The two types of surfaces commonly used in geometric modeling systems are parametric and implicit surfaces that lead to three types of surface-surface intersection problems: parametric-parametric, implicit-implicit and parametric-implicit. While differential geometry of a parametric curve can be found in textbooks such as in [412,444,76], there is little literature on differential geometry of intersection curves. Faux and Pratt [116] give a formula for the curvature of an intersection curve of two parametric surfaces. Willmore [444] describes how to obtain the unit tangent , the unit principal normal , and the unit binormal , as well as the curvature and the torsion of the intersection curve of two implicit surfaces. Hartmann [154] provides formulae for computing the curvature of intersection curves for all three types of intersection problems. They all assume transversal intersections where the tangential direction at an intersection point can be computed simply by the cross product of the normal vectors of the both surfaces.
However, when the two normals are parallel to each other, the tangent direction cannot be determined by this method. We call such intersection points tangential intersection points. Kruppa describes in his book [213] that the tangential direction of the intersection curve at a tangential intersection point corresponds to the direction from the intersection point towards the intersection of the Dupin's indicatrices of the two surfaces. Cheng [53], Markot and Magedson [264,263] give solutions for parametric surfaces at isolated tangential intersection points, based on the analysis of the plane vector field function defined by the gradient of an oriented distance function of one surface from the other. The plane field function will vanish at the tangential intersection point, and higher-order expansion of the function is required at such points to determine the marching direction for the intersection curve. Kriezis [208] and Kriezis et al. [210] determine the marching direction for tangential intersection curves based on the fact that the determinant of the Hessian matrix of the oriented distance function is zero. Luo et al. [242] present a method to trace such tangential intersection curves for parametric-parametric surfaces employing the marching method. The marching direction is obtained by solving an underdetermined system based on the equality of the differentiation of the two normal vectors and the projection of the Taylor expansion of the two surfaces onto the normal vector at the intersection point. Ye and Maekawa [458] developed algorithms to compute unit tangent vectors, curvature vectors, binormal vectors, curvatures, torsions, and algorithms to evaluate the higher order derivatives for transversal as well as tangential intersections for all three types of intersection problems.