@@ -35,7 +35,7 @@ Finally, we also want to optimize the geometry of the vertices. A very simple he
How exactly should the positions be updated? One idea is to simply replace each vertex position with its centroid. We can make the algorithm slightly more stable by moving _gently_ toward the centroid, rather than immediately snapping the vertex to the center. For instance, if _p_ is the original vertex position and _c_ is the centroid, we might compute the new vertex position as _q_ = _p_ + _w_(_c_ - _p_) where _w_ is some weighting factor between 0 and 1 (we use 1/5 in the examples below). In other words, we start out at _p_ and move a little bit in the update direction _v_ = _c_ - _p_.
Another important issue arises if the update direction _v_ has a large _normal_ component, then we'll end up pushing the surface in or out, rather than just sliding our sample points around on the surface. As a result, the shape of the surface will change much more than we'd like (try it!). To ameliorate this issue, we will move the vertex only in the _tangent_ direction, which we can do by projecting out the normal component, i.e., by replacing _v_ with _v_ - dot(_N_,_v_)_N_, where _N_ is the unit normal at the vertex. To get this normal, you will implement the method `Vertex::normal()`, which computes the vertex normal as the area-weighted average of the incident triangle normals. In other words, at a vertex i the normal points in the direction
Another important issue arises if the update direction _v_ has a large _normal_ component, then we'll end up pushing the surface in or out, rather than just sliding our sample points around on the surface. As a result, the shape of the surface will change much more than we'd like (try it!). To ameliorate this issue, we will move the vertex only in the _tangent_ direction, which we can do by projecting out the normal component, i.e., by replacing _v_ with _v_ - dot(_N_,_v_)_N_, where _N_ is the unit normal at the vertex. To get this normal, you can use the method `Vertex::normal()`, which computes the vertex normal as the area-weighted average of the incident triangle normals. In other words, at a vertex i the normal points in the direction
