6.4.1 Tangential direction

The unit tangential vector of at must lie on the common tangent plane of and . Therefore, can be represented as a linear combination of and , as well as and , as in (6.17), i.e.

Equation (6.56) consists of two linear equations with four unknowns , , since the tangent vector is constrained in the tangent plane and does not have a normal component. Since at , we find that from (6.25). Thus, from (6.29) we have

This equation is a quadratic equation in . Thus together with the unit length constraint of the tangent vector, (6.56) and (6.57) form a system of four nonlinear equations in four unknowns. This nonlinear system can be solved by representing and in terms of linear combinations of and from (6.56), and then substituting the results into (6.57). By taking the cross product of both sides of (6.56) with and , and projecting the resulting equations onto the common surface normal vector at , and can be represented as the following linear combinations of and

where

Substituting (6.58) and (6.59) into (6.57), we have

where

If we denote when or when and , and solve (6.64) for or , then can be computed as

or

There are four distinct cases to the solution of (6.64) depending upon the discriminant :

1. Isolated tangential contact point: If then (6.64) does not have any real solution. Thus, is an isolated contact point of and .
2. Tangential intersection curve: If and then (6.64) has a double root and is unique. Thus, and intersect at and at its neighborhood.
3. Branch Point: If then (6.64) has distinct roots. Thus, is a branch point of the intersection curve , i.e. there is another intersection branch crossing at .
4. Higher order contact point: If then (6.64) vanishes for any values of and . Thus, and has a contact of at least second order (i.e., curvature continuous) at . In related work by Pegna and Wolter [305], they developed mathematical criteria for curvature continuity between two surfaces. Those criteria were later generalized to arbitrary higher order continuity (contact) in [161].

When is a flat point of one of the surfaces, say , then , , all vanish, however we can still evaluate (6.64). When is a flat point of both surfaces, then the two surfaces have a contact of order at least 2 at which is addressed under case 4.

There is a geometric interpretation to the tangent direction at . Recall that the Dupin's indicatrix of a surface at point is a conic section (see Sect. 3.6). Since and intersect tangentially at , they have the same tangent-plane at . Equation (6.57) indicates that along , the Dupin's indicatrices of and at intersect. Conversely, is the vector(s) on the common tangent-plane at along which the Dupin's indicatrices of and intersect. The two Dupin's indicatrices may not intersect at all (isolated tangential contact point), or intersect at two points tangentially, or intersect transversally at four points (branch point), or overlap (higher order contact point). In the case of overlap, they must be the same at , and and are at least curvature continuous at . Figure 6.3 shows the possible combinations of Dupin's indicatrices of two surfaces for four distinct cases. Although the coordinate system of the two indicatrices are chosen to be the same for simplicity, in general they may have different orientations. At hyperbolic points the Dupin's indicatrix is a set of conjugate hyperbolas depending on which side of the tangent plane the normal section is locally lying. However, for simplicity we have only illustrated the cases for one of the conjugate hyperbolas. The Dupin's indicatrices of case 2 upper right in Fig. 6.3 are parallel to each other and do not intersect. This is the case when two surfaces and intersect tangentially at a parabolic point where they have the same principal directions. We assume without loss of generality that and parameter curves are in the directions of the principal directions, where being the principal direction with zero curvature. With these assumptions, we have and thus (6.64) reduces to

Therefore it has a double root with unique direction ( ) for , provided that . When , the two surfaces are at least curvature continuous at , and their Dupin's indicatrices overlap.

Figure 6.3: Dupin's indicatrices of two tangentially intersecting surfaces (adapted from [458])

Implicit-implicit and parametric-implicit intersection cases can be handled in a similar way. For the implicit-implicit intersection case we first equate the normal curvatures of the two implicit surfaces and using (6.34), where the unknowns are the unit tangent vector . We can eliminate one of the component, say , using (6.20) yielding the quadratic equation in and similar to (6.64).

For the parametric-implicit intersection case we equate the normal curvatures (6.29) and (6.34) of the parametric and implicit surfaces where the unknowns are , , , and . We can replace , , in terms of and using (6.17), which leads us to a quadratic equation similar to (6.64). Upon solving the quadratic equation and applying the unit length constraint, we obtain the unit tangent vector.

There are also four distinct cases for implicit-implicit and parametric-implicit intersections, depending on the discriminant of the quadratic equation.