今年考研？七夕顺便把心形线复习一下

1. 心形线公式

心脏线，也称心形线，是外摆线的一种，亦为蚶线的一种，是一个圆上的固定一点在它绕着与其相切且半径相同的另外一个圆周滚动时所形成的轨迹，因其形状像心形而得名。

极坐标下公式有：

$r=a(1-sin\theta )$ 或 $r=a(1+sin\theta )$

$r=a(1-cos\theta )$ 或 $r=a(1+cos\theta )$

Matlab绘制：

clc,clear,close all;
theta=-pi:0.001:pi;
subplot(221)
r=1-sin(theta);  %心形线方程，“桃p股”朝下
polarplot(theta,r,'r','LineWidth',1.8);
title('r=1-sin\theta')
subplot(222)
r=1+sin(theta);  %心形线方程，"桃p股”朝上
polarplot(theta,r,'r','LineWidth',1.8);
title('r=1+sin\theta')
subplot(223)
r=1-cos(theta);  %心形线方程，"桃p股"朝左
polarplot(theta,r,'b','LineWidth',1.8);
title('r=1-cos\theta')
subplot(224)
r=1+cos(theta);  %心形线方程，"桃p股"朝右
polarplot(theta,r,'b','LineWidth',1.8);
title('r=1+cos\theta')
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2. 心形线例题

1. 求由心形线$r=4(1+cos\theta )$，直线$\theta =0$ 和 $\theta =\frac{\pi }{2}$所围图形绕极轴旋转一周所得旋转体的体积.

$r=4(1+cos\theta )$的图像：

$r=4(1+cos\theta )$ 在 0 - 90°的图像：

与极坐标轴所围面积绕极轴旋转一周得到的旋转体体积详细计算步骤：

2. 改编成选择题，应该选什么呢？
   
   请将答案打在评论区~~

3. 一些花式心形线

除了做题，浪漫的Coder们还通过计算机编程作图绘制出了诸多花式心形线

Python画图：

import matplotlib.pyplot as plt   
import numpy as np                
import math

x = np.linspace(-2,2,500)
y=lambda x:np.power((x**2),(1/3))+0.99*np.sqrt(3.3-np.power(x,2))*np.sin(9.9*math.pi*x)
plt.plot(x,y(x))
plt.axis([-3,3,-2,3])
plt.show()
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from matplotlib import pyplot as plt
import numpy as np
import math
i = np.linspace(0,2*math.pi,500)
x=np.cos(i)
y=np.sin(i)+np.power(np.power(np.cos(i),2),1/3)
plt.xlim([-1,1])
plt.plot(x,y)
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一位大佬写的Matlab绘制爱心跃动效果：

clear;  clc;  close all; 
 
f = @(x, y, z)(x.^2 + 2.25*y.^2 + z.^2 - 1).^3 -  ...
    x.^2.* z.^3 - 0.1125*y.^2.*z.^3;
 
g = @(x, y, z)(sqrt(x.^2+y.^2)-2.5).^2 + z.^2 - 0.4^2;
 
t = linspace(-5, 5);
[x1, y1, z1] = meshgrid(t);
[x2, y2, z2] = meshgrid(t);
val1 = f(x1, y1, z1);
val2 = g(x2, y2, z2);
[p1, v1] = isosurface(x1, y1, z1, val1, 0);
[p2, v2] = isosurface(x2, y2, z2, val2, 0);
figure()
subplot(1, 1, 1)
h = patch('faces',p1,'vertices',v1,'facevertexcdata',jet(size(v1,1)),...
    'facecolor','w','edgecolor','flat'); hold on;
 
patch('faces',p2,'vertices',v2,'facevertexcdata',jet(size(v2,1)),...
    'facecolor','w','edgecolor','flat');
 
grid on; axis equal; axis([-3,3,-3,3,-1.5,1.5]); view(3)
 
 
warning('off');
 
T = title("$I\ Love\ U\ !$"); 
set(T,'Interpreter','latex','FontSize',24)
pic_num = 1;
for i = 1:20
    v1 = 0.98 * v1;
    set(h, 'vertices', v1); drawnow;
    F = getframe(gcf);
    I = frame2im(F);
    [I,map]=rgb2ind(I,256);
    if pic_num == 1 
        imwrite(I,map,'BeatingHeart.gif','gif','Loopcount',inf,'DelayTime',0.05);
    else
        imwrite(I,map,'BeatingHeart.gif','gif','WriteMode','append','DelayTime',0.05);
    end
    pic_num = pic_num + 1;
end
 
for i = 1:20
    v1 = v1 / 0.98;
    set(h, 'vertices', v1); drawnow;
    F = getframe(gcf);
    I = frame2im(F);
    [I,map] = rgb2ind(I,256);
    imwrite(I,map,'BeatingHeart.gif','gif','WriteMode','append','DelayTime',0.05);
    pic_num = pic_num + 1;
end
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参考：

数学的有趣图形-心形线——起源矢量 https://zhuanlan.zhihu.com/p/271674164
好玩的Matlab 3D心形代码——赖床份子 https://blog.csdn.net/weixin_48117689/article/details/124082260