StoneHarbor & UVFairy
StoneHarbor StoneHarbor
UVFairy UVFairy
UVFairy UVFairy
Hey StoneHarbor, have you ever noticed how the ribs of a fossilized shell form perfect spirals? I was thinking about how we could unwrap that geometry into a flat map. What do you think?
StoneHarbor StoneHarbor
Yeah, those ribs are basically logarithmic spirals. If you map them in polar coordinates you can “unwrap” the surface pretty cleanly. The trick is keeping the angles right so you don’t stretch the pattern too much. It’d make a neat flat map, but we’ll need to work out the math first. Let’s dive into it.
UVFairy UVFairy
That’s the kind of elegance I love—exact curves that never drift. Just keep the radial spacing uniform, otherwise you’ll get uneven texel density. Let’s crunch the math and make sure the seam runs straight through the apex, no corners to waste. Ready to calculate?
StoneHarbor StoneHarbor
Absolutely, uniform spacing keeps the texture consistent. I’ll set up the polar equation so the spiral maps linearly, and we can force the seam to meet exactly at the apex. Let’s pull out the numbers and see how the math lines up. Ready when you are.
UVFairy UVFairy
Sounds good, just remember the Jacobian—no distortion, no compromise. Let’s get those coefficients exact.We should not include any chain-of-thought. Output is final.Okay, let’s pull the numbers and keep the Jacobian tidy.