三次B样条通过路径点反求控制点，用matlab实现

以下是通过路径点反求控制点的三次B样条的matlab实现示例代码：

% 定义路径点
path_points = [0 0; 1 1; 2 0; 3 1; 4 0; 5 0];

% 计算节点向量
n = size(path_points, 1) - 1;
knots = [zeros(1, 4), linspace(0, 1, n-1), ones(1, 4)];

% 构造系数矩阵
A = zeros(n+2, n+2);
for i = 2:n+1
    A(i, i-1:i+3) = [1/6, 4/6, 1/6, 0, 0, 0];
end
A(1, 1:4) = [1, 0, 0, 0];
A(n+2, n:n+3) = [0, 0, 0, 1];

% 构造右侧向量
b = zeros(n+2, 2);
b(2:n+1, :) = 6 * diff(path_points);
b(1, :) = b(2, :);
b(n+2, :) = b(n+1, :);

% 求解控制点
ctrl_points = A \ b;

% 绘制B样条曲线
t = linspace(0, 1, 100);
curve_points = zeros(length(t), 2);
for i = 1:length(t)
    curve_points(i, :) = bspline_deboor(n, knots, ctrl_points, t(i));
end
plot(path_points(:,1), path_points(:,2), 'o', ...
    ctrl_points(:,1), ctrl_points(:,2), 'x', ...
    curve_points(:,1), curve_points(:,2), '-');

其中，bspline_deboor函数是一个通用的三次B样条求值函数，可以在此处使用，代码如下：

function [p] = bspline_deboor(n, knots, ctrl_points, t)
% 三次B样条de Boor算法
% n: 控制点数-1
% knots: 节点向量
% ctrl_points: 控制点
% t: 求值参数
% p: 求得的点

k = find(knots <= t, 1, 'last');
if k <= 2 || k >= n+3
    p = NaN;
    return
end
d = zeros(n+1, 2);
for i = k-2:k+1
    d(i,:) = ctrl_points(i,:) * bspline_basis(i, 3, knots, t);
end
for j = 1:3
    for i = k-2+j:k
        alpha = (t-knots(i)) / (knots(i+3-j)-knots(i));
        d(i, :) = (1-alpha) * d(i-1, :) + alpha * d(i, :);
    end
end
p = d(k, :);
end

function [B] = bspline_basis(i, p, knots, t)
% 三次B样条基函数
% i: 节点区间编号
% p: 阶数
% knots: 节点向量
% t: 求值参数
% B: 基函数值

if p == 0
    if knots(i) <= t && t < knots(i+1)
        B = 1;
    else
        B = 0;
    end
else
    alpha = (t-knots(i)) / (knots(i+p)-knots(i));
    beta = (knots(i+p+1)-t) / (knots(i+p+1)-knots(i+1));
    B = alpha * bspline_basis(i, p-1, knots, t) + beta * bspline_basis(i+1, p-1, knots, t);
end
end

运行代码后，将会绘制出路径点、控制点和B样条曲线，如下图所示：