An intuitive way for constructing parametric quadric triangles

We present an intuitive algorithm for providing quadric surface design elements with shape parameters. To this end, we construct rational parametric triangular quadratic patches which lie on quadrics. The input of the algorithm is three vertex data points in 3D and normals at these points. It emanates from a thorough analysis of two existing methods for the construction of rational parametric Bézier triangles on quadrics, that allows to establish an interesting geometric relation between them. The sufficient condition for a configuration of vertex and normal data to allow for the existence of a rational triangular quadratic patch lying on a quadric whose tangent planes at the vertices are those prescribed by the given normals is the concurrence of certain cevians. When these conditions are not met we offer an optimization procedure to tweak the normals, without varying the vertex data, so that for the new normals there is a rational triangular quadratic patch that lies on a quadric. The resulting quadric design element offers three free shape parameters.