6.4: Parallel Surfaces and Mean Curvature

The concept of a parallel surface to a smooth surface in three-dimensional space is analogous to that of a parallel curve to a smooth curve in the plane. In the plane, we saw that the properties of parallel curves were closely related to the notion of curvature; it is the same for surfaces in three-space. For a surface X(u,v) we define the parallel surface at distance r by

X^r(u,v) = X(u,v) + rN(u,v).

The parallel surface X^r(u,v) will be regular at all points where the partial derivative vectors are not linearly dependent, i.e. when X^r_u(u,v) × X^r_v(u,v) is not the zero vector.

We may compute the partial derivatives as follows:

X^r_u(u,v) = X_u(u,v) + rN_u(u,v)
X^r_v(u,v) = X_v(u,v) + rN_v(u,v)

X^r_u(u,v) × X^r _u(u,v) = (X_u(u,v)+rN_u(u,v)) × (X_v(u,v)+rN _v(u,v))
or
X^r_u(u,v) × X^r_u(u,v) = X_u(u,v) × X_v(u,v) + rX _u(u,v) × N_v(u,v) + rX_v(u,v) × N_u(u,v) + r²N_u(u,v) × N_v( u,v)

We have already defined N_u(u,v) × N_v(u,v) = Κ(u,v)X_u(u,v) × X_v(u,v). Since X_u(u,v) × N_v(u,v) is the cross product of two vectors in the tangent plane, it also is a multiple of X_u(u,v) × X_v(u,v) and so is X_v(u,v) × N_u(u,v). We may then define another scalar function H(u,v) over the domain of the surface by the condition

X_u(u,v) × N_v(u,v) + X_v(u,v) × N_u(u,v) = -2H(u,v)X_u(u,v) × X_v(u,v)

The scalar function H(u,v), for reasons that will become clear very soon, is called the mean curvature of the surface.

We can now express a fundamental property of parallel surfaces, namely that a parallel surface fails to be regular if and only if

X^r_u(u,v) × X^r _v(u,v) = (1-2rH(u,v)+r²Κ(u,v))X_u(u,v) × X_v(u,v) = 0

This will occur if and only if r is a root of the quadratic equation

1 - 2rH(u,v) + r²Κ(u,v) = 0

r = (H ± (H² - Κ))/Κ.

These two values are called the Radii of Curvature of the surface at the point. The reciprocals of the radii of curvature are called the Principal Curvatures. By an algebraic manipulation, the principal curvatures are given by the formulae

κ₁ = H - (H²-Κ)
κ₂=H + (H²-Κ)

Κ=κ₁κ₂     and
H=(κ₁+κ₂)/2   

Note that we are justified in writing (H²-Κ) because

H²-Κ = ¼(κ₁²+2κ₁κ₂+κ₂²) - κ₁κ₂
= ¼(κ₁-κ₂)²                

A point for which κ₁ = κ₂ is called umbilic.

This demonstration shows the surface X(u,v) and a single parallel surface X^r(u,v), as well as their respective domains. The distance between the original surface and the parallel surface is determined by the variable r. In the domain window, the grid indicates the domain of the surface and the color-coded square patch serves as the domain for the parallel surface. The third window displays the normal map of the both the original surface and the parallel surface.

The colors of the patch, oriented clockwise, are magenta, blue, white and red. When the parallel surface lies on the originl surface (i.e. when r = 0), the orientation of the square patch on the normal map is reversed; instead of having a clockwise orientation, the colors have a counterclockwise orientation. As the value of r is increased past 0, the parallel surface rises. Note that the Gauss map of the surface and the parallel surface remain the same. At a certain value of r, the parallel surface begins to fold over on itself and the orientation of the normal image changes as a result. A similar effect occurs for negative values of r.

Exercises

1. Describe the parallel surfaces of a sphere of radius R. For which values of r will the parallel surface at a distance r have singularities? What is the nature of the singularities?

2. Same question for the hyperbolic paraboloid.

3. Same question for a circular cylinder, or, more generally, an elliptical cylinder.

4. Same question for a paraboloid, ellipsoid, or hyperboloid of revolution.

5. Same question (much more difficult) for a general paraboloid, ellipsoid, or hyperboloid.