import math import numpy as np import matplotlib.pyplot as plt from mpl_toolkits.axes_grid1 import make_axes_locatable def arc_gwn_batch(a, b, bulge, P): """ 向量化批处理接口：对 (N, 2) 的点阵 P 同时计算单段圆弧的广义绕数 W。 """ a = np.asarray(a, dtype=float) b = np.asarray(b, dtype=float) P = np.asarray(P, dtype=float) half_ab = 0.5 * (a + b) bisec_vec = 0.5 * np.array([a[1] - b[1], b[0] - a[0]]) center = half_ab + bisec_vec / bulge ca = a - center cb = b - center cp = P - center D = np.linalg.norm(cp, axis=1) r = np.linalg.norm(ca) theta = 2.0 * math.atan(bulge) W = np.empty(len(P)) mask_center = D < 1e-12 mask_circle = np.abs(D - r) < 1e-12 mask_normal = ~(mask_circle) # W[mask_center] = theta / (2.0 * math.pi) W[mask_circle] = theta / (4.0 * math.pi) cp_n = cp[mask_normal] D_n = D[mask_normal] sigma = np.where(r > D_n, 1.0, -1.0) Rd = r * D_n r_p_D = r + D_n abs_r_D = np.abs(r - D_n) dot_a = ca.dot(cp_n.T) cross1 = cp_n[:, 0] * ca[1] - ca[0] * cp_n[:, 1] Ga = np.arctan2(r_p_D * cross1, abs_r_D * (Rd + dot_a)) dot_b = cb.dot(cp_n.T) cross2 = cp_n[:, 0] * cb[1] - cb[0] * cp_n[:, 1] Gb = np.arctan2(r_p_D * cross2, abs_r_D * (Rd + dot_b)) delta = Gb - Ga fix = delta * theta < 0.0 delta[fix] += math.copysign(math.pi, theta) two_pi_W = 0.5 * theta + sigma * delta W[mask_normal] = two_pi_W / (2.0 * math.pi) return W def plot_multi_arc_gwn_field(arcs, margin=0.5, nx=400, ny=400): """ 绘制多段圆弧的广义绕数之和。 参数 ---- arcs : list of tuple 每段圆弧表示为 (a, b, bulge)，其中 a, b 为长度为 2 的序列。 margin : float AABB 外扩边界。 nx, ny : int 采样分辨率。 """ arcs = [(np.asarray(a, dtype=float), np.asarray(b, dtype=float), float(bulge)) for a, b, bulge in arcs] # ---- 1. 计算所有圆弧的几何信息，确定全局 AABB ---- all_arc_x = [] all_arc_y = [] all_centers = [] all_rs = [] for a, b, bulge in arcs: half_ab = 0.5 * (a + b) bisec_vec = 0.5 * np.array([a[1] - b[1], b[0] - a[0]]) center = half_ab + bisec_vec / bulge r = np.linalg.norm(a - center) theta = 2.0 * math.atan(bulge) start_angle = math.atan2(a[1] - center[1], a[0] - center[0]) end_angle = start_angle + theta t_arc = np.linspace(start_angle, end_angle, 200) arc_x = center[0] + r * np.cos(t_arc) arc_y = center[1] + r * np.sin(t_arc) all_arc_x.append(arc_x) all_arc_y.append(arc_y) all_centers.append(center) all_rs.append(r) # 全局 AABB x_min = min(ax.min() for ax in all_arc_x) - margin x_max = max(ax.max() for ax in all_arc_x) + margin y_min = min(ay.min() for ay in all_arc_y) - margin y_max = max(ay.max() for ay in all_arc_y) + margin # ---- 2. 均匀采样并累加所有圆弧的 GWN ---- xs = np.linspace(x_min, x_max, nx) ys = np.linspace(y_min, y_max, ny) X, Y = np.meshgrid(xs, ys) points = np.stack([X.ravel(), Y.ravel()], axis=1) W = np.zeros(len(points)) for a, b, bulge in arcs: W += arc_gwn_batch(a, b, bulge, points) W = W.reshape(X.shape) # ---- 3. 三分区独立归一化 ---- eps = np.finfo(float).eps * 1e6 bound_low = -eps bound_high = eps # 负区 W_neg = W.copy() W_neg[W >= bound_low] = np.nan W_min = np.nanmin(W_neg) if not np.isnan(W_min): W_neg = (bound_low - W_neg) / (bound_low - W_min) # 零区 W_zero = W.copy() W_zero[(W < bound_low) | (W > bound_high)] = np.nan W_zero = (W_zero - bound_low) / (2.0 * eps) # 正区 W_pos = W.copy() W_pos[W <= bound_high] = np.nan W_max = np.nanmax(W_pos) if not np.isnan(W_max): W_pos = (W_pos - bound_high) / (W_max - bound_high) # ---- 4. 绘图 ---- fig, ax = plt.subplots(figsize=(10, 8)) im_neg = ax.pcolormesh(X, Y, W_neg, cmap='Blues', vmin=0, vmax=1, shading='auto') im_zero = ax.pcolormesh(X, Y, W_zero, cmap='Greys', vmin=0, vmax=1, shading='auto') im_pos = ax.pcolormesh(X, Y, W_pos, cmap='Reds', vmin=0, vmax=1, shading='auto') # 叠加所有圆弧 for idx, (a, b, bulge) in enumerate(arcs): ax.plot(all_arc_x[idx], all_arc_y[idx], 'k-', linewidth=2.5, zorder=4) ax.plot(a[0], a[1], 'go', markersize=7, zorder=5) ax.plot(b[0], b[1], 'mo', markersize=7, zorder=5) ax.plot(all_centers[idx][0], all_centers[idx][1], 'k+', markersize=8, mew=1.5, zorder=5) ax.set_aspect('equal') ax.set_xlim(x_min, x_max) ax.set_ylim(y_min, y_max) ax.set_title(f'Multi-Arc GWN Sum ({len(arcs)} arcs)') # ---- 5. 三个独立 colorbar ---- divider = make_axes_locatable(ax) cax_neg = divider.append_axes("left", size="5%", pad=0.4) cbar_neg = fig.colorbar(im_neg, cax=cax_neg, orientation='vertical') cbar_neg.ax.yaxis.set_ticks_position('left') cbar_neg.ax.yaxis.set_label_position('left') cbar_neg.set_label('Negative $W$ (normalized)', rotation=90, labelpad=10) cbar_neg.set_ticks([0, 1]) cbar_neg.set_ticklabels([f'{bound_low:.2e}', f'{W_min:.3f}' if not np.isnan(W_min) else 'N/A']) cax_pos = divider.append_axes("right", size="5%", pad=0.4) cbar_pos = fig.colorbar(im_pos, cax=cax_pos, orientation='vertical') cbar_pos.set_label('Positive $W$ (normalized)', rotation=270, labelpad=15) cbar_pos.set_ticks([0, 1]) cbar_pos.set_ticklabels([f'{bound_high:.2e}', f'{W_max:.3f}' if not np.isnan(W_max) else 'N/A']) cax_zero = divider.append_axes("bottom", size="5%", pad=0.6) cbar_zero = fig.colorbar(im_zero, cax=cax_zero, orientation='horizontal') cbar_zero.set_label(f'Near-zero $W$ (normalized) [{bound_low:.2e}, {bound_high:.2e}]', labelpad=5) cbar_zero.set_ticks([0, 1]) cbar_zero.set_ticklabels([f'{bound_low:.2e}', f'{bound_high:.2e}']) plt.tight_layout() plt.show() # ==================== 主程序 ==================== if __name__ == '__main__': # 示例 1：两段圆弧拼成近似整圆 arcs = [ ([2.0, 0.0], [0.0, 0.0], float("inf")), # 上半圆 ([0.0, 0.0], [2.0, 0.0], float("inf")), # 下半圆（注意 bulge>0 表示逆时针） ] plot_multi_arc_gwn_field(arcs, margin=0.5, nx=1024, ny=1024)