数学图形之单叶双曲面

    双曲线绕其对称轴旋转而生成的曲面即为双曲面。在数学里，双曲面是一种二次曲面。
    其中单叶双曲面可以用公式表达为:

    (x^2)/(a^2)+(y^2)/(b^2)-(z^2)/(c^2)=1 

     

    在现实中，许多发电厂的冷却塔结构是单叶双曲面形状。由于单叶双曲面是一种双重直纹曲面(ruled surface) ，它可以用直的钢梁建造。这样，会减少风的阻力．同时，也可以用最少的材料来维持结

构的完整.

    本文将展示几种生成单叶双曲面算法和切图.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

(1)

vertices = dimension1:72 dimension2:72

u = from 0 to (2*PI) dimension1
v = from (-4) to (4) dimension2

x = cosh(v)*cos(u)
z = cosh(v)*sin(u)
y = sinh(v)

(2)

单叶双曲面是一种直纹面（Ruled_surface），即它是由一族直线铺成的曲面。直线 x=1, y=t, z=2t 绕 z 轴旋转得到的单叶双曲面。下面是使用直线生成单叶双曲面的例子:

vertices = dimension1:72 dimension2:72

u = from 0 to (2*PI) dimension1
v = from (-10) to (10) dimension2

x = cos(u) - v*sin(u)
z = sin(u) + v*cos(u)
y = 2*v

(3)椭圆开口的随机单叶双曲面

vertices = dimension1:72 dimension2:72

u = from 0 to (2*PI) dimension1
v = from (-5) to (5) dimension2

a = rand2(1, 5)
b = rand2(1, 5)
c = rand2(1, 5)

x = a*sqrt(1 + v*v)*cos(u)
z = b*sqrt(1 + v*v)*sin(u)
y = c*v

(4)椭圆开口的随机单叶双曲面,另一种写法

vertices = D1:100 D2:100
u = from 0 to (2*PI) D1
v = from (-PI*0.45) to (PI*0.45) D2

a = rand2(1, 10)
b = rand2(1, 10)
c = rand2(1, 10)

x = a*sec(v)*sin(u)
y = b*tan(v)
z = c*sec(v)*cos(u)

(5)单叶双曲面的上半部分

vertices = dimension1:72 dimension2:72
u = from (PI*0.05) to (PI*0.499) dimension1
v = from 0 to (2*PI) dimension2
a = rand2(1, 5)
b = rand2(1, 5)
c = rand2(1, 5)
x = a*cosh(u)*cos(v)
z = b*cosh(u)*sin(v)
y = c*sinh(u)

(6)单叶双曲面的下半部分

vertices = dimension1:72 dimension2:72
u = from (-PI*0.499) to (-PI*0.01) dimension1
v = from 0 to (2*PI) dimension2
a = rand2(1, 5)
b = rand2(1, 5)
c = rand2(1, 5)
x = a*cosh(u)*cos(v)
z = b*cosh(u)*sin(v)
y = c*sinh(u)

(7)直线 随机旋转面

vertices = 100
u = from -10 to 10

a = rand2(-5, 5)
b = rand2(-5, 5)

x = u
y = (x + a) + b

i = rand2(-2, 2)
j = rand2(-2, 2)
k = rand2(-2, 2)

surface_slices = 72
rotate = anchor[0, 0, 0], axis[i, j, k], angle[0, 2*PI]