羅馬曲面,像是一個被捏扁的正四面體.

本文將展示羅馬曲面的生成算法和切圖,使用自己定義語法的腳本代碼生成數學圖形.相關軟件參見:數學圖形可視化工具,該軟件免費開源.QQ交流群: 367752815

維基上關于羅馬曲面的解釋如下:

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.

羅馬曲面腳本代碼:

#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^2*sin(t)*cos(t), r*sin(t)*(1-r^2)^(1/2), r*cos(t)*(1-r^2)^(1/2)], r=0..1, t=0..2*Pi, numpoints=2500)vertices = D1:160 D2:80 u = from 0 to (PI) D1 v = from 0 to (PI) D2a = sin(u) b = cos(u)c = sin(v) d = cos(v)r = 5.0x = r*r*b*d*c y = r*r*a*d*c z = r*r*b*a*d*d

我還找到幾個與羅馬曲面相關的圖形

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 D2y = 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 D2x = 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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轉載于:https://www.cnblogs.com/WhyEngine/p/3896249.html

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