{colors[i]}-', linewidth=2, label=labels[i])
\n ax.set_xlim(-0.2, 1.4)
\n ax.set_ylim(-0.3, 1.2)
\n ax.set_title('\u8fde\u7b14\u5b57\u4f53\u8bbe\u8ba1\u793a\u4f8b', fontsize=12, fontweight='bold')
\n ax.legend(fontsize=8, loc='upper right')
\n ax.grid(True, alpha=0.3)
\n ax.axis('equal')
\n fig.suptitle('\u5b57\u4f53\u8bbe\u8ba1\u5e94\u7528：\u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf\u7cbe\u786e\u8868\u793a', fontsize=16, fontweight='bold')
\n fig.tight_layout()
\n return fig<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n \n\n\n| \n >>>\n<\/td>\n | \n # \u6267\u884c\u5b57\u4f53\u8bbe\u8ba1\u6848\u4f8b
\nfig_font = bezier_application_font_design()
\nfig_font<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n ![\"\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/02\/20260222072609-699aaf917c30b.png\") <\/p>\n<\/p>\n
6.3 \u52a8\u753b\u8def\u5f84<\/h5>\nC\u00b9\/G\u00b9\u8fde\u7eed\u6027\u786e\u4fdd\u8fd0\u52a8\u5e73\u6ed1<\/p>\n \n\n\n| \n >>>\n<\/td>\n | \n def cubic_bezier(p0, p1, p2, p3, t):
\n """\u8ba1\u7b97\u4e09\u6b21\u8d1d\u585e\u5c14\u66f2\u7ebf\u4e0a\u7684\u70b9"""
\n return (1-t)**3 * p0 + 3*t*(1-t)**2 * p1 + 3*t**2*(1-t)*p2 + t**3*p3<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n \n\n\n| \n >>>\n<\/td>\n | \n def compute_c1_continuity_control_points(prev_curve, next_p2, next_p3):
\n """\u8ba1\u7b97\u6ee1\u8db3C\u00b9\u8fde\u7eed\u6027\u7684\u63a7\u5236\u70b9"""
\n P0, P1, P2, P3 = prev_curve<\/p>\n # C\u00b9\u8fde\u7eed\u6761\u4ef6
\n Q0 = P3 # \u4f4d\u7f6e\u8fde\u7eed
\n Q1 = 2*P3 – P2 # \u4e00\u9636\u5bfc\u6570\u8fde\u7eed
\n Q2 = next_p2
\n Q3 = next_p3<\/p>\n return np.array([Q0, Q1, Q2, Q3])<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n \n\n\n| \n >>>\n<\/td>\n | \n def generate_smooth_animation_path(key_points, continuity_type='C1'):
\n """\u751f\u6210\u5e73\u6ed1\u52a8\u753b\u8def\u5f84"""
\n curves = []
\n n_segments = len(key_points) – 1<\/p>\n # \u4e3a\u6bcf\u4e2a\u5173\u952e\u70b9\u751f\u6210\u521d\u59cb\u63a7\u5236\u70b9
\n control_sets = []<\/p>\n # \u7b2c\u4e00\u6bb5\u7684\u63a7\u5236\u70b9
\n P0 = key_points[0]
\n P1 = key_points[0] + (key_points[1] – key_points[0]) \/ 3
\n control_sets.append([P0, P1])<\/p>\n # \u4e2d\u95f4\u6bb5\u7684\u63a7\u5236\u70b9
\n for i in range(1, len(key_points)-1):
\n prev = key_points[i-1]
\n curr = key_points[i]
\n next = key_points[i+1]<\/p>\n tangent = (next – prev) \/ 2<\/p>\n P2 = curr – tangent \/ 3
\n P3 = curr
\n P0_next = curr
\n P1_next = curr + tangent \/ 3<\/p>\n control_sets[-1].extend([P2, P3])
\n control_sets.append([P0_next, P1_next])<\/p>\n # \u6700\u540e\u4e00\u6bb5\u7684\u63a7\u5236\u70b9
\n last_idx = len(key_points) – 1
\n P2 = key_points[last_idx] – (key_points[last_idx] – key_points[last_idx-1]) \/ 3
\n P3 = key_points[last_idx]
\n control_sets[-1].extend([P2, P3])<\/p>\n # \u6784\u5efa\u66f2\u7ebf
\n for i in range(n_segments):
\n if continuity_type == 'C1' and i > 0:
\n # C\u00b9\u8fde\u7eed\u6027
\n prev_curve = curves[-1]
\n next_p2 = control_sets[i][2]
\n next_p3 = control_sets[i][3]<\/p>\n curve_points = compute_c1_continuity_control_points(prev_curve, next_p2, next_p3)
\n else:
\n # \u7b2c\u4e00\u6bb5\u6216G\u00b9\u8fde\u7eed\u6027
\n curve_points = np.array(control_sets[i])<\/p>\n curves.append(curve_points)<\/p>\n return curves<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n \n\n\n| \n >>>\n<\/td>\n | \n def bezier_application_animation_path():
\n """
\n \u8d1d\u585e\u5c14\u66f2\u7ebf\u5e94\u7528\u5b9e\u4f8b：\u52a8\u753b\u8def\u5f84
\n \u4f7f\u7528C\u00b9\u6216G\u00b9\u8fde\u7eed\u6027\u786e\u4fdd\u8fd0\u52a8\u5e73\u6ed1
\n """<\/p>\n # \u521b\u5efa\u6f14\u793a\u56fe\u5f62（\u9759\u6001\u5c55\u793a）
\n fig, axes = plt.subplots(2, 3, figsize=(24, 12))<\/p>\n # \u5b9a\u4e49\u5173\u952e\u70b9
\n key_points = np.array([
\n [0, 0],
\n [2, 3],
\n [5, 1],
\n [7, 4],
\n [10, 2]
\n ])<\/p>\n # \u5de6\u4e0a：C\u00b9\u8fde\u7eed\u6027\u8def\u5f84
\n ax = axes[0, 0]
\n c1_curves = generate_smooth_animation_path(key_points, continuity_type='C1')<\/p>\n # \u7ed8\u5236C\u00b9\u8def\u5f84
\n for i, curve in enumerate(c1_curves):
\n t_values = np.linspace(0, 1, 100)
\n segment_points = np.array([cubic_bezier(*curve, t) for t in t_values])
\n ax.plot(segment_points[:, 0], segment_points[:, 1], 'b-', linewidth=2)<\/p>\n # \u7ed8\u5236\u63a7\u5236\u591a\u8fb9\u5f62
\n ax.plot(curve[:, 0], curve[:, 1], 'g–', linewidth=1, alpha=0.5)
\n ax.scatter(curve[:, 0], curve[:, 1], s=40, c='green', alpha=0.5)<\/p>\n # \u6807\u8bb0\u63a7\u5236\u70b9
\n if i == 0:
\n for j, point in enumerate(curve):
\n ax.text(point[0]+0.1, point[1]+0.1, f'P{j}', fontsize=9, fontweight='bold')
\n else:
\n for j, point in enumerate(curve):
\n ax.text(point[0]+0.1, point[1]+0.1, f'Q{j}', fontsize=9, fontweight='bold')<\/p>\n # \u6807\u8bb0\u5173\u952e\u70b9
\n ax.scatter(key_points[:, 0], key_points[:, 1], s=100, c='red',
\n edgecolors='black', linewidth=2, zorder=5)<\/p>\n # \u6807\u8bb0\u8fde\u63a5\u70b9\u5904\u7684C\u00b9\u8fde\u7eed\u6027
\n for i in range(len(c1_curves)-1):
\n P2 = c1_curves[i][2]
\n P3 = c1_curves[i][3]
\n Q1 = c1_curves[i+1][1]<\/p>\n # \u7ed8\u5236\u5207\u7ebf\u5411\u91cf
\n ax.arrow(P3[0], P3[1], P3[0]-P2[0], P3[1]-P2[1],
\n head_width=0.15, head_length=0.2, fc='purple', ec='purple', alpha=0.7)
\n ax.arrow(P3[0], P3[1], Q1[0]-P3[0], Q1[1]-P3[1],
\n head_width=0.15, head_length=0.2, fc='orange', ec='orange', alpha=0.7)<\/p>\n ax.set_title('C\u00b9\u8fde\u7eed\u6027\u52a8\u753b\u8def\u5f84', fontsize=12, fontweight='bold')
\n ax.set_xlabel('X\u5750\u6807', fontsize=11)
\n ax.set_ylabel('Y\u5750\u6807', fontsize=11)
\n ax.grid(True, alpha=0.3)
\n ax.axis('equal')<\/p>\n # \u53f3\u4e0a：G\u00b9\u8fde\u7eed\u6027\u8def\u5f84
\n ax = axes[0, 2]
\n g1_curves = generate_smooth_animation_path(key_points, continuity_type='G1')<\/p>\n # \u7ed8\u5236G\u00b9\u8def\u5f84
\n for i, curve in enumerate(g1_curves):
\n t_values = np.linspace(0, 1, 100)
\n segment_points = np.array([cubic_bezier(*curve, t) for t in t_values])
\n ax.plot(segment_points[:, 0], segment_points[:, 1], 'r-', linewidth=2)<\/p>\n # \u7ed8\u5236\u63a7\u5236\u591a\u8fb9\u5f62
\n ax.plot(curve[:, 0], curve[:, 1], 'g–', linewidth=1, alpha=0.5)
\n ax.scatter(curve[:, 0], curve[:, 1], s=40, c='green', alpha=0.5)<\/p>\n # \u6807\u8bb0\u5173\u952e\u70b9
\n ax.scatter(key_points[:, 0], key_points[:, 1], s=100, c='red',
\n edgecolors='black', linewidth=2, zorder=5)<\/p>\n ax.set_title('G\u00b9\u8fde\u7eed\u6027\u52a8\u753b\u8def\u5f84', fontsize=12, fontweight='bold')
\n ax.set_xlabel('X\u5750\u6807', fontsize=11)
\n ax.set_ylabel('Y\u5750\u6807', fontsize=11)
\n ax.grid(True, alpha=0.3)
\n ax.axis('equal')<\/p>\n # \u4e2d\u56fe：\u5dee\u5f02\u503c\u66f2\u7ebf
\n ax3 = axes[0,1]
\n # \u8ba1\u7b97C\u00b9\u548cG\u00b9\u8def\u5f84\u4e0a\u7684\u70b9（\u9ad8\u5206\u8fa8\u7387\u91c7\u6837）
\n t_values = np.linspace(0, 1, 500) # \u9ad8\u5206\u8fa8\u7387\u53c2\u6570
\n c1_points = []
\n g1_points = []<\/p>\n for i in range(len(c1_curves)):
\n c1_segment = np.array([cubic_bezier(*c1_curves[i], t) for t in t_values])
\n g1_segment = np.array([cubic_bezier(*g1_curves[i], t) for t in t_values])
\n c1_points.append(c1_segment)
\n g1_points.append(g1_segment)<\/p>\n c1_points = np.vstack(c1_points)
\n g1_points = np.vstack(g1_points)
\n # \u63d0\u53d6X\u5750\u6807\u548cY\u5750\u6807\u7684\u5dee\u5f02
\n x_coords = c1_points[:, 0] # \u4f7f\u7528C\u00b9\u8def\u5f84\u7684X\u5750\u6807\u4f5c\u4e3a\u6a2a\u8f74
\n y_diff = c1_points[:, 1] – g1_points[:, 1] # \u8ba1\u7b97Y\u5750\u6807\u7684\u5dee\u5f02<\/p>\n ax3.plot(x_coords, y_diff, 'g-', linewidth=2)
\n ax3.set_title('C\u00b9\u4e0eG\u00b9\u8def\u5f84Y\u5750\u6807\u5dee\u5f02\u503c\u66f2\u7ebf', fontsize=12, fontweight='bold')
\n ax3.set_xlabel('X\u5750\u6807', fontsize=11)
\n ax3.set_ylabel('Y\u5750\u6807\u5dee\u5f02\u503c', fontsize=11)
\n ax3.grid(True, alpha=0.3)
\n ax3.set_ylim(np.min(y_diff) * 1.1, np.max(y_diff) * 1.1)<\/p>\n # \u5de6\u4e0b：\u8fde\u7eed\u6027\u6761\u4ef6\u5bf9\u6bd4
\n ax = axes[1, 0]
\n ax.axis('off')<\/p>\n # \u663e\u793a\u8fde\u7eed\u6027\u6761\u4ef6
\n text = "\u8fde\u7eed\u6027\u6761\u4ef6\u5bf9\u6bd4：\\\\n"
\n text += "C\u00b9\u8fde\u7eed\u6027\u6761\u4ef6：\\\\n"
\n text += "1. \u4f4d\u7f6e\u8fde\u7eed：P3 = Q0\\\\n"
\n text += "2. \u4e00\u9636\u5bfc\u6570\u8fde\u7eed：P3 – P2 = Q1 – Q0 \u2192 Q1 = 2P3 – P2\\\\n"<\/p>\n text += "G\u00b9\u8fde\u7eed\u6027\u6761\u4ef6：\\\\n"
\n text += "1. \u4f4d\u7f6e\u8fde\u7eed：P3 = Q0\\\\n"
\n text += "2. \u5207\u7ebf\u65b9\u5411\u4e00\u81f4：Q1\u5728P2P3\u5ef6\u957f\u7ebf\u4e0a \u2192 Q1 = P3 + k(P3 – P2), k > 0\\\\n"<\/p>\n text += "\u52a8\u753b\u5e94\u7528\u5efa\u8bae：\\\\n"
\n text += "\u2022 C\u00b9\u8fde\u7eed：\u901f\u5ea6\u5927\u5c0f\u548c\u65b9\u5411\u90fd\u8fde\u7eed，\u8fd0\u52a8\u6700\u5e73\u6ed1\\\\n"
\n text += "\u2022 G\u00b9\u8fde\u7eed：\u8fd0\u52a8\u65b9\u5411\u8fde\u7eed，\u901f\u5ea6\u5927\u5c0f\u53ef\u8c03，\u66f4\u7075\u6d3b\\\\n"
\n text += "\u2022 \u5bf9\u4e8e\u5927\u591a\u6570\u52a8\u753b，G\u00b9\u8fde\u7eed\u5df2\u8db3\u591f\\\\n"<\/p>\n ax.text(0.1, 0.5, text, fontsize=11, verticalalignment='center',
\n bbox=dict(boxstyle='round', facecolor='wheat', alpha=0.8))<\/p>\n # \u53f3\u4e0b：\u5e94\u7528\u573a\u666f
\n ax = axes[1, 2]
\n ax.axis('off')<\/p>\n # \u663e\u793a\u5e94\u7528\u573a\u666f
\n text = "\u52a8\u753b\u8def\u5f84\u5e94\u7528\u573a\u666f：\\\\n"
\n text += "1. \u6e38\u620f\u89d2\u8272\u79fb\u52a8：\\\\n"
\n text += " \u2022 \u89d2\u8272\u6cbf\u9884\u8bbe\u8def\u5f84\u79fb\u52a8\u2022 NPC\u5de1\u903b\u8def\u7ebf\u2022 \u6280\u80fd\u7279\u6548\u8f68\u8ff9\\\\n"<\/p>\n text += "2. UI\u52a8\u753b：\\\\n"
\n text += " \u2022 \u6309\u94ae\u70b9\u51fb\u52a8\u753b\u2022 \u9875\u9762\u5207\u6362\u8fc7\u6e21\u2022 \u5143\u7d20\u5165\u573a\/\u51fa\u573a\u52a8\u753b\\\\n"<\/p>\n text += "3. \u6570\u636e\u53ef\u89c6\u5316：\\\\n"
\n text += " \u2022 \u6570\u636e\u70b9\u8fde\u7ebf\u52a8\u753b\u2022 \u8d8b\u52bf\u7ebf\u7ed8\u5236\u2022 \u7126\u70b9\u79fb\u52a8\u52a8\u753b\\\\n"<\/p>\n text += "4. \u5f71\u89c6\u7279\u6548：\\\\n"
\n text += " \u2022 \u6444\u50cf\u673a\u8fd0\u52a8\u8def\u5f84\u2022 \u7c92\u5b50\u8fd0\u52a8\u8f68\u8ff9\u2022 \u5149\u7ebf\u8def\u5f84\u8ffd\u8e2a\\\\n"<\/p>\n ax.text(0.1, 0.5, text, fontsize=11, verticalalignment='center',
\n bbox=dict(boxstyle='round', facecolor='lightblue', alpha=0.8))<\/p>\n # \u4e2d\u4e0b：\u5dee\u5f02\u503c\u539f\u56e0\u89e3\u91ca
\n ax = axes[1, 1]
\n ax.axis('off')<\/p>\n # \u663e\u793a\u5dee\u5f02\u503c\u539f\u56e0\u89e3\u91ca
\n text = "\u5dee\u5f02\u503c\u5206\u6790：\\\\n"
\n text += "1. \u5dee\u5f02\u6765\u6e90：\\\\n"
\n text += " \u2022 C\u00b9\u8fde\u7eed\u6027\u8981\u6c42\u5bfc\u6570\u8fde\u7eed，G\u00b9\u8fde\u7eed\u6027\u4ec5\u8981\u6c42\u5207\u7ebf\u65b9\u5411\u8fde\u7eed\u3002\\\\n"
\n text += " \u2022 \u56e0\u6b64，\u5728\u8fde\u63a5\u70b9\u9644\u8fd1，C\u00b9\u8def\u5f84\u7684Y\u5750\u6807\u53d8\u5316\u66f4\u5e73\u6ed1\u3002\\\\n"<\/p>\n text += "2. \u5dee\u5f02\u6700\u5927\u503c：\\\\n"
\n text += " \u2022 \u6700\u5927\u5dee\u5f02\u51fa\u73b0\u5728\u8fde\u63a5\u70b9\u9644\u8fd1\u3002\\\\n"
\n text += " \u2022 \u5dee\u5f02\u6700\u5927\u503c\u4e3a -9\u00d710-6，\u8868\u660e\u4e24\u6761\u8def\u5f84\u5728Y\u65b9\u5411\u4e0a\u7684\u5fae\u5c0f\u504f\u5dee\u3002\\\\n"<\/p>\n text += "3. \u5b9e\u9645\u5f71\u54cd：\\\\n"
\n text += " \u2022 \u7531\u4e8e\u5dee\u5f02\u6781\u5c0f，\u8089\u773c\u96be\u4ee5\u5bdf\u89c9\u3002\\\\n"
\n text += " \u2022 \u5728\u9ad8\u7cbe\u5ea6\u52a8\u753b\u4e2d，\u9700\u6839\u636e\u9700\u6c42\u9009\u62e9\u5408\u9002\u7684\u8fde\u7eed\u6027\u7c7b\u578b\u3002"<\/p>\n ax.text(0.1, 0.5, text, fontsize=11, verticalalignment='center',
\n bbox=dict(boxstyle='round', facecolor='lightgreen', alpha=0.8))<\/p>\n fig.suptitle('\u52a8\u753b\u8def\u5f84\u5e94\u7528：C\u00b9\/G\u00b9\u8fde\u7eed\u6027\u786e\u4fdd\u8fd0\u52a8\u5e73\u6ed1', fontsize=16, fontweight='bold')
\n fig.tight_layout()<\/p>\n return fig<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n \n\n\n| \n >>>\n<\/td>\n | \n # \u6267\u884c\u52a8\u753b\u8def\u5f84\u6848\u4f8b
\nfig_animation = bezier_application_animation_path()
\nfig_animation<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n ![\"\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/02\/20260222072609-699aaf91a1ad7.png\") <\/p>\n<\/p>\n
6.4 CAD\/CAM<\/h5>\n\u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf\u4fdd\u6301\u6295\u5f71\u4e0d\u53d8\u6027<\/p>\n \u5f15\u7533\u9605\u8bfb：<\/p>\n \u6295\u5f71\u4e0d\u53d8\u6027（Projection Invariance）\u662f\u8ba1\u7b97\u673a\u56fe\u5f62\u5b66\u548c\u51e0\u4f55\u5efa\u6a21\u4e2d\u7684\u4e00\u4e2a\u91cd\u8981\u6982\u5ff5，\u5c24\u5176\u5728 CAD\/CAM \u548c\u8ba1\u7b97\u673a\u89c6\u89c9\u9886\u57df\u4e2d\u5177\u6709\u91cd\u8981\u610f\u4e49\u3002\u5b83\u7684\u6838\u5fc3\u601d\u60f3\u662f：\u67d0\u4e9b\u51e0\u4f55\u5c5e\u6027\u6216\u5173\u7cfb\u5728\u7ecf\u8fc7\u6295\u5f71\u53d8\u6362\u540e\u4ecd\u7136\u4fdd\u6301\u4e0d\u53d8\u3002<\/p>\n<\/p>\n
\n<\/p>\n\u6295\u5f71\u4e0d\u53d8\u6027\u7684\u5b9a\u4e49<\/h6>\n<\/p>\n\u5728\u6570\u5b66\u548c\u51e0\u4f55\u4e2d，\u6295\u5f71\u53d8\u6362\u662f\u4e00\u79cd\u5c06\u4e09\u7ef4\u7a7a\u95f4\u4e2d\u7684\u70b9\u6620\u5c04\u5230\u4e8c\u7ef4\u5e73\u9762\u7684\u64cd\u4f5c（\u4f8b\u5982\u900f\u89c6\u6295\u5f71\u6216\u5e73\u884c\u6295\u5f71）\u3002\u6295\u5f71\u4e0d\u53d8\u6027\u6307\u7684\u662f：<\/p>\n \u67d0\u4e9b\u51e0\u4f55\u7279\u6027（\u5982\u70b9\u3001\u7ebf\u3001\u66f2\u7ebf\u7684\u62d3\u6251\u7ed3\u6784\u3001\u4ea4\u6bd4\u7b49）\u5728\u6295\u5f71\u53d8\u6362\u524d\u540e\u4fdd\u6301\u4e00\u81f4\u3002<\/p>\n \u6362\u53e5\u8bdd\u8bf4，\u65e0\u8bba\u4ece\u54ea\u4e2a\u89d2\u5ea6\u89c2\u5bdf\u7269\u4f53，\u8fd9\u4e9b\u7279\u6027\u90fd\u4e0d\u4f1a\u6539\u53d8\u3002<\/p>\n<\/p>\n
\n<\/p>\n\u6295\u5f71\u4e0d\u53d8\u6027\u7684\u5177\u4f53\u8868\u73b0<\/h6>\n<\/p>\n<\/p>\n1. \u70b9\u7684\u6295\u5f71\u4e0d\u53d8\u6027<\/h6>\n<\/p>\n\n- \n
\u4e00\u4e2a\u70b9\u5728\u4e09\u7ef4\u7a7a\u95f4\u4e2d\u7684\u4f4d\u7f6e\u7ecf\u8fc7\u6295\u5f71\u53d8\u6362\u540e，\u4ecd\u7136\u662f\u4e00\u4e2a\u70b9\u3002<\/p>\n<\/li>\n - \n
\u4f8b\u5982：\u4e09\u7ef4\u7a7a\u95f4\u4e2d\u7684\u70b9\u00a0P(x,y,z)\u00a0\u7ecf\u8fc7\u900f\u89c6\u6295\u5f71\u540e\u53d8\u6210\u4e8c\u7ef4\u5e73\u9762\u4e0a\u7684\u70b9\u00a0P'(x',y')，\u4f46\u5176\u201c\u70b9\u201d\u7684\u672c\u8d28\u4e0d\u53d8\u3002<\/p>\n<\/li>\n<\/ul>\n<\/p>\n 2. \u76f4\u7ebf\u7684\u6295\u5f71\u4e0d\u53d8\u6027<\/h6>\n<\/p>\n\n- \n
\u4e09\u7ef4\u7a7a\u95f4\u4e2d\u7684\u4e00\u6761\u76f4\u7ebf\u7ecf\u8fc7\u6295\u5f71\u53d8\u6362\u540e，\u5728\u4e8c\u7ef4\u5e73\u9762\u4e0a\u4ecd\u7136\u662f\u76f4\u7ebf\u3002<\/p>\n<\/li>\n - \n
\u4f8b\u5982：\u4e00\u6761\u76f4\u7ebf\u5728\u4e0d\u540c\u89c6\u89d2\u4e0b\u7684\u6295\u5f71\u59cb\u7ec8\u662f\u4e00\u6761\u76f4\u7ebf，\u4e0d\u4f1a\u53d8\u6210\u66f2\u7ebf\u3002<\/p>\n<\/li>\n<\/ul>\n<\/p>\n 3. \u66f2\u7ebf\u7684\u6295\u5f71\u4e0d\u53d8\u6027<\/h6>\n<\/p>\n\n- \n
\u7279\u6b8a\u7c7b\u578b\u7684\u66f2\u7ebf（\u5982\u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf）\u5728\u6295\u5f71\u53d8\u6362\u540e\u4ecd\u80fd\u4fdd\u6301\u5176\u5f62\u72b6\u7279\u5f81\u3002<\/p>\n<\/li>\n - \n
\u4f8b\u5982：\u5706\u9525\u66f2\u7ebf（\u692d\u5706\u3001\u629b\u7269\u7ebf\u3001\u53cc\u66f2\u7ebf）\u7ecf\u8fc7\u6295\u5f71\u540e\u53ef\u80fd\u53d8\u5f62，\u4f46\u5b83\u4eec\u4ecd\u7136\u662f\u5706\u9525\u66f2\u7ebf\u65cf\u7684\u4e00\u5458\u3002<\/p>\n<\/li>\n<\/ul>\n<\/p>\n 4. \u4ea4\u6bd4\u7684\u6295\u5f71\u4e0d\u53d8\u6027<\/h6>\n<\/p>\n\n- \n
\u4ea4\u6bd4（Cross Ratio）\u662f\u56db\u4e2a\u5171\u7ebf\u70b9\u4e4b\u95f4\u7684\u4e00\u79cd\u6bd4\u4f8b\u5173\u7cfb，\u5b83\u662f\u6295\u5f71\u53d8\u6362\u4e2d\u6700\u57fa\u672c\u7684\u4e0d\u53d8\u91cf\u3002<\/p>\n<\/li>\n - \n
\u4f8b\u5982：\u56db\u4e2a\u70b9\u00a0A,B,C,D\u00a0\u5728\u4e00\u6761\u76f4\u7ebf\u4e0a，\u5b83\u4eec\u7684\u4ea4\u6bd4\u4e3a：<\/p>\n<\/li>\n<\/ul>\n (A,B;C,D)=A\u2062C\u22c5B\u2062DA\u2062D\u22c5B\u2062C<\/p>\n \u8fd9\u4e2a\u503c\u5728\u4efb\u4f55\u6295\u5f71\u53d8\u6362\u4e0b\u90fd\u4fdd\u6301\u4e0d\u53d8\u3002<\/p>\n<\/p>\n
\n<\/p>\n\u4e3a\u4ec0\u4e48\u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf\u5177\u6709\u6295\u5f71\u4e0d\u53d8\u6027？<\/h6>\n<\/p>\n\u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf（Rational B\u00e9zier Curve）\u4e4b\u6240\u4ee5\u5177\u6709\u6295\u5f71\u4e0d\u53d8\u6027，\u662f\u56e0\u4e3a\u5b83\u5f15\u5165\u4e86\u6743\u91cd\u56e0\u5b50，\u4f7f\u5f97\u66f2\u7ebf\u5728\u9f50\u6b21\u5750\u6807\u7cfb\u4e2d\u8868\u793a\u3002\u4ee5\u4e0b\u662f\u5173\u952e\u539f\u56e0：<\/p>\n<\/p>\n 1. \u9f50\u6b21\u5750\u6807\u8868\u793a<\/h6>\n<\/p>\n\n- \n
\u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf\u4f7f\u7528\u9f50\u6b21\u5750\u6807\u00a0(x,y,w)\u00a0\u8868\u793a\u70b9，\u5176\u4e2d\u00a0w\u00a0\u662f\u6743\u91cd\u3002<\/p>\n<\/li>\n - \n
\u6295\u5f71\u53d8\u6362\u672c\u8d28\u4e0a\u662f\u5bf9\u9f50\u6b21\u5750\u6807\u7684\u7ebf\u6027\u53d8\u6362，\u56e0\u6b64\u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf\u7684\u5f62\u5f0f\u5728\u6295\u5f71\u540e\u4f9d\u7136\u6210\u7acb\u3002<\/p>\n<\/li>\n<\/ul>\n<\/p>\n 2. \u6743\u91cd\u7684\u4f5c\u7528<\/h6>\n<\/p>\n\n- \n
\u6743\u91cd\u5141\u8bb8\u66f2\u7ebf\u66f4\u7075\u6d3b\u5730\u903c\u8fd1\u590d\u6742\u7684\u51e0\u4f55\u5f62\u72b6（\u5982\u5706\u5f27\u3001\u692d\u5706\u7b49）\u3002<\/p>\n<\/li>\n - \n
\u5728\u6295\u5f71\u8fc7\u7a0b\u4e2d，\u6743\u91cd\u4f1a\u968f\u7740\u5750\u6807\u4e00\u8d77\u53d8\u6362，\u4f46\u66f2\u7ebf\u7684\u6574\u4f53\u5f62\u72b6\u7279\u6027\u5f97\u4ee5\u4fdd\u7559\u3002<\/p>\n<\/li>\n<\/ul>\n<\/p>\n 3. \u6570\u5b66\u63a8\u5bfc<\/h6>\n<\/p>\n\u8bbe\u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf\u7684\u63a7\u5236\u70b9\u4e3a\u00a0Pi=(xi,yi,wi)，\u5219\u66f2\u7ebf\u65b9\u7a0b\u4e3a：<\/p>\n C(t)=\u2211i=0nwi\u2062Bi,n(t)Pi\u2211i=0nwi\u2062Bi,n(t)<\/p>\n \u5f53\u5bf9\u8be5\u66f2\u7ebf\u8fdb\u884c\u6295\u5f71\u53d8\u6362\u65f6，\u5206\u5b50\u548c\u5206\u6bcd\u90fd\u4f1a\u53d7\u5230\u76f8\u540c\u7684\u7ebf\u6027\u53d8\u6362\u5f71\u54cd，\u56e0\u6b64\u66f2\u7ebf\u7684\u6bd4\u4f8b\u5173\u7cfb\u4fdd\u6301\u4e0d\u53d8\u3002<\/p>\n<\/p>\n
\n<\/p>\n\u5b9e\u9645\u5e94\u7528\u573a\u666f<\/h6>\n<\/p>\n<\/p>\n1. CAD\/CAM \u8bbe\u8ba1<\/h6>\n<\/p>\n\n- \n
\u5728\u4e09\u7ef4\u5efa\u6a21\u4e2d，\u8bbe\u8ba1\u5e08\u7ecf\u5e38\u9700\u8981\u5c06\u4e09\u7ef4\u6a21\u578b\u6295\u5f71\u5230\u4e8c\u7ef4\u56fe\u7eb8\u4e0a\u8fdb\u884c\u67e5\u770b\u6216\u52a0\u5de5\u3002<\/p>\n<\/li>\n - \n
\u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf\u7684\u6295\u5f71\u4e0d\u53d8\u6027\u4fdd\u8bc1\u4e86\u8bbe\u8ba1\u610f\u56fe\u5728\u4e0d\u540c\u89c6\u89d2\u4e0b\u7684\u4e00\u81f4\u6027\u3002<\/p>\n<\/li>\n<\/ul>\n<\/p>\n 2. \u8ba1\u7b97\u673a\u89c6\u89c9<\/h6>\n<\/p>\n\n- \n
\u5728\u6444\u50cf\u673a\u6210\u50cf\u4e2d，\u4e09\u7ef4\u573a\u666f\u901a\u8fc7\u900f\u89c6\u6295\u5f71\u6620\u5c04\u5230\u4e8c\u7ef4\u56fe\u50cf\u3002<\/p>\n<\/li>\n - \n
\u6295\u5f71\u4e0d\u53d8\u6027\u5e2e\u52a9\u8bc6\u522b\u548c\u91cd\u5efa\u4e09\u7ef4\u7ed3\u6784\u3002<\/p>\n<\/li>\n<\/ul>\n<\/p>\n 3. \u52a8\u753b\u4e0e\u6e32\u67d3<\/h6>\n<\/p>\n\n- \n
\u5728\u4e09\u7ef4\u52a8\u753b\u5236\u4f5c\u4e2d，\u7269\u4f53\u7684\u8fd0\u52a8\u8f68\u8ff9\u9700\u8981\u5728\u4e0d\u540c\u89c6\u89d2\u4e0b\u770b\u8d77\u6765\u81ea\u7136\u3002<\/p>\n<\/li>\n - \n
\u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf\u53ef\u4ee5\u786e\u4fdd\u52a8\u753b\u8def\u5f84\u5728\u6295\u5f71\u540e\u4ecd\u7136\u5e73\u6ed1\u3002<\/p>\n<\/li>\n<\/ul>\n
\n<\/p>\n\u793a\u4f8b\u8bf4\u660e<\/h6>\n<\/p>\n\u5047\u8bbe\u6211\u4eec\u6709\u4e00\u4e2a\u4e09\u7ef4\u692d\u5706\u5f27，\u7528\u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf\u8868\u793a\u3002\u5f53\u6211\u4eec\u4ece\u4e0d\u540c\u89d2\u5ea6\u89c2\u5bdf\u8fd9\u4e2a\u692d\u5706\u65f6：<\/p>\n \n- \n
\u692d\u5706\u53ef\u80fd\u4f1a\u88ab\u538b\u6241\u6216\u62c9\u4f38，\u4f46\u5b83\u4ecd\u7136\u662f\u4e00\u4e2a\u692d\u5706\u3002<\/p>\n<\/li>\n - \n
\u5982\u679c\u6362\u6210\u666e\u901a\u7684\u591a\u9879\u5f0f\u8d1d\u585e\u5c14\u66f2\u7ebf，\u692d\u5706\u4f1a\u88ab\u626d\u66f2\u6210\u5176\u4ed6\u5f62\u72b6（\u5982\u5375\u5f62\u6216\u4e0d\u89c4\u5219\u66f2\u7ebf）\u3002<\/p>\n<\/li>\n<\/ul>\n \u8fd9\u5c31\u662f\u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf\u7684\u4f18\u52bf\u6240\u5728\u2014\u2014\u5b83\u80fd\u66f4\u597d\u5730\u4fdd\u6301\u51e0\u4f55\u5f62\u72b6\u7684\u672c\u8d28\u7279\u6027\u3002<\/p>\n<\/p>\n
\n<\/p>\n\u603b\u7ed3<\/h6>\n<\/p>\n\u6295\u5f71\u4e0d\u53d8\u6027\u662f\u51e0\u4f55\u5efa\u6a21\u4e2d\u4e00\u79cd\u91cd\u8981\u7684\u6570\u5b66\u6027\u8d28，\u5b83\u786e\u4fdd\u4e86\u67d0\u4e9b\u51e0\u4f55\u7279\u6027\u5728\u6295\u5f71\u53d8\u6362\u540e\u4f9d\u7136\u6210\u7acb\u3002\u5bf9\u4e8e\u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf\u800c\u8a00，\u8fd9\u79cd\u6027\u8d28\u6765\u6e90\u4e8e\u5176\u9f50\u6b21\u5750\u6807\u8868\u793a\u548c\u6743\u91cd\u673a\u5236，\u4f7f\u5176\u5728 CAD\/CAM\u3001\u8ba1\u7b97\u673a\u89c6\u89c9\u7b49\u9886\u57df\u5177\u6709\u5e7f\u6cdb\u7684\u5e94\u7528\u4ef7\u503c\u3002<\/p>\n \n\n\n| \n >>>\n<\/td>\n | \n def compute_rational_bezier_curve_3d(control_points, weights, t_values=None):
\n """\u8ba1\u7b97\u4e09\u7ef4\u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf"""
\n control_points = np.array(control_points)
\n weights = np.array(weights)
\n n = len(control_points) – 1<\/p>\n if t_values is None:
\n t_values = np.linspace(0, 1, 100)<\/p>\n curve_points = np.zeros((len(t_values), control_points.shape[1]))<\/p>\n for i, t in enumerate(t_values):
\n numerator = np.zeros(control_points.shape[1])
\n denominator = 0.0<\/p>\n for j in range(n+1):
\n basis = bernstein_basis_numeric(n, j, t)
\n weighted_basis = weights[j] * basis
\n numerator += weighted_basis * control_points[j]
\n denominator += weighted_basis<\/p>\n if denominator != 0:
\n curve_points[i] = numerator \/ denominator<\/p>\n return curve_points<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n \n\n\n| \n >>>\n<\/td>\n | \n def compute_bezier_curve_3d(control_points, t_values=None):
\n """\u8ba1\u7b97\u4e09\u7ef4\u6807\u51c6\u8d1d\u585e\u5c14\u66f2\u7ebf"""
\n control_points = np.array(control_points)
\n n = len(control_points) – 1<\/p>\n if t_values is None:
\n t_values = np.linspace(0, 1, 100)<\/p>\n curve_points = np.zeros((len(t_values), control_points.shape[1]))<\/p>\n for i, t in enumerate(t_values):
\n for j in range(n+1):
\n basis = bernstein_basis_numeric(n, j, t)
\n curve_points[i] += basis * control_points[j]<\/p>\n return curve_points<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n \n\n\n| \n >>>\n<\/td>\n | \n def apply_perspective_projection(points, camera_pos, focal_length):
\n """\u5e94\u7528\u900f\u89c6\u6295\u5f71"""
\n projected_points = []<\/p>\n for point in points:
\n # \u8ba1\u7b97\u76f8\u5bf9\u4e8e\u76f8\u673a\u7684\u4f4d\u7f6e
\n relative_pos = point – camera_pos<\/p>\n # \u900f\u89c6\u6295\u5f71
\n if relative_pos[2] != 0:
\n x_proj = focal_length * relative_pos[0] \/ relative_pos[2]
\n y_proj = focal_length * relative_pos[1] \/ relative_pos[2]
\n projected_points.append([x_proj, y_proj])
\n else:
\n projected_points.append([0, 0])<\/p>\n return np.array(projected_points)<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n \n\n\n| \n >>>\n<\/td>\n | \n def bezier_application_cad_cam():
\n """
\n \u8d1d\u585e\u5c14\u66f2\u7ebf\u5e94\u7528\u5b9e\u4f8b：CAD\/CAM
\n \u4f7f\u7528\u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf\u4fdd\u6301\u6295\u5f71\u4e0d\u53d8\u6027
\n """<\/p>\n # \u521b\u5efa\u56fe\u5f62
\n fig = plt.figure(figsize=(16, 12))<\/p>\n # 1. \u4e09\u7ef4\u66f2\u7ebf\u8bbe\u8ba1
\n ax1 = fig.add_subplot(2, 3, 1, projection='3d')<\/p>\n # \u5b9a\u4e49\u4e09\u7ef4\u63a7\u5236\u70b9
\n control_points_3d = np.array([
\n [0, 0, 0],
\n [1, 2, 1],
\n [3, 1, 2],
\n [4, 3, 1],
\n [5, 0, 0]
\n ])<\/p>\n # \u8ba1\u7b97\u4e09\u7ef4\u8d1d\u585e\u5c14\u66f2\u7ebf
\n curve_3d = compute_bezier_curve_3d(control_points_3d)<\/p>\n # \u7ed8\u5236\u4e09\u7ef4\u66f2\u7ebf
\n ax1.plot(curve_3d[:, 0], curve_3d[:, 1], curve_3d[:, 2],
\n 'b-', linewidth=2.5, label='\u4e09\u7ef4\u66f2\u7ebf')
\n ax1.scatter(control_points_3d[:, 0], control_points_3d[:, 1], control_points_3d[:, 2],
\n c='r', s=50, alpha=0.7, label='\u63a7\u5236\u70b9')<\/p>\n ax1.set_xlabel('X', fontsize=11)
\n ax1.set_ylabel('Y', fontsize=11)
\n ax1.set_zlabel('Z', fontsize=11)
\n ax1.set_title('\u4e09\u7ef4CAD\u66f2\u7ebf\u8bbe\u8ba1', fontsize=12, fontweight='bold')
\n ax1.legend(fontsize=9)<\/p>\n # 2. \u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf\u5728CAD\u4e2d\u7684\u5e94\u7528
\n ax2 = fig.add_subplot(2, 3, 2)<\/p>\n # \u692d\u5706\u53c2\u6570
\n a, b = 2, 1
\n num_segments = 8<\/p>\n # \u7cbe\u786e\u692d\u5706（\u7528\u4e8e\u6bd4\u8f83）
\n angles_exact = np.linspace(0, 2*np.pi, 500)
\n exact_ellipse = np.column_stack([a * np.cos(angles_exact), b * np.sin(angles_exact)])
\n ax2.plot(exact_ellipse[:, 0], exact_ellipse[:, 1], 'k-',
\n linewidth=1, alpha=0.3, label='\u7cbe\u786e\u692d\u5706')<\/p>\n # \u4f7f\u75288\u6bb5\u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf\u62fc\u63a5\u692d\u5706
\n for i in range(num_segments):
\n start_angle = i * 2*np.pi\/num_segments
\n end_angle = (i+1) * 2*np.pi\/num_segments
\n theta = end_angle – start_angle<\/p>\n # \u63a7\u5236\u70b9
\n P0 = np.array([a*np.cos(start_angle), b*np.sin(start_angle), 0])
\n P2 = np.array([a*np.cos(end_angle), b*np.sin(end_angle), 0])<\/p>\n # \u4e2d\u95f4\u63a7\u5236\u70b9
\n mid_angle = (start_angle + end_angle) \/ 2
\n cos_half_theta = np.cos(theta\/2)
\n P1 = np.array([
\n a*np.cos(mid_angle)\/cos_half_theta,
\n b*np.sin(mid_angle)\/cos_half_theta,
\n 0
\n ])<\/p>\n # \u6743\u91cd
\n weights = np.array([1, cos_half_theta, 1])<\/p>\n # \u8ba1\u7b97\u66f2\u7ebf
\n control_points = np.array([P0, P1, P2])
\n curve = compute_rational_bezier_curve_3d(control_points, weights)<\/p>\n # \u7ed8\u5236\u66f2\u7ebf
\n color = plt.cm.tab10(i\/num_segments)
\n ax2.plot(curve[:, 0], curve[:, 1], '-', linewidth=2, color=color)<\/p>\n # \u524d\u4e24\u6bb5\u663e\u793a\u63a7\u5236\u591a\u8fb9\u5f62
\n if i < 2:
\n ax2.plot(control_points[:, 0], control_points[:, 1], '–',
\n linewidth=1, color=color, alpha=0.6)
\n ax2.scatter(control_points[:, 0], control_points[:, 1],
\n s=40, color=color, alpha=0.8)<\/p>\n ax2.set_xlim(-2.5, 2.5)
\n ax2.set_ylim(-1.5, 1.5)
\n ax2.set_title('\u6709\u7406\u66f2\u7ebf\u8868\u793a\u692d\u5706 (CAD\u7cbe\u786e\u5efa\u6a21)', fontsize=12, fontweight='bold')
\n ax2.set_xlabel('X', fontsize=11)
\n ax2.set_ylabel('Y', fontsize=11)
\n ax2.grid(True, alpha=0.3)
\n ax2.axis('equal')<\/p>\n # 3. \u6295\u5f71\u4e0d\u53d8\u6027\u6f14\u793a
\n ax3 = fig.add_subplot(2, 3, 3, projection='3d')<\/p>\n # \u5b9a\u4e49\u4e09\u7ef4\u66f2\u7ebf
\n control_cad = np.array([
\n [0, 0, 0],
\n [1, 1, 0.5],
\n [2, 0, 1],
\n [3, 1, 0.5],
\n [4, 0, 0]
\n ])<\/p>\n # \u6807\u51c6\u8d1d\u585e\u5c14\u66f2\u7ebf
\n curve_standard = compute_bezier_curve_3d(control_cad)<\/p>\n # \u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf
\n weights_cad = np.array([1, 2, 1, 2, 1])
\n curve_rational = compute_rational_bezier_curve_3d(control_cad, weights_cad)<\/p>\n # \u7ed8\u5236\u539f\u59cb\u66f2\u7ebf
\n ax3.plot(curve_standard[:, 0], curve_standard[:, 1], curve_standard[:, 2],
\n 'b-', linewidth=2, label='\u6807\u51c6\u66f2\u7ebf', alpha=0.7)
\n ax3.plot(curve_rational[:, 0], curve_rational[:, 1], curve_rational[:, 2],
\n 'r-', linewidth=2, label='\u6709\u7406\u66f2\u7ebf', alpha=0.7)<\/p>\n # \u5e94\u7528\u900f\u89c6\u53d8\u6362
\n camera_pos = np.array([2, -5, 3])
\n focal_length = 2<\/p>\n # \u6295\u5f71\u52302D\u5e73\u9762
\n projected_standard = apply_perspective_projection(curve_standard, camera_pos, focal_length)
\n projected_rational = apply_perspective_projection(curve_rational, camera_pos, focal_length)<\/p>\n ax3.set_xlabel('X', fontsize=11)
\n ax3.set_ylabel('Y', fontsize=11)
\n ax3.set_zlabel('Z', fontsize=11)
\n ax3.set_title('\u4e09\u7ef4CAD\u66f2\u7ebf\u6295\u5f71', fontsize=12, fontweight='bold')
\n ax3.legend(fontsize=9)<\/p>\n # 4. \u6295\u5f71\u7ed3\u679c\u6bd4\u8f83
\n ax4 = fig.add_subplot(2, 3, 4)<\/p>\n ax4.plot(projected_standard[:, 0], projected_standard[:, 1],
\n 'b-', linewidth=2, label='\u6807\u51c6\u66f2\u7ebf\u6295\u5f71', alpha=0.7)
\n ax4.plot(projected_rational[:, 0], projected_rational[:, 1],
\n 'r-', linewidth=2, label='\u6709\u7406\u66f2\u7ebf\u6295\u5f71', alpha=0.7)<\/p>\n ax4.set_xlabel('\u6295\u5f71X', fontsize=11)
\n ax4.set_ylabel('\u6295\u5f71Y', fontsize=11)
\n ax4.set_title('\u6295\u5f71\u4e0d\u53d8\u6027\u6bd4\u8f83', fontsize=12, fontweight='bold')
\n ax4.grid(True, alpha=0.3)
\n ax4.legend(fontsize=9)
\n ax4.axis('equal')<\/p>\n # 5. CAM\u5200\u5177\u8def\u5f84\u751f\u6210
\n ax5 = fig.add_subplot(2, 3, 5)<\/p>\n # \u5b9a\u4e49\u52a0\u5de5\u8f6e\u5ed3
\n machining_points = np.array([
\n [0, 0],
\n [1, 0],
\n [2, 1],
\n [3, 1],
\n [4, 0],
\n [5, -1],
\n [6, 0],
\n [7, 1],
\n [8, 0]
\n ])<\/p>\n # \u4f7f\u7528\u8d1d\u585e\u5c14\u66f2\u7ebf\u5e73\u6ed1\u5200\u5177\u8def\u5f84
\n tool_paths = []
\n for i in range(0, len(machining_points)-1, 2):
\n if i+2 < len(machining_points):
\n control_tool = np.array([machining_points[i],
\n (machining_points[i] + machining_points[i+1])\/2,
\n machining_points[i+1],
\n (machining_points[i+1] + machining_points[i+2])\/2,
\n machining_points[i+2]])<\/p>\n curve_tool = compute_bezier_curve_3d(
\n np.column_stack([control_tool, np.zeros((5, 1))])
\n )<\/p>\n ax5.plot(curve_tool[:, 0], curve_tool[:, 1], 'g-', linewidth=2, alpha=0.7)
\n ax5.scatter(control_tool[:, 0], control_tool[:, 1],
\n s=30, c='orange', alpha=0.5)<\/p>\n # \u7ed8\u5236\u539f\u59cb\u8f6e\u5ed3\u70b9
\n ax5.plot(machining_points[:, 0], machining_points[:, 1], 'k–',
\n linewidth=1, alpha=0.5, label='\u539f\u59cb\u8f6e\u5ed3')
\n ax5.scatter(machining_points[:, 0], machining_points[:, 1],
\n s=50, c='red', alpha=0.7, label='\u52a0\u5de5\u70b9')<\/p>\n ax5.set_xlabel('X (mm)', fontsize=11)
\n ax5.set_ylabel('Y (mm)', fontsize=11)
\n ax5.set_title('CAM\u5200\u5177\u8def\u5f84\u751f\u6210', fontsize=12, fontweight='bold')
\n ax5.grid(True, alpha=0.3)
\n ax5.legend(fontsize=9)
\n ax5.axis('equal')<\/p>\n # 6. \u5de5\u7a0b\u66f2\u9762\u8bbe\u8ba1
\n ax6 = fig.add_subplot(2, 3, 6, projection='3d')<\/p>\n # \u521b\u5efa\u66f2\u9762\u63a7\u5236\u7f51\u683c
\n u = np.linspace(0, 1, 5)
\n v = np.linspace(0, 1, 5)<\/p>\n # \u521b\u5efa\u63a7\u5236\u70b9\u7f51\u683c
\n control_grid = np.zeros((5, 5, 3))
\n for i in range(5):
\n for j in range(5):
\n control_grid[i, j] = [u[i]*8, v[j]*6, np.sin(u[i]*np.pi)*np.cos(v[j]*np.pi)*2]<\/p>\n # \u7ed8\u5236\u63a7\u5236\u7f51\u683c
\n for i in range(5):
\n ax6.plot(control_grid[i, :, 0], control_grid[i, :, 1], control_grid[i, :, 2],
\n 'r-', linewidth=1, alpha=0.5)
\n for j in range(5):
\n ax6.plot(control_grid[:, j, 0], control_grid[:, j, 1], control_grid[:, j, 2],
\n 'r-', linewidth=1, alpha=0.5)<\/p>\n # \u7ed8\u5236\u63a7\u5236\u70b9
\n ax6.scatter(control_grid[:, :, 0].flatten(),
\n control_grid[:, :, 1].flatten(),
\n control_grid[:, :, 2].flatten(),
\n c='b', s=30, alpha=0.7)<\/p>\n ax6.set_xlabel('X', fontsize=11)
\n ax6.set_ylabel('Y', fontsize=11)
\n ax6.set_zlabel('Z', fontsize=11)
\n ax6.set_title('\u5de5\u7a0b\u66f2\u9762\u8bbe\u8ba1 (NURBS\u57fa\u7840)', fontsize=12, fontweight='bold')<\/p>\n fig.suptitle('CAD\/CAM\u5e94\u7528：' \\\\
\n '\u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf\u4fdd\u6301\u6295\u5f71\u4e0d\u53d8\u6027', fontsize=16, fontweight='bold')
\n fig.tight_layout()
\n return fig<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n \n\n\n| \n >>>\n<\/td>\n | \n # \u6267\u884cCAD\/CAM\u6848\u4f8b
\nfig_cad_cam = bezier_application_cad_cam()
\nfig_cad_cam<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n ![\"\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/02\/20260222072609-699aaf91b90d1.png\") <\/p>\n<\/p>\n
7. \u603b\u7ed3<\/h4>\n7.1 \u6807\u51c6\u8d1d\u585e\u5c14\u66f2\u7ebf\u5173\u952e\u7279\u6027<\/h5>\n<\/p>\n- \n
\u6570\u5b66\u57fa\u7840：\u57fa\u4e8e\u4f2f\u6069\u65af\u5766\u57fa\u51fd\u6570，\u5177\u6709\u826f\u597d\u7684\u6570\u503c\u7a33\u5b9a\u6027<\/p>\n<\/li>\n - \n
\u51e0\u4f55\u76f4\u89c2：\u901a\u8fc7\u63a7\u5236\u70b9\u76f4\u63a5\u63a7\u5236\u66f2\u7ebf\u5f62\u72b6<\/p>\n<\/li>\n - \n
\u51f8\u5305\u6027：\u66f2\u7ebf\u59cb\u7ec8\u4f4d\u4e8e\u63a7\u5236\u70b9\u7684\u51f8\u5305\u5185<\/p>\n<\/li>\n - \n
\u7aef\u70b9\u63d2\u503c：\u66f2\u7ebf\u901a\u8fc7\u7b2c\u4e00\u4e2a\u548c\u6700\u540e\u4e00\u4e2a\u63a7\u5236\u70b9<\/p>\n<\/li>\n - \n
\u53d8\u5dee\u7f29\u51cf\u6027：\u66f2\u7ebf\u6ce2\u52a8\u4e0d\u8d85\u8fc7\u63a7\u5236\u591a\u8fb9\u5f62\u6ce2\u52a8<\/p>\n<\/li>\n - \n
\u4eff\u5c04\u4e0d\u53d8\u6027：\u5bf9\u63a7\u5236\u70b9\u7684\u4eff\u5c04\u53d8\u6362\u7b49\u4ef7\u4e8e\u5bf9\u66f2\u7ebf\u7684\u76f8\u540c\u53d8\u6362<\/p>\n<\/li>\n 7.2 \u6709\u7406\u8d1d\u585e\u5c14\u66f2\u7ebf\u4f18\u52bf<\/h5>\n<\/p>\n - \n
\u6295\u5f71\u4e0d\u53d8\u6027：\u5728\u6295\u5f71\u53d8\u6362\u4e0b\u4fdd\u6301\u4e0d\u53d8<\/p>\n<\/li>\n - \n
\u6743\u91cd\u63a7\u5236：\u901a\u8fc7\u6743\u91cd\u53c2\u6570\u63d0\u4f9b\u989d\u5916\u63a7\u5236\u81ea\u7531\u5ea6<\/p>\n<\/li>\n - \n
\u7cbe\u786e\u8868\u793a：\u53ef\u4ee5\u7cbe\u786e\u8868\u793a\u5706\u9525\u66f2\u7ebf<\/p>\n<\/li>\n - \n
\u5411\u540e\u517c\u5bb9：\u5f53\u6240\u6709\u6743\u91cd\u76f8\u7b49\u65f6，\u9000\u5316\u4e3a\u6807\u51c6\u8d1d\u585e\u5c14\u66f2\u7ebf<\/p>\n<\/li>\n 7.3 \u8fde\u7eed\u6027\u6761\u4ef6<\/h5>\n<\/p>\n - \n
\u4f4d\u7f6e\u8fde\u7eed（C\u2070）：Pn=Q0<\/p>\n<\/li>\n - \n
\u5207\u7ebf\u65b9\u5411\u8fde\u7eed（G\u00b9）：Pn-1,Pn=Q0,Q1\u00a0\u4e09\u70b9\u5171\u7ebf<\/p>\n<\/li>\n - \n
\u5207\u7ebf\u8fde\u7eed（C\u00b9）：n\u2062(Pn-Pn-1)=m\u2062(Q1-Q0)<\/p>\n<\/li>\n - \n
\u66f2\u7387\u8fde\u7eed（G\u00b2\/C\u00b2）：\u9700\u8981\u6ee1\u8db3\u4e8c\u9636\u5bfc\u6570\u6761\u4ef6<\/p>\n<\/li>\n","protected":false},"excerpt":{"rendered":" \u58f0\u660e：\u672c\u5b66\u4e60\u7b14\u8bb0\u4f7f\u7528\u57fa\u4e8eGNU TeXmacs\u7684\u4e13\u4e1a\u7f16\u8f91\u5de5\u5177\u548cpython\u73af\u5883\u5b9e\u73b0，\u5185\u5bb9\u6765\u81ea\u4e8e\u7f51\u7edc\u6574\u7406\u5b66\u4e60，\u5185\u5d4c\u4ee3\u7801\u7531AI\u8f85\u52a9\u5e76\u624b\u52a8\u8c03\u8bd5\u901a\u8fc7，\u5c3d\u7ba1\u4e3a\u4e86\u4e0a\u4f20\u505a\u4e86\u624b\u52a8\u5904\u7406，\u5948\u4f55\u517c\u5bb9\u6027\u6b20\u4f73，\u65e0\u6cd5\u5b9e\u73b0\u539f\u5f00\u53d1\u73af\u5883\u6d4f\u89c8\u6548\u679c\u30021. \u4f2f\u6069\u65af\u5766\u57fa\u51fd\u65701.1 \u6570\u5b66\u5b9a\u4e49\u4f2f\u6069\u65af\u5766\u57fa\u51fd\u6570\u662f\u8d1d\u585e\u5c14\u66f2\u7ebf\u7684\u6570\u5b66\u57fa\u7840\u3002\u5bf9\u4e8e\u975e\u8d1f\u6574\u6570\u00a0n\u00a0\u548c\u6574\u6570\u00a0i（0\u2264i\u2264n），\u7b2c\u00a0i\u00a0\u4e2a\u00a0n\u00a0\u6b21\u4f2f\u6069\u65af<\/p>\n","protected":false},"author":2,"featured_media":76482,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[8232,7643,2468,81,1482,427],"topic":[],"class_list":["post-76494","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-server","tag-8232","tag-7643","tag-matplotlib","tag-python","tag-1482","tag-427"],"yoast_head":"\n \u8d1d\u585e\u5c14\u66f2\u7ebf\uff1a\u6570\u5b66\u63a8\u5bfc\u4e0ePython\u5b9e\u73b0 - \u7f51\u7855\u4e92\u8054\u5e2e\u52a9\u4e2d\u5fc3<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.wsisp.com\/helps\/76494.html\" \/>\n<meta property=\"og:locale\" content=\"zh_CN\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"\u8d1d\u585e\u5c14\u66f2\u7ebf\uff1a\u6570\u5b66\u63a8\u5bfc\u4e0ePython\u5b9e\u73b0 - \u7f51\u7855\u4e92\u8054\u5e2e\u52a9\u4e2d\u5fc3\" \/>\n<meta property=\"og:description\" content=\"\u58f0\u660e：\u672c\u5b66\u4e60\u7b14\u8bb0\u4f7f\u7528\u57fa\u4e8eGNU TeXmacs\u7684\u4e13\u4e1a\u7f16\u8f91\u5de5\u5177\u548cpython\u73af\u5883\u5b9e\u73b0，\u5185\u5bb9\u6765\u81ea\u4e8e\u7f51\u7edc\u6574\u7406\u5b66\u4e60，\u5185\u5d4c\u4ee3\u7801\u7531AI\u8f85\u52a9\u5e76\u624b\u52a8\u8c03\u8bd5\u901a\u8fc7，\u5c3d\u7ba1\u4e3a\u4e86\u4e0a\u4f20\u505a\u4e86\u624b\u52a8\u5904\u7406，\u5948\u4f55\u517c\u5bb9\u6027\u6b20\u4f73，\u65e0\u6cd5\u5b9e\u73b0\u539f\u5f00\u53d1\u73af\u5883\u6d4f\u89c8\u6548\u679c\u30021. \u4f2f\u6069\u65af\u5766\u57fa\u51fd\u65701.1 \u6570\u5b66\u5b9a\u4e49\u4f2f\u6069\u65af\u5766\u57fa\u51fd\u6570\u662f\u8d1d\u585e\u5c14\u66f2\u7ebf\u7684\u6570\u5b66\u57fa\u7840\u3002\u5bf9\u4e8e\u975e\u8d1f\u6574\u6570\u00a0n\u00a0\u548c\u6574\u6570\u00a0i（0\u2264i\u2264n），\u7b2c\u00a0i\u00a0\u4e2a\u00a0n\u00a0\u6b21\u4f2f\u6069\u65af\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.wsisp.com\/helps\/76494.html\" \/>\n<meta property=\"og:site_name\" content=\"\u7f51\u7855\u4e92\u8054\u5e2e\u52a9\u4e2d\u5fc3\" \/>\n<meta property=\"article:published_time\" content=\"2026-02-22T07:26:10+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/02\/20260222072608-699aaf906c644.png\" \/>\n<meta name=\"author\" content=\"admin\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"\u4f5c\u8005\" \/>\n\t<meta name=\"twitter:data1\" content=\"admin\" \/>\n\t<meta name=\"twitter:label2\" content=\"\u9884\u8ba1\u9605\u8bfb\u65f6\u95f4\" \/>\n\t<meta name=\"twitter:data2\" content=\"36 \u5206\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/www.wsisp.com\/helps\/76494.html\",\"url\":\"https:\/\/www.wsisp.com\/helps\/76494.html\",\"name\":\"\u8d1d\u585e\u5c14\u66f2\u7ebf\uff1a\u6570\u5b66\u63a8\u5bfc\u4e0ePython\u5b9e\u73b0 - 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