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:
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]
