8.3 Stationary points of curvature of explicit surfaces

We can apply the procedures discussed in Sect. 8.2 to obtain the stationary points of curvature of explicit surfaces [248]. Locally any surface can be expressed as a graph of a differentiable function [76]. Given a point on the parametric surface , we can set an orthogonal Cartesian coordinate system such that -plane coincides with the tangent plane of at and -axis is along the normal at . It follows that in the neighborhood of any parametric surface can be represented in the form , where is a differentiable function with .

We can Taylor expand about as follows

where is a remainder term with and . If we take into account that , we can consider

as the second order approximation of .

If we denote , , and , , as coefficients of the first and second fundamental forms of the surface, and assume further that and axes are directed along the principal directions at , assuming is not an umbilic, then [76]. It follows that , since (see Equation (3.65)). Although we have assumed is not an umbilic, we can show that will also vanish when the point is an umbilic (see Sect. 9.2). Also the principal curvatures at when can be expressed as follows:

where the minus signs are due to convention (b) of the normal curvature (see Fig. 3.7 (b)).

If we set and and assuming that is a nonplanar point, the surface can be written locally as a second order approximation in the nonparametric form given by

Its corresponding parametric form is

Equation (8.77) or (8.78) represents an explicit quadratic surfaces which can be categorized into four types according to combinations of and as listed in Table 8.1. The four types of explicit quadratic surfaces are depicted in Fig. 8.9.

Table 8.1. Four types of explicit quadratic surfaces according to and (adapted from [248])

Signs of and	Types of surfaces	Types of points at
	Hyperbolic paraboloid	Hyperbolic point
and	Elliptic paraboloid	Elliptic point
	Paraboloid of revolution	Umbilical point
or	Parabolic cylinder	Parabolic point

Since any regular surface can be locally approximated in the neighborhood of a point by an explicit quadratic surface to the second order, we examine the stationary points of curvatures of explicit quadratic surfaces as representatives of explicit surfaces. We can apply the procedures in Sect. 8.2 to evaluate the stationary points of curvatures of explicit quadratic surfaces. We will only examine the stationary points of principal curvatures, since those of Gaussian and mean curvatures can be found in a similar way. In the sequel we assume that and without loss of generality. It follows that at (0,0,0) (8.76) holds and the -axis will be the direction of maximum principal curvature and -axis will be the direction for the minimum principal curvature. The surface is a hyperbolic paraboloid when , an elliptic paraboloid when , a paraboloid of revolution when , and a parabolic cylinder when . The paraboloid and the parabolic cylinder can be considered as degenerate cases of the elliptic paraboloid.

Gaussian curvature (see (3.66)) and mean curvature (see (3.67) in Table 3.2) are readily evaluated

and hence the principal curvatures become

The stationary points of principal curvature in the domain must satisfy the simultaneous equations (8.66). The equations not only find the stationary points of principal curvatures but also find the locations of the umbilics (see Sect. 8.2.3 and [255]). For explicit quadratic surface , , , , (see (8.61) - (8.65)) are given by

We can reduce the two simultaneous bivariate irrational equations (8.66) involving polynomials and square roots of polynomials into a system of three nonlinear polynomial equations in three variables through the introduction of auxiliary variables [255,254] (see Sect. 4.5). As this is a system of low degree polynomial equations, we can solve these equations analytically using a symbolic manipulation program such as MATHEMATICA [446], MAPLE [51].

Since the maximum principal curvature can be obtained in a similar manner, we will only focus on the minimum principal curvature. Provided we find all the real roots, we check first if the roots are umbilics or not by substituting the roots into . If the roots do not satisfy this equation, the points are not umbilics and we can use Theorem 7.3.1 to classify the stationary points of minimum principal curvature. To apply the extrema theory of functions to the minimum principal curvature function, the second order partial derivatives of the minimum principal curvatures are required; however, we avoid to present these here, since they are extremely lengthy. If the points are umbilics, we need to use a specialized criterion [257] (see Theorem 9.5.1), to check if the point is a local extremum of the principal curvature, since the curvature function is not differentiable at an umbilic.

Table 8.2: Classification of roots according to types of explicit quadratic surfaces

Types of surfaces	Hyperbolic	Elliptic paraboloid
	paraboloid
Signs of and
# of real roots	1	3
Roots	(0,0)	(0,0)
Classification	Minimum	Minimum	Lemon type umbilics
at roots
Types of surfaces	Paraboloid	Parabolic cylinder
	of revolution
Signs of and
# of real roots	1
Roots	(0,0)	Along -axis
Classification	Minimum	Minima
at roots

All the real roots of (8.66) and their classifications are listed in Table 8.2. The hyperbolic paraboloid has only one real root , which corresponds to point (0,0,0) on the surface, and gives a minimum of the minimum principal curvature function with according to the extrema theory. Since there is no other extremum, this minimum is also a global minimum. The elliptic paraboloid has three real roots and . The first root corresponds to a non-umbilical point (0,0,0) on the surface, while the other two real roots correspond to generic lemon type umbilical points on the surface as shown in Fig. 9.1. Using the extrema theory of functions, it follows that the root gives a minimum of the minimum principal curvature function with . Using the criterion in Theorem 9.5.1, we can find that the roots corresponding to the two lemon type umbilics do not provide extrema of the minimum principal curvature function. Consequently the minimum is a global minimum.

As approaches , two generic umbilics merge to one non-generic umbilic at , and the surface reduces to a paraboloid of revolution as illustrated in Fig. 9.1. The paraboloid of revolution has only one real root , which corresponds to point (0,0,0) on the surface, and is a non-generic umbilical point as mentioned above. Since the root corresponds to an umbilical point, we cannot use the extrema theory of functions, nor can we use the criterion in Theorem 9.5.1, since all the second derivatives of (see (9.25)) vanish. But it is apparent that the paraboloid of revolution has a global minimum of the minimum principal curvature function at the umbilic with , since the paraboloid of revolution can be constructed by rotating a parabola, which has a global minimum of its curvature at the origin, around the -axis. Here we are employing the sign convention (b) (see Fig. 3.7 (b) and Table 3.2) of the curvature.

In the case of a parabolic cylinder, the minimum principal curvature reduces to a simple univariate function

It is easy to show that (8.87) has a global minimum at . Therefore the parabolic cylinder has global minima with value along the -axis. These discussions lead to the following lemma [248]:

Lemma 8.3.1. The minimum principal curvature function of explicit quadratic surfaces, except for the parabolic cylinder, has only one extremum at with value , which corresponds to the point on the surface, and it is a global minimum. The parabolic cylinder has global minima at , which corresponds to the -axis on the surface, and has no other extrema.

Similarly we can deduce the following lemma [248].

Lemma 8.3.2. The maximum principal curvature function of explicit quadratic surfaces, except for the parabolic cylinder has only one extremum at with value , which corresponds to the point on the surface, and it is a global minimum for elliptic paraboloid and paraboloid of revolution, while it is a global maximum for hyperbolic paraboloid (note that for hyperbolic paraboloid is negative). The maximum principal curvature function of parabolic cylinders is zero everywhere.

Note that Lemmata 8.3.1 and 8.3.2 are based on convention (b) of the normal curvature (see Fig. 3.7 (b)), and will be used in Sect. 11.3.4.