Actually #raytracing a #torus using ray… by pavel

function u(t) {

c.width=1920 // clear the canvas
for(i=0;i<9;i++)
x.fillRect(400+i*100+S(t)*300,400,50,200) // draw 50x200 rects

} // 122/140

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    u(t) is called 60 times per second.
    t: Elapsed time in seconds.
    S: Shorthand for Math.sin.
    C: Shorthand for Math.cos.
    T: Shorthand for Math.tan.
    R: Function that generates rgba-strings, usage ex.: R(255, 255, 255, 0.5)
    c: A 1920x1080 canvas.
    x: A 2D context for that canvas.

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pavel

function u(t) {

for(let i=2e3,f=(x,n,c=8-(t+6)*n)=>~n&(x*x*n*n+(8+(t-12)*n)**2+c*c-5)**2/16+c*c>1?f(x,n+=.002):c;i--;)x.fillRect(i,t*180,f(i/135-7,.7),3)

Apr 23 2020 7:27 AM UTC

}// 137/140

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Actually #raytracing a #torus using ray marching

Show more comments…
u/Xen

Sweeet!
u/tomxor

Yes! we need more explicit surfaces on dwitter :)
u/Odog8

which expression is the DFT?
u/tomxor

I'm confused... this isn't explicit? it recurses.
u/pavel

This recurses just to advance ray length.
u/pavel

The closed form expression is x*x*n*n+(8+(t-12)*n)**2+c*c-5)**2/16+c*c-1 where c=8-(t+6)*n and n is the free variable.
u/tomxor

Ah, so is this technically ray casting? I have to ask because I still don't intuitively get all of the math.
u/tomxor

Sorry I meant ray marching
u/pavel

Yes. The math constructs a ray from (0,8,8) looking at (0,0,0), varied a bit depending on the pixel being drawn, its length is a function of n. I plug this into the equation for a torus and increment n from .7 to -1 until I find a zero of the equation.
u/pavel

Here is a version without much golfing that makes things clearer for(let i=2e3,f=(o,n,x=o*n,y=t/3*n-n,z=4-n)=>z>-1&(x*x+y*y+z*z+3)**2>16*(x*x+y*y)?f(o,n+.01):z;i--;)x.fillRect(i,t*180,f(i/540-16/9,3),3)
u/pavel

(x*x+y*y+z*z+3)**2>16*(x*x+y*y) is directly from https://en.wikipedia.org/wiki/Torus
u/pavel

with R=2 and r=1
u/pavel

This example shoots rays from (0,0,4). In order to rotate the shape you have to move where the rays come from, the camera!
u/tomxor

Thanks for the explanation 👍


      u(t) is called 60 times per second.
      t: elapsed time in seconds.
      c: A 1920x1080 canvas.
      x: A 2D context for that canvas.
      S: Math.sin
      C: Math.cos
      T: Math.tan
      R: Generates rgba-strings, ex.: R(255, 255, 255, 0.5)

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