Barycentric Coordinates Calculator
Express a 2D point as weights of a triangle's vertices — for interpolation, hit-testing and rasterization.
Barycentric coordinates express any point as a weighted blend of the triangle's vertices summing to 1. They power the two pillars of rasterization: the inside test (all weights ≥ 0) and attribute interpolation (blend color/UV/depth/normals by these same weights). Negative weight = outside that edge.
Formula
About Barycentric Coordinates Calculator
Barycentric coordinates express a point as a weighted average of a triangle's three vertices, with the weights summing to one. They're the mathematical core of GPU rasterization: the inside-the-triangle test reduces to 'are all three weights non-negative?', and every per-pixel attribute — color, texture UV, depth, interpolated normals — is computed by blending the vertices' values with these same weights. This calculator computes the barycentric weights of any 2D point relative to a triangle and tells you whether the point lies inside, the exact operation a GPU performs millions of times per frame.
How to use Barycentric Coordinates Calculator
- 1Enter your values into Barycentric Coordinates Calculator — sensible, domain-typical defaults are pre-filled so you see a real result immediately.
- 2The result recomputes live using the formula shown on the page; there is no button to press.
- 3Adjust any input to compare scenarios, then read the worked example to see the substituted numbers.
Why use Barycentric Coordinates Calculator?
- ✓Computes Barycentric Coordinates instantly in your browser — no sign-up, no upload, no server round-trip.
- ✓100% free and unlimited, with the exact formula shown: P = u.
- ✓Runs entirely client-side, so every value you enter stays private on your device.
- ✓Live recompute as you type, with a worked example and authoritative references for trust.
Frequently asked questions
How do barycentric coordinates test if a point is in a triangle?+
Compute the three weights; the point is inside the triangle if and only if all three are non-negative (and they always sum to 1). A negative weight means the point is on the far side of the edge opposite that vertex. This is the elegant, branch-light inside test that GPU rasterizers and ray-triangle intersection use.
Why are they used for interpolation in graphics?+
Because they provide the natural blend weights for the triangle's vertices. To find a pixel's color, UV or depth, you take the vertices' values and weight them by the pixel's barycentric coordinates. This gives smooth, linear (affine) interpolation across the face — exactly how Gouraud shading and texture mapping fill triangles between vertex data.
What's perspective-correct interpolation?+
Plain barycentric interpolation in screen space distorts attributes under perspective (textures appear to swim). The fix divides each attribute by its vertex's clip-space w, interpolates, then divides back — 'perspective-correct' interpolation. GPUs do this automatically. Screen-space barycentric weights still drive it; the per-attribute correction handles the foreshortening.
Can barycentric coordinates be negative or exceed 1?+
Yes — for points outside the triangle. Negative means beyond the opposite edge; a weight above 1 means beyond a vertex. They remain a valid affine coordinate system over the whole plane, which is why the simple sign test cleanly separates inside from outside. Inside the triangle, all three lie in [0, 1].
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