以下是使用法向量加权最小二乘法将离散点及其法向量拟合成为多项式自由曲面的MATLAB代码：
- % 数据点和法向量
- P = [x1 y1 z1; x2 y2 z2; ...];
- N = [nx1 ny1 nz1; nx2 ny2 nz2; ...];
- % 定义多项式自由曲面的阶数
- m = 3;
- n = 3;
- % 计算每个数据点对应的权重
- W = zeros(size(P,1),1);
- for i=1:size(P,1)
- W(i) = exp(-norm(N(i,:))^2);
- end
- % 求解最小二乘问题，得到系数矩阵C、D、E和F
- A = zeros((m+1)*(n+1), (m+1)*(n+1));
- b = zeros((m+1)*(n+1), 1);
- for i=1:size(P,1)
- p = P(i,:);
- w = W(i);
- n = N(i,:);
- [B, dBdU, dBdV] = Bernstein(m,n,p(1),p(2));
- N = [n(1)*eye(3) n(2)*eye(3) n(3)*eye(3)];
- F = kron(N,B)*w;
- A = A + kron(dBdU,dBdU') + kron(dBdV,dBdV')*w;
- b = b + F(:);
- end
- X = A\b;
- C = reshape(X(1:(m+1)*(n+1)), m+1, n+1);
- D = reshape(X((m+1)*(n+1)+1:2*(m+1)*(n+1)), m+1, n+1);
- E = reshape(X(2*(m+1)*(n+1)+1:3*(m+1)*(n+1)), m+1, n+1);
- F = reshape(X(3*(m+1)*(n+1)+1:end), m+1, n+1);
- % 生成网格点坐标
- u = linspace(min(P(:,1)),max(P(:,1)),50);
- v = linspace(min(P(:,2)),max(P(:,2)),50);
- [U,V] = meshgrid(u,v);
- XYZ = zeros(numel(u), numel(v), 3);
- B = Bernstein(m,n,U,V);
- % 计算自由曲面上每个网格点的坐标
- for i=1:numel(u)
- for j=1:numel(v)
- XYZ(i,j,:) = [B(:,:,i,j)*C(:) B(:,:,i,j)*D(:) B(:,:,i,j)*E(:)] + F;
- end
- end
- % 绘制自由曲面
- surf(XYZ(:,:,1), XYZ(:,:,2), XYZ(:,:,3))
其中，Bernstein函数是定义在(u,v)处的第(i,j)个Bezier基函数，返回一个3乘以1的列向量。dBdU和dBdV是Bernstein基函数对u和v的一阶导数，返回一个1乘以3的行向量。
--- 光学专家Gpt |
