Question

I have three vertices which make up a plane/polygon in 3D Space, v0, v1 & v2.

To calculate barycentric co-ordinates for a 3D point upon this plane I must first project both the plane and point into 2D space.

After trawling the web I have a good understanding of how to calculate barycentric co-ordinates in 2D space, but I am stuck at finding the best way to project my 3D points into a suitable 2D plane.

It was suggested to me that the best way to achieve this was to “drop the axis with the smallest projection”. Without testing the area of the polygon formed when projected on each world axis (xy, yz, xz) how can I determine which projection is best (has the largest area), and therefore is most suitable for calculating the most accurate barycentric co-ordinate?

Answer 1

After much discussion there is actually a pretty simple way to solve the original problem of knowing which axis to drop when projecting to 2D space. The answer is described in 3D Math Primer for Graphics and Game Development as follows…

"A solution to this dilemma is to
choose the plane of projection so as
to maximize the area of the projected
triangle. This can be done by
examining the plane normal; the
coordinate that has the largest
absolute value is the coordinate that
we will discard. For example, if the
normal is [–1, 0, 0], then we would
discard the x values of the vertices
and p, projecting onto the yz plane."

My original solution which involved computing a score per axis (using sub deltas) is flawed as it is possible to generate a zero score for all three axis, in which case the axis to drop remains undetermined.

Using the normal of the collision plane (which can be precomputed for efficiency) to determine which axis to drop when projecting into 2D is therefore the best approach.