Triangle Barycentrics CMSC 435634 Barycentric Coordinates Weighted average

Triangle Barycentrics CMSC 435/634

Barycentric Coordinates • Weighted average of vertex positions – Weights a, b, g ; a + b + g = 1 – • Same weight can interpolate other data –

Barycentric Coordinates • Each coordinate is 1 at its vertex • 0 at both other vertices – and on the line between them

Computing Barycentrics (1) • Ratio of (signed) distance from edge

Computing Barycentrics (2) • Ratio of (signed) triangle areas • Since • Everything but h’s cancel

Computing Barycentrics (2 a) • Area with cross product • Dot with normal for sign

Computing Barycentrics (2 b) • Skip normalization

Computing Barycentrics (2 c) • Area from matrix determinant

Computing Barycentrics (2 d) • Area by Green’s Theorem

Computing Barycentrics (3) • System of equations • a, b, and g are linear in X and Y

Computing Barycentrics (3) • Each barycentric is – Equal to 1 at one vertex – Equal to 0 at the other two

Computing Barycentrics (3) • This defines a system of three equations • or

Computing Barycentrics (3) • Solve for coefficients for all three:

Computing Barycentrics (3) • Matrix Inverse

Computing Barycentrics (3) • Solve for a coefficients

Computing Barycentrics (3) • At

Computing Barycentrics • Bottom Line: – Lots of ways to compute them – All algebraically equivalent • Use the one that you find easiest