title: Bézier Triangles and N-Patches
body: <p>I'm trying to make a smooth surface out of a polygon mesh. So to make the needed subdivisions, I want to use N-Patches, which consist of Bézier triangles. As I understand, Bézier triangles can be made out of triangle polygon meshes, which are made out of triangle primitives, as seen her:<img src="https://i.stack.imgur.com/5EmIj.gif" alt="enter image description here"></p>
  
  <p>This, however, complicates the task. For example, if I provide a polygon mesh made out of triangle primitives, how do I apply the Bézier triangles and hence the N-patches to the polygon mesh information? For a cubic Bézier triangles, do I need to bind all of the polygon triangles in a group of 9 triangles, where each group represents a bigger triangle, so I get the 9 border control points, with which I can get the inner needed control points, with which to perform a de Casteljau algorithm for the cubic Bézier triangle, as seen here: </p>
  
  <p><img src="https://i.stack.imgur.com/L5CNW.gif" alt="enter image description here"></p>
  
  <p>This doesn't seem practical and in many cases unusable, e.g. the UV map of the polygon mesh must constructed ouf of these 9 vertices triangles (a group of 9 basic triangles).</p>
  
  <p>This seems the only logical use of Bézier triangles to me, but I think the Bézier Triangles should be constructed out of/for each basic polygon mesh triangles, but this doesn't make sense. If you construct inner control points of a basic polygon mesh triangle with de Casteljau for Bézier triangles, what is there to control/smooth anyway? The basic triangles, out of which the polygon mesh is made of, are completely flat, as all 3 points of the triangle are guaranteed to be on the same plane - so the surface over a triangle will be flat even after using the Bézier triangle, which is the correct behavior. That's why I reasoned  the Bézier Triangles need to be created out of many basic triangles.</p>
  
  <p>Some quick reference: <a href="http://www.gamasutra.com/view/feature/131389/b%C3%A9zier_triangles_and_npatches.php?print=1" rel="nofollow noreferrer">http://www.gamasutra.com/view/feature/131389/b%C3%A9zier_triangles_and_npatches.php?print=1</a></p>