Spherical Coordinat System

Goal:

• f(0) = 0
• f(1) = inf
• f(-1) = -inf

try1 : f(x) = 1/(1-x)-1

• f(0) = 0
• f(1) = inf
• f(-1) = fail

f(x) = 1/(1-x)-1

try2 : f(x) = tan(x*PI/2)

• f(0) = 0
• f(1) = inf
• f(-1) = -inf

inverse of function : f-1(x)= atan(x)/(PI/2)

OK! But calculation time complexity is high and inverse of function is also.

f(x) = tan(x*PI/2)

f-1(x)= atan(x)/(PI/2)

try3 : reshape of try1

• f(x) = 1/(1-abs(x))-1

f(x) = 1/(1-abs(x))-1

• f(x) = sign(x)*(1/(1-|x|) - 1)

It seems good! But it can be simplified.

f(x) = sign(x)*(1/(1-|x|) — 1)

sign(x) = x/|x| = |x|/x

• f(x) = (x/|x|)*(1/(1-|x|) - 1)
• f(x) = (x/|x|)*((1 - 1 + |x|)/(1-|x|))
• f(x) = (x/|x|)*(|x|/(1-|x|))
• f(x) = (x/(1-|x|)) nice!!

what about inverse?

• f(x) = y = (x/(1-|x|))
• f-1(y) = x

in our case :

• sign(x) = sign(y) = x/|x| = |x|/x = y/|y| = |y|/y
• y = f(x) = (x/(1-|x|)) = 1/(1/x - sign(x))
• 1/y = 1/x - sign(x)
• 1/y + sign(x) = 1/x
• (1+y*sign(x))/y = 1/x
• x = y/(1+y*sign(x))
• x = y/(1+y*sign(y))
• x = y/(1+y*|y|/y)
• f-1(y) = y/(1+|y|)
• f-1(x) = x/(1+|x|) niceeeee!

in summary :

• f(x) = x/(1-|x|)
• f-1(x) = x/(1+|x|)

generalized :

• f(x) = ((x*|x|)^(n-1)) / (1-(|x|^n))
• f-1(x) = sign(x) * ((x/(1+|x|))^(1/n))

plot of g(x) = 1/x in spherical coordinate system (center: [0,0])

plot of g(x) = 1/x in spherical coordinate system (singularity point)

https://www.shadertoy.com/view/mssXR7

dirty explain :)