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The key point on this slide is that the triangle's edges b-a and c-a form a non-orthonormal basis for points in the triangle. The offset of any point in the triangle from vertex (e.g., point x) from vertex a is given by a linear combination these vectors.

The slide also shows that another way to think about a point x in the triangle is as a linear combination of the triangle's vertex positions, where the weights of this combination, $\alpha,\beta,\gamma$ sum to 1 and are called x's barycentric coordinates.


Barycentric Coordinates in Action!

For assignment 2, we have implemented a 3D mesh visualizer that uses barycentric coordinates to determine whether a user is selecting a face, edge, or vertice!