3.3 Second fundamental form II (curvature)

Figure 3.6: Definition of normal curvature

In order to quantify the curvatures of a surface , we consider a curve on which passes through point as shown in Fig. 3.6. The unit tangent vector and the unit normal vector of the curve at point are related by (2.20) as follows:

where is the normal curvature vector and is the geodesic curvature vector which are the components of the curvature vector of in the surface normal direction and in the direction perpendicular to in the surface tangent plane. Thus, the normal curvature vector can be expressed as

where is called the normal curvature of the surface at in the direction . In other words, is the magnitude of the projection of onto the surface normal at , with a sign determined by the orientation of the surface normal at .

By differentiating along the curve with respect to we obtain

thus

where

Since and are perpendicular to , we have and , and hence we have an alternative expression for , and

Computation of curvatures at points where the surface representation is degenerate (see Sect. 1.3.6) is given in [453].

The numerator of (3.26) is the second fundamental form , i.e.

and , , are called second fundamental form coefficients. Therefore the normal curvature is given by

where is the direction of the tangent line to at . We can observe that at a given point on the surface depends only on which leads to the following theorem due to Meusnier.

Theorem 3.3.1. All curves lying on a surface passing through a given point with the same tangent line have the same normal curvature at this point.

Using this theorem we can say that the normal curvature is positive when the center of the curvature of the normal section curve, which is a curve through cut out by a plane that contains and is on the same side of the surface normal (see Fig. 3.7 (a)). Sometimes the positive normal curvature is defined in the opposite direction, i.e. the center of curvature of the normal section curve is on the opposite side of the surface normal as illustrated in Fig. 3.7 (b). In such cases (3.23) (3.30) become

The latter convention is often used in the area of offset curves and surfaces in the context of NC machining. Throughout this book we refer to the first convention as convention (a) and to the second one as convention (b). We have listed all the equations, which involve changes due to this convention in the last page of this chapter.

Figure 3.7: Definition of positive normal curvature: (a) , (b)

Figure 3.8: Geometrical illustration of the second fundamental form

Suppose is a point on a surface and is a point in the neighborhood of and is the surface containing and , as in Fig. 3.8. Now suppose and are the points and , then Taylor's expansion gives

Therefore

Thus using (3.28), (3.29), the projection of onto is

where the higher order terms are neglected and since , we get

Thus is equal to twice the distance from to the tangent plane of the surface at within second order terms. We want to observe in which situation is positive and negative or in other words we want to examine in which side of the tangent plane lies. When , (3.35) becomes , which can be considered as a quadratic equation in terms of or . If we solve for , assuming , we obtain

which leads us to the following four cases:

• If , there is no real root. This means there is no intersection between the surface and its tangent plane except at point . Point is called elliptic point (Fig. 3.9(a)). For example, an ellipsoid consists entirely of elliptic points.
• If and , there are double roots. The surface intersects its tangent plane with one line , which passes through point . Point is called parabolic point (Fig. 3.9(b)). For example, a circular cylinder consists entirely of parabolic points.
• If , there are two roots. The surface intersects its tangent plane with two lines , which intersect at point Point is called hyperbolic point (Fig. 3.9(c)). For example, a hyperboloid of revolution consists entirely of hyperbolic points.
• If , the surface and the tangent plane have a contact of higher order than in the preceding cases. Point is called a flat or planar point.

If and , we can solve for instead of . If and , we have , thus the iso-parametric lines , will be the two intersection lines.

Figure 3.9: (a) Elliptic point; (b) parabolic point; (c) hyperbolic point