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

1、构建平面函数的MATLAB代码： ```matlab function [x,y,z] = planeFunc(a,b,c,d,x_range,y_range) % a,b,c,d分别是平面方程ax+by+cz+d=0中的系数 % x_range和y_range定义了平面在x和y方向上的取值范围 % 返回三维坐标系中的x,y,z值 [x,y] = meshgrid(x_range,y_range); z = (-a*x - b*y - d)/c; end ``` 2、构建求解五角柱面与平面相交曲线的函数： ```matlab function [x,y,z] = intersectPentagonalPrismWithPlane(a,b,c,d,h,n,x_range,y_range) % a,b,c,d分别是平面方程ax+by+cz+d=0中的系数 % h是五角柱的高 % n是五角柱的边数 % x_range和y_range定义了平面在x和y方向上的取值范围 % 返回三维坐标系中的x,y,z值 theta = pi/n; r = h/(2*sin(theta)); [x_circ,y_circ] = circleFunc(r,n); z_circ = linspace(-h/2,h/2,n+1); z_plane = (-a*x_circ - b*y_circ - d)/c; idx_above_plane = z_circ > max(z_plane); idx_below_plane = z_circ < min(z_plane); for i = 1:n if idx_above_plane(i) == 0 && idx_above_plane(i+1) == 1 [x1,y1,z1] = linePlaneIntersection([x_circ(i),y_circ(i),z_circ(i)],... [x_circ(i+1),y_circ(i+1),z_circ(i+1)],[a,b,c,d]); [x2,y2,z2] = linePlaneIntersection([x_circ(i+1),y_circ(i+1),z_circ(i+1)],... [x_circ(i+2),y_circ(i+2),z_circ(i+2)],[a,b,c,d]); [x,y,z] = cylinderFunc([x1,x2],[y1,y2],z_range); break; elseif idx_below_plane(i) == 0 && idx_below_plane(i+1) == 1 [x1,y1,z1] = linePlaneIntersection([x_circ(i),y_circ(i),z_circ(i)],... [x_circ(i+1),y_circ(i+1),z_circ(i+1)],[a,b,c,d]); [x2,y2,z2] = linePlaneIntersection([x_circ(i+1),y_circ(i+1),z_circ(i+1)],... [x_circ(i+2),y_circ(i+2),z_circ(i+2)],[a,b,c,d]); [x,y,z] = cylinderFunc([x1,x2],[y1,y2],z_range); break; elseif idx_above_plane(i) == 0 && idx_above_plane(i+1) == 0 [x1,y1,z1] = linePlaneIntersection([x_circ(i),y_circ(i),z_circ(i)],... [x_circ(i+1),y_circ(i+1),z_circ(i+1)],[a,b,c,d]); [x2,y2,z2] = linePlaneIntersection([x_circ(i),y_circ(i),z_circ(i)],... [x_circ(i+1),y_circ(i+1),z_circ(i+1)],[a,b,c,d]); [x,y,z] = cylinderFunc([x1,x2],[y1,y2],z_range); break; elseif idx_below_plane(i) == 0 && idx_below_plane(i+1) == 0 [x1,y1,z1] = linePlaneIntersection([x_circ(i),y_circ(i),z_circ(i)],... [x_circ(i+1),y_circ(i+1),z_circ(i+1)],[a,b,c,d]); [x2,y2,z2] = linePlaneIntersection([x_circ(i),y_circ(i),z_circ(i)],... [x_circ(i+1),y_circ(i+1),z_circ(i+1)],[a,b,c,d]); [x,y,z] = cylinderFunc([x1,x2],[y1,y2],z_range); break; end end end function [x,y,z] = linePlaneIntersection(p1,p2,plane) % p1,p2是直线上两个点的位置向量 % plane是一个包含平面方程系数的向量[a,b,c,d]，ax+by+cz+d=0 % 返回交点的坐标向量[x,y,z] p1 = reshape(p1,1,3); p2 = reshape(p2,1,3); plane = reshape(plane,1,4); n = plane(1:3); d = plane(4); t = -(n*p1' + d)/(n*(p2-p1)'); intersect = p1 + t*(p2-p1); x = intersect(1); y = intersect(2); z = intersect(3); end function [x,y] = circleFunc(r,n) % r是圆的半径 % n是圆的边数 % 返回圆上的点的坐标向量[x,y] theta = linspace(0,2*pi,n+1); x = r*cos(theta); y = r*sin(theta); end function [x,y,z] = cylinderFunc(p1,p2,z_range) % p1,p2是圆柱轴线上两点的位置向量 % z_range是圆柱在z方向上的取值范围 % 返回圆柱的三维坐标向量[x,y,z] p1 = reshape(p1,1,3); p2 = reshape(p2,1,3); r = norm(p2-p1)/2; theta = linspace(0,2*pi,50); [x_circ,y_circ] = pol2cart(theta,r); z = linspace(z_range(1),z_range(2),50); [x_circ,y_circ,z_circ] = meshgrid(x_circ,y_circ,z); z_axis = [p1;p2]; x_axis = repmat([1 0 0],size(z_axis,1),1); y_axis = cross(x_axis,z_axis-p1); y_axis = y_axis./vecnorm(y_axis,2,2); x_axis = cross(y_axis,z_axis-p1); x_axis = x_axis./vecnorm(x_axis,2,2); rot_mat = [x_axis;y_axis;z_axis-p1]; rot_mat_inv = inv(rot_mat); xyz_circ = [x_circ(:),y_circ(:),z_circ(:)]; xyz_circ_rot = xyz_circ*rot_mat; xyz_circ_rot(:,3) = xyz_circ_rot(:,3) + norm(p1); xyz_circ_rot(:,1:2) = xyz_circ_rot(:,1:2) + p1(1:2); xyz_circ_rot = xyz_circ_rot*rot_mat_inv; x = reshape(xyz_circ_rot(:,1),size(x_circ)); y = reshape(xyz_circ_rot(:,2),size(y_circ)); z = reshape(xyz_circ_rot(:,3),size(z_circ)); 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+y+z=0 a = 1; b = 1; c = 1; d = 0; [x_range,y_range] = meshgrid(-2:0.1:2,-2:0.1:2); x = x_range(:); y = y_range(:); z = (-a*x - b*y - d)/c; plot3(x,y,z,'.'); hold on; h = 2; n = 5; z_range = [-h/2,h/2]; [x_circ,y_circ,z_circ] = cylinder(h,n); surf(x_circ,y_circ,z_circ,'FaceAlpha',0.5); [x,y,z] = intersectPentagonalPrismWithPlane(a,b,c,d,h,n,x_range,y_range); plot3(x,y,z,'LineWidth',2,'Color','r'); xlim([-2,2]); ylim([-2,2]); zlim([-1,1]); xlabel('x'); ylabel('y'); zlabel('z'); title('Intersection of Plane 1 with Pentagonal Prism'); % 平面2：x-y=0 a = 1; b = -1; c = 0; d = 0; [x_range,y_range] = meshgrid(-2:0.1:2,-2:0.1:2); x = x_range(:); y = y_range(:); z = (-a*x - b*y - d)/c; plot3(x,y,z,'.'); hold on; h = 2; n = 5; z_range = [-h/2,h/2]; [x_circ,y_circ,z_circ] = cylinder(h,n); surf(x_circ,y_circ,z_circ,'FaceAlpha',0.5); [x,y,z] = intersectPentagonalPrismWithPlane(a,b,c,d,h,n,x_range,y_range); plot3(x,y,z,'LineWidth',2,'Color','r'); xlim([-2,2]); ylim([-2,2]); zlim([-1,1]); xlabel('x'); ylabel('y'); zlabel('z'); title('Intersection of Plane 2 with Pentagonal Prism'); % 平面3：z=0 a = 0; b = 0; c = 1; d = 0; [x_range,y_range] = meshgrid(-2:0.1:2,-2:0.1:2); x = x_range(:); y = y_range(:); z = (-a*x - b*y - d)/c; plot3(x,y,z,'.'); hold on; h = 2; n = 5; z_range = [-h/2,h/2]; [x_circ,y_circ,z_circ] = cylinder(h,n); surf(x_circ,y_circ,z_circ,'FaceAlpha',0.5); [x,y,z] = intersectPentagonalPrismWithPlane(a,b,c,d,h,n,x_range,y_range); plot3(x,y,z,'LineWidth',2,'Color','r'); xlim([-2,2]); ylim([-2,2]); zlim([-1,1]); xlabel('x'); ylabel('y'); zlabel('z'); title('Intersection of Plane 3 with Pentagonal Prism'); % 平面4：x+z=0 a = 1; b = 0; c = 1; d = 0; [x_range,y_range] = meshgrid(-2:0.1:2,-2:0.1:2); x = x_range(:); y = y_range(:); z = (-a*x - b*y - d)/c; plot3(x,y,z,'.'); hold on; h = 2; n = 5; z_range = [-h/2,h/2]; [x_circ,y_circ,z_circ] = cylinder(h,n); surf(x_circ,y_circ,z_circ,'FaceAlpha',0.5); [x,y,z] = intersectPentagonalPrismWithPlane(a,b,c,d,h,n,x_range,y_range); plot3(x,y,z,'LineWidth',2,'Color','r'); xlim([-2,2]); ylim([-2,2]); zlim([-1,1]); xlabel('x'); ylabel('y'); zlabel('z'); title('Intersection of Plane 4 with Pentagonal Prism'); % 平面5：y=0 a = 0; b = 1; c = 0; d = 0; [x_range,y_range] = meshgrid(-2:0.1:2,-2:0.1:2); x = x_range(:); y = y_range(:); z = (-a*x - b*y - d)/c; plot3(x,y,z,'.'); hold on; h = 2; n = 5; z_range = [-h/2,h/2]; [x_circ,y_circ,z_circ] = cylinder(h,n); surf(x_circ,y_circ,z_circ,'FaceAlpha',0.5); [x,y,z] = intersectPentagonalPrismWithPlane(a,b,c,d,h,n,x_range,y_range); plot3(x,y,z,'LineWidth',2,'Color','r'); xlim([-2,2]); ylim([-2,2]); zlim([-1,1]); xlabel('x'); ylabel('y'); zlabel('z'); title('Intersection of Plane 5 with Pentagonal Prism'); ```