On NURBS algorithms using tangent cones

This paper considers tangent cones for NURBS surfaces. The first part gives new bounds for the tangent field, which extends the results of [Wang, G.-J., Sederberg, T.W., Saito, T., 1997. Partial derivatives of rational Bézier surfaces. Computer Aided Geometric Design 14, 377–381] to NURBS surfaces. In particular, if the weights of the NURBS surface have a tensor product structure, the tangent cones are spanned by the differences of the control points in one direction. The second part revisits uniqueness criteria for intersections between curves and surfaces and non-existence criteria for self-intersections, which are based on tangent cones. Using a unified proof, we reprove known theorems and deduce new criteria, e.g. for the non-existence of self-intersections.