polar / cart

Almost every radial effect — kaleidoscopes, spirals, petal patterns, ripples that twist — follows the same shape: convert to polar, distort, convert back.

Synesthesia’s _toPolar handles the first leg. The entry here is really about the second one, because the return trip is the half that keeps getting rewritten from scratch in every scene.

vec2 polar(vec2 c) { return vec2(atan(c.y, c.x), length(c)); }
vec2 cart(vec2 p)  { return p.y * vec2(cos(p.x), sin(p.x)); }

The convention here is .x = angle, .y = radius, and the two functions are exact inverses of each other. That last part matters more than it sounds: if you half-remember the ordering and write the inverse by hand, you get a scene that’s subtly sheared and you’ll spend twenty minutes hunting it.

Usage

The whole idiom in five lines:

vec2 p = polar(_uvc);      // p.x = angle, p.y = radius

p.x += p.y * 2.0;          // twist: rotate more the further out you go
p.y += sin(p.x * 6.0) * 0.05;  // ripple the radius by angle

vec2 uv = cart(p);

Mirror the angle to get a kaleidoscope:

vec2 p = polar(_uvc);
float slices = 8.0;
p.x = abs(mod(p.x, 6.28318 / slices) - 3.14159 / slices);
vec2 uv = cart(p);

Notes

• atan(y, x) returns -π..π, not 0..2π. If you mod the angle for radial repetition, that discontinuity at ±π shows up as a seam — add π first, or use abs() for a mirrored fold like the kaleidoscope above.
• At the exact origin, atan(0, 0) is undefined and the radius is 0. It’s one pixel and usually invisible, but if you’re dividing by radius anywhere, guard it.
• Distorting p.x (angle) twists; distorting p.y (radius) ripples in and out. Doing both is where most interesting radial patterns live.
• Use _toPolar on its own when you only need the one-way trip. Take the pair when you’re coming back, so the two halves are guaranteed to agree.