6.3.2 Curvature and curvature vector

The curvature vector of the intersection curve at , being perpendicular to , must lie in the normal plane spanned by and . Thus we can express it as

where and are the coefficients that we need to determine. The normal curvature at in direction is the projection of the curvature vector onto the unit surface normal vector at given by

By projecting (6.24) onto the normals of both surfaces (see Fig. 6.1) we have

where is the angle between and and is evaluated by

Solving the coefficients and from linear system (6.26), and substituting into (6.24) yields

It follows that if we can evaluate the two normal curvatures and at , we are able to obtain the curvature vector of the intersection curve at from (6.28). Note that (6.28) does not depend on the type of surfaces. Let us first derive the normal curvature for a parametric surface. Recall that the curvature vector of the intersection curve is also given by (6.18) considered as a curve on the parametric surface. The normal curvature is obtained by projecting (6.18) onto the unit surface normal

where , , are the second fundamental form coefficients (3.28).

We still need to evaluate , to compute (6.29). Since we know the unit tangent vector of the intersection curve from (6.23), we can find and by taking the dot product on both hand sides of (6.17) with and , which leads to a linear system

where , , are the first fundamental form coefficients given in (3.12). Thus,

where , since we are assuming regular surfaces (see (6.16)). Similarly we can compute the normal curvature of the implicit surface by using (6.21). The projection of curvature vector onto the unit normal vector of the surface, from (6.21), is given by

where , , are the three components of given by (6.23).

Consequently, the curvature of the intersection curve at can be calculated using (6.3), (6.27) and (6.28) as follows: