6.2 More differential geometry of curves

Let , , or in vector form be the intersection curve with arc length parametrization. Then from (2.5) and (2.20), we have

where is the unit tangent vector and is the curvature vector, which is the rate of change of the tangent vector. From (6.2) it follows that

Now let us evaluate the third derivative by differentiating (6.2)

where we can replace by the second equation of the Frenet-Serret formulae (2.56) yielding

Since the vectors , , are a right-handed orthonormal triplet, the torsion can be obtained from (6.5) as

provided that the curvature does not vanish.

Classical differential geometry textbooks [412,206,444,76] do not cover the case , which is addressed below following Ye and Maekawa [458]. When (6.2) does not define the unit principal normal vector. To obtain the principal normal vector at points where , higher order derivatives of the curve are involved. If , then the curve is a straight line, and the Frenet frame of the curve is not defined. We assume here that occurs only at isolated points. In such case, (2.56) is valid. If and , the third order derivative (6.4) reduces to

which defines the unit principal normal vector where is obtained from . If and , we need to evaluate the fourth order derivative by differentiating (6.4) yielding

where . In general, if and , then

where .

The evaluation of torsion when the curvature vanishes can be performed as follows. If and , we need to evaluate the fourth order derivative of , i.e. . This can be obtained by differentiating (6.5) and replacing , , using the Frenet-Serret formulae which results in:

In this case (6.10) further reduces to

thus

Similarly we have

and hence, if and , then becomes

In general, if and , then [458]

Let , , and , , or in vector form, and , be the two parametric surfaces. Also, let us denote the two implicit surfaces as and . We assume that these surfaces are all regular. In other words

The unit normal vector of a parametric surface and an implicit surface are given by (3.3) and (3.9).

So far, we have studied the intersection curve independent of the two intersecting surfaces. However, the intersecting curve can also be viewed as a curve on the two intersecting surfaces. A curve , in the -plane defines a curve on a parametric surface , while a curve with constraint defines a curve on an implicit surface .

We can easily derive the first three derivatives of the intersection curve , , as a curve on a parametric surface using the chain rule:

Similarly we can evaluate , and as follows: