6.4.2 Curvature and curvature vector

The curvature vector (see (6.2)) of the intersection curve at can be expressed as in (6.18) as follows:

To obtain the curvature vector = , we need to determine the coefficients and . Equation (6.69) introduces two constraints on the four unknowns, since the normal components of both sides of (6.69) are the same (see (6.57)). This can be seen clearly if we rewrite (6.69) as follows

where

From (6.70), can be expressed by as follows:

where and are coefficients defined in (6.60) through (6.63), and are coefficients defined as follows:

We still need two more equations to solve for . One additional equation can be obtained by differentiating from at three times (see (6.19)) and projecting the resulting vector equation onto the normal vector , i.e. (see (6.38), (6.39)):

Another additional equation can be obtained from the fact that the curvature vector is perpendicular to the tangent vector , i.e.

Upon substituting (6.72) and (6.73) into (6.76) we can solve the linear system (6.76) and (6.77) for and , and hence, the curvature vector can be computed from (6.69), and the curvature follows immediately from (6.34).

The curvature vector of the implicit-implicit intersection case, as well as the parametric-implicit case, can be obtained by a similar procedure. We need to evaluate for the implicit-implicit case by solving a linear system of three equations. The first linear equation in is derived using (6.21). The second equation is given by equating the projection of the third derivative onto the unit surface normal vector, i.e. (6.42). Finally, the third equation is obtained from the fact that the curvature vector is perpendicular to the tangent vector, i.e. .

The parametric-implicit case can be obtained by solving a linear system of two equations in and . The first linear equation in is derived by equating the projection of the third derivative vector of onto the unit normal vector, i.e. (6.38) and (6.42). The first and second derivatives and appear in (6.43) - (6.45) are replaced in terms of , , and using (6.17) and (6.18). The second equation is given by (6.77).