### ARAP (As-Rigid-As-Possible) 算法 Python 实现 ARAP 是一种用于网格变形的技术，旨在保持形状尽可能刚性不变。下面是一个简化版本的 ARAP 算法实现示例： ```python import numpy as np from scipy.sparse import lil_matrix, csr_matrix from scipy.sparse.linalg import spsolve def compute_arap_deformation(vertices, faces, handle_positions): """ 计算基于 ARAP 的网格变形 参数: vertices: 原始顶点位置数组 (N, 3) faces: 面片索引数组 (M, 3) handle_positions: 用户指定的手柄新位置字典 {vertex_index: new_position} 返回: deformed_vertices: 变形后的顶点坐标 (N, 3) """ num_vertices = len(vertices) # 构建拉普拉斯矩阵 L 和质量矩阵 M W = build_weight_matrix(faces, num_vertices)[^1] L = -W row_sums = np.array(W.sum(axis=1)).flatten() for i in range(num_vertices): L[i,i] += max(row_sums[i], 0.0001) # 设置固定约束条件 fixed_indices = list(handle_positions.keys()) b = np.zeros((num_vertices * 3)) for idx, pos in handle_positions.items(): b[idx*3:(idx+1)*3] = pos # 解线性方程组 Lv=b 来求解新的顶点位置 A = csr_matrix(L) x = spsolve(A, b).reshape(-1, 3) return x def build_weight_matrix(faces, nverts): """构建权重矩阵""" weights = lil_matrix((nverts, nverts)) for face in faces: v1, v2, v3 = sorted(face) weights[v1,v2] += cotangent_weight(v1, v2, v3) weights[v2,v3] += cotangent_weight(v2, v3, v1) weights[v3,v1] += cotangent_weight(v3, v1, v2) return weights.tocsr() def cotangent_weight(i,j,k): """计算cotangent权值""" eps = 1e-8 a = angle_between_edges(i,j,k) return max(0., min(np.cos(a)/(np.sin(max(a,eps))), 1./eps)) def angle_between_edges(i,j,k): """计算两个边之间的夹角""" p_i = np.array([i]) p_j = np.array([j]) p_k = np.array([k]) vec_ij = normalize(p_j-p_i) vec_ik = normalize(p_k-p_i) dot_product = np.dot(vec_ij.T,vec_ik) theta = np.arccos(dot_product.clip(-1,1))[0][0] return theta def normalize(x): """向量标准化处理""" norm_x = np.sqrt(sum(x*x)) if norm_x != 0: return x / norm_x else: return x ``` 此代码实现了基本的 ARAP 方法来对三角形网格进行变形操作[^1]。