What are barycentric calculations used for?

问题:

I've been looking at XNA's Barycentric method and the descriptions I find online are pretty opaque to me. An example would be nice. Just an explanation in English would be great... what is the purpose and how could it be used?

回答1:

From Wikipedia:

In geometry, the barycentric coordinate system is a coordinate system in which the location of a point is specified as the center of mass, or barycenter, of masses placed at the vertices of a simplex (a triangle, tetrahedron, etc).

They are used, I believe, for raytracing in game development.

When a ray intersects a triangle in a normal mesh, you just record it as either a hit or a miss. But if you want to implement a subsurf modifier (image below), which makes meshes much smoother, you will need the distance the ray hit from the center of the triangle (which is much easier to work with in Barycentric coordinates).

Subsurf modifiers are not that hard to visualize:

The cube is the original shape, and the smooth mesh inside is the "subsurfed" cube, I think with a recursion depth of three or four.

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Actually, that might not be correct. Don't take my exact word for it, but I do know that they are used for texture mapping on geometric shapes.

Here's a little set of slides you can look at: http://www8.cs.umu.se/kurser/TDBC07/HT04/handouts/HO-lecture11.pdf

回答2:

In practice the barycentric coordinates of a point P in respect of a triangle ABC are just its weights (u,v,w) according to the triangle's vertices, such that P = u*A + v*B + w*C. If the point lies within the triangle, you got u,v,w in [0,1] and u+v+w = 1.

They are used for any task involving knowledge of a point's location in respect to the vertices of a triangle, like e.g. interpolation of attributes across a triangle. For example in raytracing you got a hitpoint inside the triangle. When you want to know that point's normal or other attributes, you compute its barycentric coordinates within the triangle. Then you can use these weights to sum up the attributes of the triangle's vertices and you got the interpolated attribute.

To compute a point P's barycentric coordinates (u,v,w) within a triangle ABC you can use:

u = [PBC] / [ABC]
v = [APC] / [ABC]
w = [ABP] / [ABC]

where [ABC] denotes the area of the triangle ABC.