To the Graduate Council: I am submitting herewith a thesis written by ...

Chapter 5: Analysis and Results 60We define N v as the normal of surface S at a vertex v. We compute the normal at avertex using the normals of the triangles that contain the vertex. The normal of atriangle is the normal of the plane that fits the three points and is given by Equation5.2. We compute the vertex normal as the average of these normals weighted by areaof the triangles involved.Nvi( vi− v ) × ( v=||( v − v ) × ( v − 11 n vv= N in i=0N ;i( i+1)mod n( i+1)mod n− v )− v )||(5.2)NN = vv|| Nv||(5.3)We show a small section of a triangle mesh in Figure 5.1 to understand the definitionsbetter. The blue colored point in the middle is the vertex at which we would like tocompute the curvature. Points in red are the neighbor points and the lines connectingthe vertex v and its neighbors are the triangles that determine the surface. N v is thenormal at the vertex that we have defined in Equation 5.3.The paraboloid fitting method [Kresk, 1998] at each vertex is computed by translatingthe vertex under consideration to the origin and its neighbors are rotated so that thevertex normal coalesces with the z axis. The osculating paraboloid of the form z= ax 2 +bxy + cy 2 is assumed to contain these transformed points. The coefficients a, b, c arefound by solving a least square fit to v and the neighboring vertices [ v n−1i ] i=0 . The total andmean curvatures are computed using the formula in Equation 5.4.2κ = 4ac − b ; H = a + c(5.4)N vv 1v 2v 3v 4αvv 6v 5Figure 5.1: Neighborhood of a vertex in a triangle mesh.