平面与五角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、构建求解判断五角柱面n边与平面相交曲线的函数 3、绘制其中5种平面与五角柱面相交的曲线
时间: 2023-09-18 08:05:34 浏览: 104
### 回答1:
1、构建平面函数的MATLAB代码:
```matlab
function [x,y,z] = planeFunc(a,b,c,d,xRange,yRange)
% a,b,c,d为平面方程参数,xRange和yRange为指定的平面范围
[x,y] = meshgrid(xRange, yRange);
z = (-a*x - b*y - d)/c;
end
```
2、构建求解判断五角柱面n边与平面相交曲线的函数:
```matlab
function [x,y,z] = pentagonCylinderIntersection(n,r,h,a,b,c,d,xRange,yRange)
% n为五角柱面n边,r为五角柱半径,h为五角柱高度,a,b,c,d为平面方程参数,xRange和yRange为指定的平面范围
theta = 0:2*pi/n:2*pi*(1-1/n);
x = r*cos(theta);
y = r*sin(theta);
z = linspace(0,h,n+1);
[X,Y,Z] = cylinder([r r],n);
Z = h*Z;
for i = 1:n
patch(x(i)+X,Y(i)+Z(i),Z(i),'b');
end
[xp,yp,zp] = planeFunc(a,b,c,d,xRange,yRange);
C = (a*x + b*y + c*z + d);
idx = find(abs(C) < 1e-10);
x = xp(idx);
y = yp(idx);
z = zp(idx);
end
```
3、绘制其中5种平面与五角柱面相交的曲线:
```matlab
% 平面1:x=0
a = 1;
b = 0;
c = 0;
d = 0;
xRange = -1:0.1:1;
yRange = -1:0.1:1;
[x,y,z] = pentagonCylinderIntersection(5,1,2,a,b,c,d,xRange,yRange);
figure();
plot3(x,y,z,'r');
axis equal;
title('平面1与五角柱相交');
% 平面2:y=0
a = 0;
b = 1;
c = 0;
d = 0;
xRange = -1:0.1:1;
yRange = -1:0.1:1;
[x,y,z] = pentagonCylinderIntersection(5,1,2,a,b,c,d,xRange,yRange);
figure();
plot3(x,y,z,'g');
axis equal;
title('平面2与五角柱相交');
% 平面3:z=0
a = 0;
b = 0;
c = 1;
d = 0;
xRange = -1:0.1:1;
yRange = -1:0.1:1;
[x,y,z] = pentagonCylinderIntersection(5,1,2,a,b,c,d,xRange,yRange);
figure();
plot3(x,y,z,'b');
axis equal;
title('平面3与五角柱相交');
% 平面4:x+y+z=0
a = 1;
b = 1;
c = 1;
d = 0;
xRange = -1:0.1:1;
yRange = -1:0.1:1;
[x,y,z] = pentagonCylinderIntersection(5,1,2,a,b,c,d,xRange,yRange);
figure();
plot3(x,y,z,'m');
axis equal;
title('平面4与五角柱相交');
% 平面5:x-y=0
a = 1;
b = -1;
c = 0;
d = 0;
xRange = -1:0.1:1;
yRange = -1:0.1:1;
[x,y,z] = pentagonCylinderIntersection(5,1,2,a,b,c,d,xRange,yRange);
figure();
plot3(x,y,z,'c');
axis equal;
title('平面5与五角柱相交');
```
### 回答2:
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:
function plane = createPlane(a, b, c, d, alpha, beta, gamma)
% a, b, c, d 分别为平面方程 Ax + By + Cz + D = 0 中的系数
% alpha, beta, gamma 分别为旋转角度,单位为弧度
% 构建旋转矩阵
R_x = [1 0 0; 0 cos(alpha) -sin(alpha); 0 sin(alpha) cos(alpha)];
R_y = [cos(beta) 0 sin(beta); 0 1 0; -sin(beta) 0 cos(beta)];
R_z = [cos(gamma) -sin(gamma) 0; sin(gamma) cos(gamma) 0; 0 0 1];
R = R_x * R_y * R_z;
% 构建平面方程的参数向量
normal_vector = R * [a; b; c];
plane = [normal_vector; d];
end
2、构建求解判断五角柱面n边与平面相交曲线的函数:
function intersection = planePentagonalCylinderIntersection(a, b, c, d, pentagon_center, pentagon_radius, pentagon_height)
% a, b, c, d 分别为平面方程 Ax + By + Cz + D = 0 中的系数
% pentagon_center 为五角柱的底面中心坐标,pentagon_radius为五角柱的底面半径,pentagon_height 为五角柱的高度
% 计算平面与五角柱的焦点
distance = abs(dot([a; b; c], pentagon_center) + d) / norm([a; b; c]);
intersection = pentagon_center - distance * [a; b; c];
% 判断焦点是否在五角柱体内部
if norm(intersection - pentagon_center) <= pentagon_radius && intersection(3) >= pentagon_center(3) && intersection(3) <= pentagon_center(3) + pentagon_height
disp('相交');
else
disp('不相交');
end
end
3、绘制其中5种平面与五角柱面相交的曲线。
clear;
clc;
% 五角柱的底面中心坐标
pentagon_center = [0; 0; 0];
% 五角柱的底面半径
pentagon_radius = 1;
% 五角柱的高度
pentagon_height = 2;
% 平面1
plane1 = createPlane(1, 0, 0, 0, 0, 0, 0);
planePentagonalCylinderIntersection(plane1(1), plane1(2), plane1(3), plane1(4), pentagon_center, pentagon_radius, pentagon_height);
% 平面2
plane2 = createPlane(0, 1, 0, 0, 0, pi/4, 0);
planePentagonalCylinderIntersection(plane2(1), plane2(2), plane2(3), plane2(4), pentagon_center, pentagon_radius, pentagon_height);
% 平面3
plane3 = createPlane(0, 0, 1, 0, pi/4, pi/3, 0);
planePentagonalCylinderIntersection(plane3(1), plane3(2), plane3(3), plane3(4), pentagon_center, pentagon_radius, pentagon_height);
% 平面4
plane4 = createPlane(1, 1, 1, 0, pi/2, pi/2, pi/2);
planePentagonalCylinderIntersection(plane4(1), plane4(2), plane4(3), plane4(4), pentagon_center, pentagon_radius, pentagon_height);
% 平面5
plane5 = createPlane(1, 2, 3, 4, pi/6, pi/6, pi/6);
planePentagonalCylinderIntersection(plane5(1), plane5(2), plane5(3), plane5(4), pentagon_center, pentagon_radius, pentagon_height);
### 回答3:
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:
```matlab
function plane_func(parameters)
% 解析输入参数
px = parameters(1); % 平面的x坐标
py = parameters(2); % 平面的y坐标
pz = parameters(3); % 平面的z坐标
alpha = parameters(4); % 平面的绕x轴旋转角度
beta = parameters(5); % 平面的绕y轴旋转角度
gamma = parameters(6); % 平面的绕z轴旋转角度
% 构造旋转矩阵
Rx = [1, 0, 0;
0, cosd(alpha), -sind(alpha);
0, sind(alpha), cosd(alpha)];
Ry = [cosd(beta), 0, sind(beta);
0, 1, 0;
-sind(beta), 0, cosd(beta)];
Rz = [cosd(gamma), -sind(gamma), 0;
sind(gamma), cosd(gamma), 0;
0, 0, 1];
% 构造平面方程
syms x y z
p = [x; y; z]; % 平面上的点P
n = [0; 0; 1]; % 平面的法向量n
q = [px; py; pz]; % 平面上的一点Q
p_transformed = (Rx * Ry * Rz * p) + q; % 平面旋转、平移后的点P'
% 输出平面方程
disp("平面方程:");
disp(n' * (p - p_transformed)); % n·(P - P')
end
```
2、构建求解判断五角柱面n边与平面相交曲线的函数:
```matlab
function intersection_curve = pentagonal_prism_plane_intersection(pentagon_points, parameters)
% 解析输入参数
px = parameters(1); % 平面的x坐标
py = parameters(2); % 平面的y坐标
pz = parameters(3); % 平面的z坐标
alpha = parameters(4); % 平面的绕x轴旋转角度
beta = parameters(5); % 平面的绕y轴旋转角度
gamma = parameters(6); % 平面的绕z轴旋转角度
% 构造旋转矩阵
Rx = [1, 0, 0;
0, cosd(alpha), -sind(alpha);
0, sind(alpha), cosd(alpha)];
Ry = [cosd(beta), 0, sind(beta);
0, 1, 0;
-sind(beta), 0, cosd(beta)];
Rz = [cosd(gamma), -sind(gamma), 0;
sind(gamma), cosd(gamma), 0;
0, 0, 1];
% 平面相关的参数
syms t real
n = [0; 0; 1]; % 平面的法向量n
p = [px; py; pz]; % 平面上的一点Q
p_transformed = (Rx * Ry * Rz * p); % 平面旋转、平移后的点P'
% 五角柱相关的参数
pentagon_center = mean(pentagon_points); % 五角柱底面中心点
pentagon_normal = cross(pentagon_points(3,:) - pentagon_points(1,:), pentagon_points(4,:) - pentagon_points(1,:)); % 五角柱侧面的法向量
pentagon_height = norm(pentagon_points(2,:) - pentagon_points(5,:)); % 五角柱的高度
% 计算平面与五角柱的交点
intersection_points = [];
for i = 1:5
p0 = pentagon_points(i,:); % 五角柱底面的顶点
p1 = pentagon_points(mod(i, 5) + 1,:); % 顶点p0的下一个顶点
p0_transformed = (Rx * Ry * Rz * p0')'; % 五角柱旋转、平移后的顶点p0'
p1_transformed = (Rx * Ry * Rz * p1')'; % 五角柱旋转、平移后的顶点p1'
% 计算平面与五角柱底面边的交点
t_value = solve(n' * (p - (p0_transformed + t * (p1_transformed - p0_transformed))), t);
if isreal(t_value)
intersection_point = subs(p + t * n, t, t_value);
intersection_points = [intersection_points; intersection_point];
end
% 计算平面与五角柱侧面边的交点
t_value = solve(n' * (p - (p0_transformed + t * pentagon_normal)), t);
if isreal(t_value) && t_value >= 0 && t_value <= pentagon_height
intersection_point = subs(p + t * n, t, t_value);
intersection_points = [intersection_points; intersection_point];
end
end
% 排除重复的交点
intersection_points = unique(intersection_points, 'rows');
% 输出交点坐标
disp("相交点的坐标:");
disp(intersection_points);
% 绘制相交曲线
figure;
scatter3(intersection_points(:,1), intersection_points(:,2), intersection_points(:,3), 'filled');
xlabel('x');
ylabel('y');
zlabel('z');
title('五角柱面与平面相交的曲线');
grid on;
intersection_curve = intersection_points;
end
```
3、绘制其中5种平面与五角柱面相交的曲线:
```matlab
% 定义五角柱底面的五个顶点
pentagon_points = [1, 1, 0;
2, 2, 0;
3, 1, 0;
2.5, 0, 0;
1.5, 0, 0];
% 绘制五角柱
figure;
fill3(pentagon_points(:,1), pentagon_points(:,2), pentagon_points(:,3), 'b');
hold on;
quiver3(pentagon_points(1,1), pentagon_points(1,2), pentagon_points(1,3), pentagon_points(2,1)-pentagon_points(1,1), pentagon_points(2,2)-pentagon_points(1,2), pentagon_points(2,3)-pentagon_points(1,3), 'r');
quiver3(pentagon_points(2,1), pentagon_points(2,2), pentagon_points(2,3), pentagon_points(3,1)-pentagon_points(2,1), pentagon_points(3,2)-pentagon_points(2,2), pentagon_points(3,3)-pentagon_points(2,3), 'r');
quiver3(pentagon_points(3,1), pentagon_points(3,2), pentagon_points(3,3), pentagon_points(4,1)-pentagon_points(3,1), pentagon_points(4,2)-pentagon_points(3,2), pentagon_points(4,3)-pentagon_points(3,3), 'r');
quiver3(pentagon_points(4,1), pentagon_points(4,2), pentagon_points(4,3), pentagon_points(5,1)-pentagon_points(4,1), pentagon_points(5,2)-pentagon_points(4,2), pentagon_points(5,3)-pentagon_points(4,3), 'r');
quiver3(pentagon_points(5,1), pentagon_points(5,2), pentagon_points(5,3), pentagon_points(1,1)-pentagon_points(5,1), pentagon_points(1,2)-pentagon_points(5,2), pentagon_points(1,3)-pentagon_points(5,3), 'r');
xlabel('x');
ylabel('y');
zlabel('z');
title('五角柱面');
grid on;
% 构建输入参数
parameters = [1.5, 1, 1, 45, 30, 60];
% 构建五角柱面与平面相交的函数
pentagonal_prism_plane_intersection(pentagon_points, parameters);
```
修改parameters和pentagon_points的值,可以绘制不同位置和角度的平面与五角柱面相交的曲线。
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