barycentric hecking interpolation!!
a little thread on a neat way to interpolate across a triangle!
say you have three points, each with a color - how do you get the blended color for an arbitrary point inside? 🤔
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byerp (coining this term now~) uses the area of the triangle on the *opposite* side of each vertex, as a fraction of the total area, to determine how much influence each vertex has!
❤💙💚
these three influences, or, weights, together form the barycentric coordinate of that point
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we can then multiply each weight with their corresponding color, and add them all up, to get the final blended color at that coordinate!
since models in games are made of triangles, this is actually how mesh data like normals, UVs, vertex colors etc. is interpolated by your GPU!
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that's not to say it's perfect - this can only interpolate the data on the triangle, but there are cases where you want to use 4 points, where two barycentric interpolations can't really get you an accurate result :c
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implemented proper bilinear color interpolation for my quad primitive and holy heck - it's *wild* how wrong barycentric is on even simple shapes like this 
how do games get away with this it's all a facade gosh
how do games get away with this it's all a facade goshShow this thread
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anyhow that's it, it's getting super late and I need to feed thor~
as usual, if you want to support what I do, please consider supporting me on Patreon!
it really does help me stay afloat and do things like this 💖
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I'd do something like this:
- calc dist to each point
- divide each dist by the sum of all distances (to get normalized dist)
- each color weight is color * (1-normDist)
- add weighted colors together
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this doesn't work as it's not a balanced weighted sum anymore, the weights have to add up to 1, which, they don't using this method
consider the case where evaluating the point at vertex A
A*(1-0)+
B*(1-.5)+
C*(1-.5)
=
A*(1)+
B*(.5)+
C*(.5)
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One neat thing about barycentric coordinates is how easy they make it to find the points of intersection between a ray tangent to a given triangle, and the edges of the triangle!! Its great for tracing geodesics on triangle meshes!!!
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off topic. Do you have any recommends or refs for doing something like sliding verts along edges? Trying to think how to do it, but also be able to rotate them via a projected plane between them. So a edge loop through a cylinder , if you rotate, the verts slide on those edges.
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Very cool. I needed something like this for a project. I put together an only-slightly-broken version on observablehq.com/@amoeba/byerpi. Mouse isn't quite right but it still works. Thanks for sharing.
