这是一个姓Boy的人发现的,所以取名为Boy surface.该图形与罗马图形有点相似,都是三分的图形.它甚至可以说是由罗马曲面变化而成的.
本文将展示几种Boy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
In geometry, Boy's surface is an immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901 (he discovered it on assignment from David Hilbert to prove that the projective plane could not be 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.shtml vertices = D1:100 D2:100 u = from 0 to (PI) D1 v = from 0 to (PI) D2 a = 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.html vertices = D1:100 D2:100 u = from (-PI/2) to (PI/2) D1 v = from 0 to PI D2 a = 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/d y = y/d z = z/d
boy surface 03
我在这套公式上花费的时间最多,因为它最复杂,而且也不完美.它的曲面没能闭合.我研究了很久才发现,当r取无穷大时,该曲面才会闭合.
#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 D2:100 t = from 0 to (PI*2) D1 r = from 0 to 10 D2 c = cos(t) s = sin(t) i = (r*c) ^ 2 j = (r*s) ^ 2 m = 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)
