Barycentric Coordinates in Triangle Computation
Learn about barycentric coordinates in triangle computation, including their weighted averages, properties, and methods for calculation. Images and explanations provided for easy understanding.
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Presentation Transcript
Triangle Barycentrics CMSC 435/634
Barycentric Coordinates Weighted average of vertex positions Weights Same weights 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 h d
Computing Barycentrics (2) Ratio of (signed) triangle areas Since All but heights cancel
Computing Barycentrics (2a) Area with cross product Dot with normal for sign
Computing Barycentrics (2b) Skip normalization
Computing Barycentrics (2c) Area from matrix determinant Any non-zero 2D projection
Computing Barycentrics (2d) Area by Green s Theorem
Computing Barycentrics (3) System of equations , , and 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 coefficients
Computing Barycentrics Bottom Line: Lots of ways to compute them All algebraically equivalent Use the one that you find easiest
