SageMath로 Costa 곡면 그리기

양성덕
20210413

세 번째 시도 당시 알지 못했던 기법을 알게 되서 이제 다음과 같이 그림을 완성한다. 이러면 Weber가 그린 방법을 그대로 따른 셈이 된다.

참고 문헌
[2] Matthias Weber, Costa's Minimal Surface, Mathematica Notebook File

var('rho')

rho = ( gamma(3/4)/sqrt(2)/gamma(5/4) ).n()

Phi1(w) = 2*rho*I*sqrt(w)*hypergeometric((1/4,3/2,), (5/4,), w^2)
Phi2(w) = -2/(3*rho)*I*w*sqrt(w)*hypergeometric((1/2,3/4,), (7/4,), w^2)

omega1(z) = -1/2*( Phi1(z) - Phi2(z))
omega2(z) =  I/2*( Phi1(z) + Phi2(z))
omega3(z) =  1/2*log(-1+z) - 1/2*log(1+z)

F(z) = ( omega1(z).real(), omega2(z).real(), omega3(z).real() ) 

p(w) = sqrt( 1 + exp(w) )

var('y');

points = [(p(x+I*y).real(), p(x+I*y).imag()) for x in srange(-1,3,0.1) for y in srange(0*pi, pi, 0.05*pi, include_endpoint=True)] 

list_plot(points, aspect_ratio=1)

xlist=  [-3.5,-3.0,-2.5, -2.0,-1.5, -1.0,-0.5,-0.3,-0.2,-0.1,-0.05,-0.03,0.0,0.03,0.05,0.1,0.2,0.4,0.6,0.8,1.0,1.2,1.5,2.0 ,2.5,3.0 ,3.5,4.0,5.0,6.0 ];
xlen= len(xlist);


ylist= list(pi*vector([0.001,0.05,0.08,0.16,0.24,0.32,0.4,0.48,0.56,0.64,0.72, 0.8,0.88,0.95,0.98,0.99,1.0]));
ylen=len(ylist);

var('x y');

X = lambda x: xlist[ int(round(x)) -1] if x<0.5 or x>xlen + 0.4 else xlist[ int(round(x)) -1]
Y = lambda y: ylist[ int(round(y)) -1] if y<0.5 or y>ylen + 0.4 else ylist[ int(round(y)) -1]

f = (lambda x,y: omega1(p(X(x)+ I*Y(y))).real(), 
     lambda x,y: omega2(p(X(x)+ I*Y(y))).real(),
     lambda x,y: omega3(p(X(x)+ I*Y(y))).real()
     )
gr1 = parametric_plot3d( f, (x, 1,xlen),(y,1,ylen))

gr1.show(frame=False)

한 조각에서 전체 그리기

gr2 = gr1.transform( scale=(-1,1,-1), rot=(0,0,1,pi/2))

gr12 = gr1.transform(scale=(-1,1,1))
gr13 = gr1.transform(scale=(-1,-1,1))
gr14 = gr1.transform(scale=(1,-1,1))

gr22 = gr2.transform(scale=(-1,1,1))
gr23 = gr2.transform(scale=(-1,-1,1))
gr24 = gr2.transform(scale=(1,-1,1))

gr = gr1 + gr2 + gr12 + gr13 + gr14 + gr22 + gr23 + gr24

gr.show(frame=False)