三维凹多面体分解为凸多面体的MATLAB实现
要将一个面数小于10的凹多面体分解为多个凸多面体，我们可以使用凸分解算法。以下是MATLAB实现方案：
方法概述
- 表示凹多面体：使用顶点和面列表表示输入的多面体
- 检测凹性：通过计算面法线和顶点位置关系确定凹边
- 分解策略：使用增量式凸分解或基于凹边的分割方法
- 实现分解：通过添加切割平面将凹多面体分割为凸部分
MATLAB代码实现
function convexParts = decomposeConcavePolyhedron(vertices, faces)% 检查输入多面体是否是凹的if isConvex(vertices, faces)convexParts = {vertices};return;end% 初始化凸部分集合convexParts = {};% 获取凹边concaveEdges = findConcaveEdges(vertices, faces);% 如果没有找到凹边，说明已经是凸的if isempty(concaveEdges)convexParts = {vertices};return;end% 选择第一个凹边进行分割edge = concaveEdges(1,:);% 找到分割平面[cutPlane, connectedFaces] = findCuttingPlane(vertices, faces, edge);% 使用分割平面将多面体分成两部分[vertices1, faces1, vertices2, faces2] = splitPolyhedron(vertices, faces, cutPlane);% 递归分解两个部分convexParts1 = decomposeConcavePolyhedron(vertices1, faces1);convexParts2 = decomposeConcavePolyhedron(vertices2, faces2);% 合并结果convexParts = [convexParts1, convexParts2];
endfunction isConvex = isConvex(vertices, faces)% 计算每个面的法向量和中心点normals = zeros(length(faces), 3);centers = zeros(length(faces), 3);for i = 1:length(faces)faceVertices = vertices(faces{i}, :);v1 = faceVertices(2,:) - faceVertices(1,:);v2 = faceVertices(3,:) - faceVertices(1,:);normals(i,:) = cross(v1, v2);normals(i,:) = normals(i,:) / norm(normals(i,:));centers(i,:) = mean(faceVertices, 1);end% 检查所有顶点是否在所有面的同一侧isConvex = true;for i = 1:length(faces)for j = 1:size(vertices, 1)if all(j ~= faces{i}) % 不检查当前面的顶点vec = vertices(j,:) - centers(i,:);if dot(vec, normals(i,:)) < -1e-6isConvex = false;return;endendendend
endfunction concaveEdges = findConcaveEdges(vertices, faces)% 找到所有边edges = [];for i = 1:length(faces)face = faces{i};for j = 1:length(face)v1 = face(j);v2 = face(mod(j, length(face)) + 1);edges = [edges; sort([v1, v2])];endendedges = unique(edges, 'rows');% 检查每条边是否是凹的concaveEdges = [];for i = 1:size(edges, 1)edge = edges(i,:);adjacentFaces = findAdjacentFaces(faces, edge);if length(adjacentFaces) == 2% 计算两个相邻面的法向量face1 = faces{adjacentFaces(1)};face2 = faces{adjacentFaces(2)};v1_1 = vertices(face1(2),:) - vertices(face1(1),:);v1_2 = vertices(face1(3),:) - vertices(face1(1),:);normal1 = cross(v1_1, v1_2);normal1 = normal1 / norm(normal1);v2_1 = vertices(face2(2),:) - vertices(face2(1),:);v2_2 = vertices(face2(3),:) - vertices(face2(1),:);normal2 = cross(v2_1, v2_2);normal2 = normal2 / norm(normal2);% 计算二面角dihedralAngle = atan2(norm(cross(normal1, normal2)), dot(normal1, normal2));% 如果二面角大于180度（π弧度），则是凹边if dihedralAngle > pi - 1e-6concaveEdges = [concaveEdges; edge];endendend
endfunction adjacentFaces = findAdjacentFaces(faces, edge)% 找到共享给定边的面adjacentFaces = [];for i = 1:length(faces)face = faces{i};for j = 1:length(face)v1 = face(j);v2 = face(mod(j, length(face)) + 1);if (v1 == edge(1) && v2 == edge(2)) || (v1 == edge(2) && v2 == edge(1))adjacentFaces = [adjacentFaces, i];break;endendend
endfunction [cutPlane, connectedFaces] = findCuttingPlane(vertices, faces, edge)% 简化方法：使用垂直于凹边并通过中点的平面% 实际应用中可能需要更复杂的策略v1 = vertices(edge(1),:);v2 = vertices(edge(2),:);midpoint = (v1 + v2) / 2;edgeVector = v2 - v1;% 找到与凹边相邻的面adjacentFaces = findAdjacentFaces(faces, edge);% 计算两个相邻面的法向量face1 = faces{adjacentFaces(1)};face2 = faces{adjacentFaces(2)};% 计算面法线v1_1 = vertices(face1(2),:) - vertices(face1(1),:);v1_2 = vertices(face1(3),:) - vertices(face1(1),:);normal1 = cross(v1_1, v1_2);normal1 = normal1 / norm(normal1);v2_1 = vertices(face2(2),:) - vertices(face2(1),:);v2_2 = vertices(face2(3),:) - vertices(face2(1),:);normal2 = cross(v2_1, v2_2);normal2 = normal2 / norm(normal2);% 切割平面法线是两个面法线的平均cutNormal = normal1 + normal2;cutNormal = cutNormal / norm(cutNormal);% 确保切割平面垂直于凹边cutNormal = cross(cutNormal, edgeVector);cutNormal = cutNormal / norm(cutNormal);% 创建平面方程: ax + by + cz + d = 0d = -dot(cutNormal, midpoint);cutPlane = [cutNormal, d];connectedFaces = adjacentFaces;
endfunction [vertices1, faces1, vertices2, faces2] = splitPolyhedron(vertices, faces, plane)% 简化实现：实际需要更复杂的几何操作% 这里只是一个框架，实际实现需要处理：% 1. 找到平面与多面体的交线% 2. 将多面体分成两部分% 3. 为每部分创建新的顶点和面列表% 由于实现复杂，这里仅返回原始多面体作为示例% 实际应用中应考虑使用现有的计算几何库或更完整的实现vertices1 = vertices;faces1 = faces;vertices2 = vertices;faces2 = faces;warning('Simplified implementation - actual splitting not performed');
end
使用示例
% 定义一个简单的凹多面体（例如一个凹陷的立方体）
vertices = [0 0 0;2 0 0;2 2 0;0 2 0;0 0 2;2 0 2;1 1 1; % 凹陷顶点0 2 2
];faces = {[1 2 3 4], % 底面[1 2 6 5], % 前面[2 3 7 6], % 右侧面（凹面）[3 4 8 7], % 后面[4 1 5 8], % 左侧面[5 6 7 8] % 顶面
};% 执行分解
convexParts = decomposeConcavePolyhedron(vertices, faces);% 可视化结果
figure;
hold on;
axis equal;
grid on;
view(3);colors = lines(length(convexParts));
for i = 1:length(convexParts)verts = convexParts{i};% 这里需要根据分解后的顶点重建面，简化示例中省略% 实际应用中需要正确处理面信息if size(verts, 1) > 0plot3(verts(:,1), verts(:,2), verts(:,3), 'o', 'Color', colors(i,:));k = convhull(verts(:,1), verts(:,2), verts(:,3));trisurf(k, verts(:,1), verts(:,2), verts(:,3), 'FaceColor', colors(i,:), 'FaceAlpha', 0.5);end
end
title('凹多面体分解为凸多面体');
注意事项
- 上述代码是一个框架实现，实际的分割操作(
splitPolyhedron函数)需要更复杂的几何计算。 - 对于生产环境，建议考虑使用专业的计算几何库，如CGAL的MATLAB接口。
- 分解结果的质量取决于分割平面的选择策略。
- 对于面数小于10的多面体，这种递归方法通常是有效的。
更可靠的替代方案
对于实际应用，可以考虑以下替代方案：
- 使用MATLAB的
boundary函数或convhull函数进行凸包计算 - 将数据导出到专业3D建模软件进行分解
- 使用CGAL库的凸分解算法并通过MEX接口集成到MATLAB中
这个实现提供了基本框架，但可能需要根据具体应用场景进行调整和完善。
资料
CvxNet: Learnable Convex Decomposition (CVPR 2020 论文)
matlab几何基础
