On the Triangulation of Convex Polygons Presenting T{Vertices

A technique to triangulate planar convex polygons presenting T{vertices is described. Simple strip or fan tessellation of a polygon with T{vertices can result in zero{area triangles and compromise the rendering process. Our technique splits such a polygon into one triangle strip and, at most, one triangle fan. The utility is particularly useful in multiresolution or adaptive representation of polygonal surfaces and surfaces' simpli cation. Triangulating a planar convex polygon is a simple, but important task. Though modern graphic languages provide for the planar convex polygon as one of the elementary geometric primitives, better performances are obtained by breaking down polygons into compact sequences of triangles. Triangle{strip (Fig. 1.a) and triangle{fan (Fig. 1.b) are compact and e cient methods to represent triangles' sequences and are provided as output primitives in most graphics libraries (OpenGL [7] and, partially, PHIGS [2], for example). The triangulation becomes a little more complicated when one or more vertices of the polygon are T{vertices, i.e. there exist three or more aligned vertices. This problem arise, for example, in the adaptive representation of polygonal surfaces ([6], [3]). One of the simplest algorithm to avoid cracks (small holes between polygons at di erent levels of resolution) is to move some of the vertices of the lower resolution polygons onto the boundary of the higher ones, so introducing T{vertices. In these cases, naively creating a triangle strip or fan, as per Fig. 1, can yield degenerate (zero-area) triangles. In this note, we present a simple technique which permits to split convex polygons presenting T{vertices into one triangle{strip and, possibly, one triangle{fan. We assume that the user is acquainted with the existence of T{ vertices and with their position on the polygon's boundary (this is not a limi-