Triangle:

Let the triangle defined by (u_i, v_j), (u_{i + 1}, v_j) and (u_i, v_{j + 1}). These three points are mapped to three points on the surface p(u_i, v_j), p(u_{i + 1}, v_j) and p(u_i, v_{j + 1}) along with normal vectors n (u_i, v_j), n (u_{i + 1}, v_j) and n (u_i, v_{j + 1}). We have now three vertices each of which contain a normal vector. These six pieces of information are adequate to render the triangle smoothly. As a consequence, we have a method for rendering a parametric surface. Or, put it in another way, we contain a method for producing a set of triangles that approximate the given parametric surface. This approximation cannot be a good one since it could have too many triangles and some of these triangles can be in wrong positions or too small. However, this situation may be developed by an "adaptive" technique. An adaptive technique adjusts dynamically the sizes, the number & the positions of triangles.

The Differential geometry is utilized for the measurements of lengths & areas, the specification of directions & angles & the definition of curvature on a surface. The parametric surface P (u, v) is directly amenable to differential analysis. There are intrinsic differential characteristics of a surface such like the unit normal and the principal curvatures and directions that are independent of parameterization. These characteristics need introducing few parametric derivatives.