数学图形之心形

明天是七夕,中国的情人节,为了应节,这一篇提供几个心形曲面的算法.

说到心形,我想到一个笑话,有个女老师在黑板上画了个心形图案,问学生加:"这是什么?"学生们回答:"屁股."老师气哭了,去找校长理论.校长来了就批评学生:"你们怎么又的把老师给气哭了呢?"然后看到了黑板上的图案继续说道:"你们也太调皮了,怎么还在黑板上画了个屁股呢!"
好笑吧,其实我也有类似的经历,有人曾对我说:"你怎么拿个屁股做博客园的头像呢?"我只好无奈的回答:"这不是屁股,这是心形,而且这个心形是我用程序画出来的."关于我的心形头像生成的算法,请看我之前发的博客:屏保:画线圈LineFlower, 3D屏保: 线圈, WhyDemo: 画线圈LineFlower.

我前几天还发过一篇博客:数学图形(1.4)心形线

关于心形曲面,我查找一些资料,但都是隐式方程的曲面,即给出一个数学方程式,然后说这个方程所表示的图形即是一个心形曲面.对于隐式方程,我以后有机会再详细介绍.我从网上下了一个隐式方程解析的代码.不太好用,运算时间很长,图形的顶点分布有些错乱.如下图所示:

我的软件工具目前只能处理显示方程的曲面.下面将发布我所写的几个心形曲面的算法脚本代码:

(1)heart(圆变)
这是我在球曲面的基础上变换出来的.

vertices = dimension1:80 dimension2:160
a = from 0 to (2*PI) dimension1
b = from (-PI*0.5) to (PI*0.5) dimension2
r = 10.0
c = sqrt(abs(a - PI))*1.5
x = r*cos(b)*sin(a)*c
y = -r*cos(b)*cos(a)*c
z = r*sin(b)*0.75

u = a
v = b*2
y = y - array_min(y)*0.5

(2)尖尖的心形

曲线:

vertices = 1000

r = 10.0
t = from 0 to (2*PI)
p = sin(t)*sqrt(abs(cos(t)))/(sin(t) + 7/5) - 2*sin(t) + 2
x = p*sin(t)
y = p*cos(t)

曲面:

vertices = D1:360 D2:100

r = 10.0

u = from 0 to (2*PI) D1
v = from (-PI/2) to (PI/2) D2

p = sin(u)*sqrt(abs(cos(u)))/(sin(u) + 7/5) - 2*sin(u) + 2

x = p*cos(u)
y = p*sin(u) + 1.5

x = x*cos(v)
y = y*cos(v)
z = 0.8*sin(v)

(3)完美的心形
这是我目前生成的最完美的心形

曲线:

vertices = 360
u = from (-PI) to (PI)

w = abs(u)
p = w*sqrt(w/(1 + w))

x = p*sin(u)
y = (p*cos(u) + 1)*0.9

平面:

vertices = D1:360 D2:100
u = from (-PI) to (PI) D1
v = from 1 to 100 D2

w = abs(u)
p = w*sqrt(w/(v + w))

x = p*sin(u)
y = p*cos(u)*0.9

曲面:

vertices = D1:360 D2:64

u = from (-PI) to (PI) D1
v = from (PI/2) to (-PI/2) D2

w = abs(u)
p = w*sqrt(w/(1 + w))
b = 0.5

x = p*sin(u)*cos(v)
y = (p*cos(u) + 0.9)*0.9*cos(v)
z = b*sin(v)

 如果你觉得这个心形还是像屁股,那你与下面的图形比较下,就会发现不像了.

(4)像屁股的心形

#http://xahlee.info/surface/kuen/kuen.html

vertices = D1:100 D2:100

u = from  (-1.4325*PI) to (1.4325*PI) D1
v = from (-10) to (10) D2

w = (cosh[v]^2 + u^2)

x = 2*cosh[v]*(cos[u] + u*sin[u])
z = 2*cosh[v]*(-u*cos[u] + sin[u])
y =  v - (2*sinh[v]*cosh[v])


x = x/w
y = y/w
z = z/w