数学图形之Breather surface

这是一种挺漂亮的曲面图形,可惜没有找到太多的相关解释.

In differential equations, a breather surface is a mathematical surface relating to breathers.

其数学公式很复杂,参数方程为:

egin{align} x & {} = -u+frac{2left(1-a^2 ight)cosh(au)sinh(au)}{aleft(left(1-a^2 ight)cosh^2(au)+a^2\,sin^2left(sqrt{1-a^2}v ight) ight)} \ \ y & {} = frac{2sqrt{1-a^2}cosh(au)left(-sqrt{1-a^2}cos(v)cosleft(sqrt{1-a^2}v ight)-sin(v)sinleft(sqrt{1-a^2}v ight) ight)}{aleft(left(1-a^2 ight)cosh^2(au)+a^2\,sin^2left(sqrt{1-a^2}v ight) ight)} \ \ z & {} = frac{2sqrt{1-a^2}cosh(au)left(-sqrt{1-a^2}sin(v)cosleft(sqrt{1-a^2}v ight)+cos(v)sinleft(sqrt{1-a^2}v ight) ight)}{aleft(left(1-a^2 ight)cosh^2(au)+a^2\,sin^2left(sqrt{1-a^2}v ight) ight)} end{align}

where 0 < a < 1.

维基的相关网址为:http://en.wikipedia.org/wiki/Breather_surface

使用自己定义语法的脚本代码生成数学图形.相关软件参见:数学图形可视化工具,该软件免费开源.QQ交流群: 367752815

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

vertices = D1:100 D2:100

u = from -13.2 to 13.2 D1
v = from -37.4 to 37.4 D2

b = 0.4
r = 1 - b*b
w = sqrt(r)

d = b*((w*cosh[b*u])^2 + (b*sin[w*v])^2)

y = -u + (2*r*cosh[b*u]*sinh[b*u])/d
z = (2*w*cosh[b*u]*(-(w*cos[v]*cos[w*v]) - sin[v]*sin[w*v]))/d
x = (2*w*cosh[b*u]*(-(w*sin[v]*cos[w*v]) + cos[v]*sin[w*v]))/d

使用随机数

#http://en.wikipedia.org/wiki/Breather_surface

vertices = D1:100 D2:100

u = from -13.2 to 13.2 D1
v = from -37.4 to 37.4 D2

a = rand2(0.1, 0.9)
w = sqrt(1 - a*a)

d = a*((w*cosh[a*u])^2 + (a*sin[w*v])^2)

y = -u + (2*(1 - a*a)*cosh[a*u]*sinh[a*u])/d
z = (2*w*cosh[a*u]*(-(w*cos[v]*cos[w*v]) - sin[v]*sin[w*v]))/d
x = (2*w*cosh[a*u]*(-(w*sin[v]*cos[w*v]) + cos[v]*sin[w*v]))/d