| OCR Text |
Show 11 boundaries are then the solution of two (potent ially nonlinear) equations, F=O and Fi=O. In the case of planar surfaces these intersections will always be straight lines. These linear edges are usually represented explicitly rather than in this implicit fashion. For parametric surf2ces, the boundaries are defined in the parameter space. Typically they are formed by the four lines u=O, u=1, v=O, and v=1. This gives a four sided patch containing those points for which u and v are in the range [0,1]. Some parametric surfaces have been used which are triangular, having the boundaries u=O, v=O, u+v=l. In general, in the parametric case, the boundary curves are par2metric curves in three dimensions. Plane nd ine Intersections The operation of discretizing a surface for display primarily involves intersecting it with a ray from the eye through each of the picture elements on the screen. In the course of doing this it is usual to first intersect the surface with a plane generating some form of space curve. For algebraic surfaces this will have an explicit representation as an algebraic curve. For parametric surfaces there may be no closed form solution. Normal Vectors Once a point has been identified as being on a surface, within its boundaries, and lying on the viewing ray, intensity calculations must be performed to draw it on a |
