Differential Geometry - Computer Graphics and Geometric Modeling - page 610

(3) If K = 0, then the Dupin indicatrix consists of either two parallel lines if one
principal curvature is not zero or is empty if both principal curvatures are zero.

Proof. This is an immediate consequence of Euler's theorem and Theorem 9.9.9(1).

Theorem 9.9.13 motivates the following definition of an elliptic, hyperbolic, or
parabolic point on a surface. Another common definition of these terms can be found
in Exercise 9.9.6.

Definition. Let p be a point of a surface S and let K be the Gauss curvature at p .
The point p is called

(1) elliptic if K > 0,
(2) hyperbolic if K < 0,
(3) parabolic if K = 0 but D n ( p ) π 0 (only one principal curvature vanishes), and
(4) planar or flat if D n ( p ) = 0 (both principal curvatures vanish).

Using Theorem 9.9.9 one can make the following observations. See Figure 9.19
again. At an elliptic point both principal curvatures have the same sign. This
means that all curves through that point must have their normal vector point to the
same side of tangent plane. Spheres and ellipsoids are examples of such surfaces (see
Figure 9.19(a)). At a hyperbolic point p , the principal curvatures have opposite
signs and so one can find two curves through p whose normals at p point to opposite
sides of the tangent plane. A saddle surface, such as the hyperbolic paraboloid

2 2

zx y =-

(see Figure 9.19(b)) and the point (0,0,0), is an example. The cylinder (Figure 9.19(c))
is an example of a parabolic point. One of the principal curvatures is zero, but
the other is not. Points of a plane are example of planar points, but nonplanar sur-
faces can have such points. For example, (0,0,0) is a planar point of the surface of
revolution where the curve z = x 4 is revolved about the z-axis (Exercise 9.9.1). Finally,
we should mention the classical interpretation of the Dupin indicatrix, namely, if we
take the tangent plane at a point p and intersect it with the surface S after moving it
slightly in the normal direction, then the intersection curve will be the same sort of
curve as the Dupin indicatrix. Figure 9.23(a) and (b) shows the intersection with the
moved plane X at an elliptic and parabolic point p , respectively. See [DoCa76] or
[MilP77].

Definition. A point on a surface where the principal normal curvatures are equal
(k 1 = k 2 ) is called an umbilical point .

Planar points are umbilical points.

9.9.14. Theorem. If every point of a connected surface S is an umbilical points,
then S is either contained in a sphere or in a plane.

Proof. See [DoCa76].