Boy surface in mathematical graphics

This was discovered by a boy surnamed.Boy surface.This image is similar to a Rome image and is a three-point image. It can be said that it is made of a Rome surface change.

This article will show the generation algorithms and cut diagrams of several boy surfaces, and generate mathematical graphics using script code with custom syntax. for related software, see: Mathematical graphics visualization tool. This software is free and open-source. QQ chat group: 367752815

In geometry,Boy's surfaceIs an immersion of the real projective plane in 3-dimen1_space found by Werner boy in 1901 (he discovered it on assignment from David Hilbert to prove that the projective planeCocould notBe immersed in 3-space). Unlike the Roman surface and the Cross-cap, it has no singularities (I. e. Pinch-points), but it does self-intersect.

 

Boy surface 01

#http://www.mathcurve.com/surfaces/boy/boy.shtmlvertices = D1:100 D2:100u = from 0 to (PI) D1v = from 0 to (PI) D2a = sin(u)b = cos(u)c = sin(v)d = cos(v)m = sqrt(2)k = rand2(0.1, 1.4)t = b/(m - k*sin(2*u)*cos(3*v))x = t*(b*cos(2*v) + m*a*d)y = t*(b*sin(2*v) - m*a*c)z = 3*t*b

Boy surface 02

#http://mathworld.wolfram.com/BoySurface.htmlvertices = D1:100 D2:100u = from (-PI/2) to (PI/2) D1v = from 0 to PI D2a = SQRT2*pow(cos(v), 2)d = 2 - SQRT2*sin(3*u)*sin(2*v)x = a*cos(2*u) + cos(u)*sin(2*v)y = a*sin(2*u) - sin(u)*sin(2*v)z = 3*pow(cos(v), 2)x = x/dy = y/dz = z/d

Boy surface 03

I spent the most time on this formula because it is the most complex and imperfect. its surface is not closed. I have studied it for a long time and found that when R is infinite, the surface will be closed.

#http://www.ipfw.edu/departments/coas/depts/math/coffman/steinersurface.html#x = (1+r^2*cos(t)^2)*(sqrt(2)-sqrt(2)*r^2*cos(t)^2+2*r*sin(t)) / (3*(1+r^2*cos(t)^2)*(1+r^2*cos(t)^2+r^2*sin(t)^2) + 3*sqrt(2)*r^2*cos(t)*sin(t)*(3-r^2*cos(t)^2))#y = 2*(1+r^2*cos(t)^2)*(sqrt(2)*r*cos(t) - r^2*cos(t)*sin(t)) /  (3*(1+r^2*cos(t)^2)*(1+r^2*cos(t)^2+r^2*sin(t)^2) + 3*sqrt(2)*r^2*cos(t)*sin(t)*(3-r^2*cos(t)^2))#z = (1+r^2*cos(t)^2)^2 / ((1+r^2*cos(t)^2)*(1+r^2*cos(t)^2+r^2*sin(t)^2) + sqrt(2)*r^2*cos(t)*sin(t)*(3-r^2*cos(t)^2))vertices = D1:100 D2:100t = from 0 to (PI*2) D1r = from 0 to 10 D2c = cos(t)s = sin(t)i = (r*c) ^ 2j = (r*s) ^ 2m = sqrt(2)a = (1+i)*(1+i+j)b = m*r*r*c*s*(3-i)x = (1+i)*(m-m*i+2*r*s) / (3*a + 3*b)y = 2*(1+i)*(m*r*c - r*r*s*c) / (3*a + 3*b)z = (1+i)*(1+i) / (a + b)