Barycentric Mappings

As in the previous chapter, let us denote points in R by p = (x, y, z) and points in R by u = (u, v). An edge is then defined as the convex hull of (or, equivalently, the line segment between) two distinct points and a triangle as the convex hull of three non-collinear points. We will denote edges and triangles in R with capital letters and those in R with small letters, for example, e = [u1,u2] and T = [p1,p2,p3]. A triangle mesh ST is the union of a set of surface triangles T = {T1, . . . , Tm} which intersect only at common edges E = {E1, . . . , El} and vertices V = {p1, . . . ,pn+b}. More specifically, the set of vertices consists of n interior vertices VI = {p1, . . . ,pn} and b boundary vertices VB = {pn+1, . . . ,pn+b}. Two distinct vertices pi,pj ∈ V are called neighbours, if they are the end points of some edge E = [pi,pj ] ∈ E , and for any pi ∈ V we let Ni = {j : [pi,pj ] ∈ E} be the set of indices of all neighbours of pi. A parameterization f of ST is usually specified the other way around, that is, by defining the inverse parameterization g = f−1. This mapping g is uniquely determined by specifying the parameter points ui = g(pi) for each vertex pi ∈ V and demanding that g is continuous and linear for each triangle. In this setting, g|T is the linear map from a surface triangle T = [pi,pj ,pk] to the corresponding parameter triangle t = [ui,uj ,uk] and f |t = (g|T ) is the inverse linear map from t to T . The parameter domain Ω finally is the union of all parameter triangles (see Figure 1).