平面与五棱柱相交曲线求解与三维图绘制: 1、构建根据输入参数改变位置和角度的平面函数的MATLAB代码 2、构建求解判断五棱柱面n边与平面相交曲线的函数 3、绘制其中5种平面与五棱柱面相交的曲线
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Matlab里的直线、曲线、平面、曲面相交问题.docx
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1、构建平面函数的MATLAB代码:
```matlab
function [x, y, z] = plane_func(a, b, c, d, x_range, y_range)
% a, b, c, d: 平面方程系数
% x_range, y_range: 平面的x、y取值范围
% x, y, z: 平面上的点坐标
% 构建网格矩阵
[X, Y] = meshgrid(x_range, y_range);
% 根据平面方程计算z
z = (-a*X - b*Y - d) / c;
% 返回平面上的点坐标
x = X;
y = Y;
end
```
2、构建求解判断五棱柱面n边与平面相交曲线的函数:
```matlab
function [x, y, z] = intersect_func(vertices, plane_coeffs, x_range, y_range)
% vertices: 五棱柱顶点坐标,共 10 个点,第 1 个点为顶点,其余 9 个点按顺序围成五边形底面
% plane_coeffs: 平面方程系数
% x_range, y_range: 五棱柱的 x、y 取值范围
% x, y, z: 五棱柱面 n 边与平面相交的曲线
% 计算五棱柱面 n 的方程系数
n = cross(vertices(2,:) - vertices(1,:), vertices(3,:) - vertices(1,:));
n = n / norm(n);
d = -dot(n, vertices(1,:));
plane_coeffs_n = [n, d];
% 计算五棱柱面 n 与平面的交点
[xn, yn] = meshgrid(x_range, y_range);
zn = (-plane_coeffs_n(1)*xn - plane_coeffs_n(2)*yn - plane_coeffs_n(4)) / plane_coeffs_n(3);
% 计算五棱柱底面的坐标
x_bottom = vertices(2:6, 1)';
y_bottom = vertices(2:6, 2)';
z_bottom = vertices(2:6, 3)';
% 计算五棱柱底面上的所有线段及其法向量
for i = 1:5
v1 = vertices(1,:);
v2 = vertices(i+1,:);
l = v2 - v1;
l_norm = l / norm(l);
n = cross(l_norm, n);
n_norm = n / norm(n);
d = -dot(n_norm, v1);
plane_coeffs_l(i,:) = [n_norm, d];
% 计算线段与平面的交点
t = -(plane_coeffs(1)*v1(1) + plane_coeffs(2)*v1(2) + plane_coeffs(3)*v1(3) + plane_coeffs(4)) / ...
(plane_coeffs(1)*l_norm(1) + plane_coeffs(2)*l_norm(2) + plane_coeffs(3)*l_norm(3));
x_l(i,:) = v1(1) + t*l_norm(1);
y_l(i,:) = v1(2) + t*l_norm(2);
z_l(i,:) = v1(3) + t*l_norm(3);
end
% 计算五棱柱面 n 与底面上的所有线段的交点
for i = 1:5
% 计算交点
[xx, yy] = meshgrid([xn(1,1), xn(end,end)], [yn(1,1), yn(end,end)]);
zz = (-plane_coeffs_l(i,1)*xx - plane_coeffs_l(i,2)*yy - plane_coeffs_l(i,4)) / plane_coeffs_l(i,3);
[x_int, y_int, z_int] = intersections(xn, yn, zn, xx, yy, zz);
% 过滤掉不在线段上的交点
x_l_i = x_l(i,:);
y_l_i = y_l(i,:);
z_l_i = z_l(i,:);
for j = 1:length(x_int)
if ~inpolygon(x_int(j), y_int(j), x_l_i, y_l_i)
x_int(j) = NaN;
y_int(j) = NaN;
z_int(j) = NaN;
end
end
% 拼接所有的交点
if i == 1
x = x_int;
y = y_int;
z = z_int;
else
x = [x; x_int];
y = [y; y_int];
z = [z; z_int];
end
end
end
```
3、绘制其中5种平面与五棱柱面相交的曲线:
```matlab
% 五棱柱的顶点坐标
vertices = [0, 0, 1;
1, 0, 0;
0.5, 0.87, 0;
-0.5, 0.87, 0;
-1, 0, 0;
-0.5, -0.87, 0;
0.5, -0.87, 0;
0, 0, -1;
1, 0, 0;
0.5, 0.87, 0];
% 平面方程系数
plane_coeffs_list = [
1, 0, 0, 0.5;
0, 1, 0, -0.5;
0, 0, 1, 0;
1, 1, 0, 0;
1, 1, 1, 0;
];
x_range = linspace(-2, 2, 100);
y_range = linspace(-2, 2, 100);
% 绘制图形
figure;
hold on;
for i = 1:size(plane_coeffs_list, 1)
% 构建平面
[X, Y, Z] = plane_func(plane_coeffs_list(i,1), plane_coeffs_list(i,2), plane_coeffs_list(i,3), plane_coeffs_list(i,4), x_range, y_range);
% 计算五棱柱面与平面的交点
[x, y, z] = intersect_func(vertices, plane_coeffs_list(i,:), x_range, y_range);
% 绘制平面和交点
surf(X, Y, Z, 'FaceAlpha', 0.5);
plot3(x, y, z, 'r.', 'MarkerSize', 10);
end
axis equal;
xlabel('X');
ylabel('Y');
zlabel('Z');
```
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