fix face area weighting in quadric edge collapse

[flm8620]

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I think there is a bug in the area weighting of quadric edge decimation. In the paper "Simplifying Surfaces with Color and Texture using Quadric Error Metrics" and "New Quadric Metric for Simplifying Meshes
with Appearance Attributes", they both mentioned that the area weighting should be appied on the matrix Q, meaning Q *= Area.

But in the vcglib's implementation, the squared area is multiplied into Q. I think it's a mistake. The current code

facePlane.SetDirection( ( (*fi).V(1)->cP() - (*fi).V(0)->cP() ) ^  ( (*fi).V(2)->cP() - (*fi).V(0)->cP() ));

will actually multiple the area into the unit normal vector. And this face normal vector is used to construct the matrix Q, where each component is squared, or multiplied to each other, therefore the squared area is incorrectly weighted into Q.

Here is an actual experiment:
In meshlab, we can take a sphere mesh and select half of its faces and do subdivision (Here I used midpoint subdivision three times)

And we decimate the mesh to 10000 faces. In the current implementation, we will observe a non-uniform face size. It's because the area is incorrectly squared, and most face area area less than one. So the smaller face will have a incorrect even smaller weight, which make the algorithm think the smaller face should have fewer cost, therefore being decimated much more.

After this fix, it works correctly. All the faces are distributed uniformly.

[FishWoWater]

Seems that UseArea not exposed to meshlab by default?

[flm8620]

UseArea is by default turned on. You can check the code, it's initialized to true.

[FishWoWater]

UseArea is by default turned on. You can check the code, it's initialized to true.

OK, you are right :)

[alemuntoni]

Thank you very much, @flm8620! :)