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Old 12-12-2010, 10:52 AM   #4
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The short answer is a model made up of quads will smooth, that is to say subdivide, more evenly then a mesh made up of triangles.

The reason, which is a much longer answer, is because of the mathematical algorithm used by most 3D applications to subdivide works best on an all quad mesh. The smoothing method is called Catmull-Clark, named after Ed Catmull, and Jim Clark (one of the founders of SGI) in 1978.

The algorithm is quite simple to describe. It is recursive. Lets look at two examples the quad and the tri-angle. Imaging you have a single quad polygon that has four vertices, one at each corner and four edges connecting these verts. Now add one vertex at the centroid of the quad and draw an edge from the center of each of the polygons original 4 edges to the vertex at the center. What do you get?

A: a new polygonal surface with 4 quad polygons. Now recurse, by repeating the process on the four new polygonal faces. So for iteration two you get 4x4 = 16 quads, iteration 3 you get 16x4 = 64 quads and on and on it goes. With each iteration you get a smoother and smoother surface. Furthermore, the closer the starting quad is to planar and square the better the result.

Now lets consider the triangle. The application will use the same algorithm. So we start with a triangle with 3 verts and 3 edges and place a new vert at the centroid of the triangle and connect an edge from that point to the middle of each of the three edges and what do we get?

A: New polygonal object with 3 QUAD faces! But the resulting quads are more diamond shaped, which are referred to in mathematical terms as a deltoid or a kite. So for iteration 1 we get 3 diamond shaped quads. For iteration 2 we get 3*4 = 12 quads. Iteration 3 = 12*4 = 48 quads. But the important thing is the original three quads are kite shaped and do not smooth evenly and with each iteration this uneven smoothing can become compounded.

There are algorithms that work on triangular meshes better the Catmull-Clark but they are not widely used so quads remain the preferred base mesh for any model that is intended to be smoothed.

Also, one must also consider that edge loops run more evenly through a quad mesh then a triangular mesh. I could go into the mathematical proof of that as well but it would take more text then most would care to stomach so I will have to ask that you just take me at my word on it.

"If I have seen further it is by standing on the shoulders of giants." Sir Isaac Newton, 1675

Last edited by ctbram : 12-12-2010 at 11:05 AM.
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