数学图形之罗马曲面(RomanSurface)
罗马曲面,像是一个被捏扁的正四面体.
本文将展示罗马曲面的生成算法和切图,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
维基上关于罗马曲面的解释如下:
The Roman surface or Steiner surface (so called because Jakob Steiner was in Rome when he thought of it) is a self-intersecting mapping of the real projective plane into three-dimensional space, with an unusually high degree of symmetry. This mapping is not an immersion of the projective plane; however, the figure resulting from removing six singular points is one.
The simplest construction is as the image of a sphere centered at the origin under the map f(x,y,z) = (yz,xz,xy). This gives an implicitformula of
Also, taking a parametrization of the sphere in terms of longitude (θ) and latitude (φ), gives parametric equations for the Roman surface as follows:
- x = r2 cos θ cos φ sin φ
- y = r2 sin θ cos φ sin φ
- z = r2 cos θ sin θ cos2 φ.
罗马曲面脚本代码:
#http://www.ipfw.edu/departments/coas/depts/math/coffman/steinersurface.html
#Steiner's Roman Surface. Three double lines, six pinch points, and a triple point.
#plot3d([r^*sin(t)*cos(t), r*sin(t)*(-r^)^(/), r*cos(t)*(-r^)^(/)], r=.., t=..*Pi, numpoints=) vertices = D1: D2:
u = from to (PI) D1
v = from to (PI) D2 a = sin(u)
b = cos(u) c = sin(v)
d = cos(v) r = 5.0 x = r*r*b*d*c
y = r*r*a*d*c
z = r*r*b*a*d*d
我还找到几个与罗马曲面相关的图形
The three double lines of Steiner's Roman Surface coincide
vertices = D1:100 D2:100
t = from 0 to (PI*2) D1
r = from 0 to 1 D2 y = 1-r^2+(r^2)*(sin(t)^2)
x = (r^2)*(sin(t)^2) + 2*(r^2)*sin(t)*cos(t)
z = sqrt((1-r^2)/2) * r * (sin(t)+cos(t)) x = x*5
y = y*5
z = z*5
Two of the three double lines in Steiner's Roman Surface
vertices = D1:100 D2:100
t = from 0 to (PI*2) D1
r = from 0 to 1 D2 x = 2*r*cos(t)*sqrt(1-r^2)
y = 2*r*sin(t)*sqrt(1-r^2)
z = 1-2*r*r*(cos(t)^2) x = x*5
y = y*5
z = z*5
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