3.5.2 Implicit surfaces

Using (3.66), (3.67) for an explicit surface, we can derive equations for the Gaussian and mean curvature of an implicit surface . At a point where , can be expressed as a function of and , say [166]. In such cases variables and are independent but is a function of both and . Since constantly satisfies the equation , the partial differentiation of with respect to the independent variable (by holding fixed) must vanish [166]. Thus,

where on the left-hand side is considered as being expressed in terms of and only and is held constant in the differentiation with respect to , while on the right-hand side is considered as being expressed in terms of , , and , are held constant in the differentiation. Similarly we have

Consequently we have

The second order partial derivatives , , are provided by differentiating (3.70). For example,

Similarly, we have

Equations (3.70) to (3.73) may be substituted into (3.63) and (3.65) to obtain the first and second fundamental form coefficients, and into (3.66) and (3.67) to compute the Gaussian and mean curvature of an implicit surface. If , alternate formulae may be found by cyclic permutation of , , .

Table 3.1: Classification of implicit quadrics

Implicit Quadrics
Ellipsoid	1	1	1	1
Hyperboloid of One Sheet	1	1	-1	1
	1	-1	1	1
	-1	1	1	1
Hyperboloid of Two Sheets	1	-1	-1	1
	-1	1	-1	1
	-1	-1	1	1
Elliptic Cone	1	1	-1	0
	1	-1	1	0
	-1	1	1	0
Elliptic Cylinder	1	1	0	1
	1	0	1	1
	0	1	1	1
Hyperbolic Cylinder	1	-1	0	1
	-1	1	0	1
	1	0	-1	1
	-1	0	1	1
	0	1	-1	1
	0	-1	1	1

For every quadric surface, it is possible to find a suitable 3-D rotation such that the cross terms , and cancel out in (1.15). If a quadric surface has a center^3.2, its axes can be translated to the center as origin so that the equation of the quadric surface does not have any first degree terms [79]. Therefore after these transformations the implicit quadrics, ellipsoids, hyperboloids of one and two sheets, elliptic cones, elliptic cylinders and hyperbolic cylinders can be expressed in a standard form

where , and take values either -1, 0 or 1 and takes values either 0 or 1, depending on the classification of quadrics (see Table 3.1).

By evaluating (3.70), (3.71), (3.72) and (3.73) for given in (3.74), and substituting into (3.66) and (3.67), we obtain

where satisfy . The principal curvatures can be obtained by substituting (3.75) and (3.77) into

where we will not show the substituted expression because it is too cumbersome.

The curvatures of a hyperbolic cylinder ( , , )

can be obtained by evaluating (3.75), (3.76) and (3.77) resulting

where .

Similarly, the curvatures of an ellipsoid ( )

are evaluated as

where . Here we note that in the derivation of the mean curvature in (3.82), we used (3.81) to simplify the expression. For the case of a sphere of radius , (3.81) simplifies to , and (3.82) and (3.83) simplify to , , which shows that a sphere is made of entirely nonflat umbilics (see Sects. 9.1 and 9.2). The negative sign comes from the sign convention of the curvature (see Fig. 3.7 and Table 3.2).

Finally, the curvatures of an elliptic cone ( , and )

excluding the apex (0,0,0) are given by

where . Here we also used (3.84) to simplify the expression of mean curvature in (3.85).

Footnotes

... center^3.2

A center of a quadric surface is defined as a point bisecting every chord passing through it [79]. Here chord is a line which joins two points on a surface. Ellipsoids and hyperboloids have centers, while paraboloids do not have centers. The elliptic/hyperbolic cylinder is a limiting case of the ellipsoid/hyperboloid and the elliptic cone is asymptotic to hyperboloids of one and two sheets.