San José State University |
---|

applet-magic.comThayer WatkinsSilicon Valley & Tornado Alley USA |
---|

Relationships Involving the "Volume" of a Region,the "Area" of its Boundary Surface, and the Average Distance from an Interior Point to that Boundary |
---|

There are some interesting relationships between regions and their boundaries that extend over several dimensions.

Let *A* be a convex two dimensional regional surrounding the origin of polar coordinates and *C* its perimeter.
Suppose the coordinates of the perimeter are given by r=R(θ).

The area A of *A* can be considered to be made up of the areas of a set of triangles. They are not necessarily isoceles triangles.
One side is of length R(θ) and the other longer by the amount dR=(dR/dθ)dθ.
The average of the two edges is then R(θ)+½((dR/dθ)dθ.
Each triangle has a base width of R(θ)dθ. The area of each triangle is then

but the product of

infinitesimals is

insignificant

compared to terms

involving only dθ

and may be dropped

So the area of each triangle is

Thus the total area is

A = ½∫

The circumference C of the perimeter is given by summation of the lengths of the edges along the perimeter; i.e.,

The Mean Value Theorem says that

where __x__ is an average of x over the
interval of integration.

Then by the Mean Value Theorem

where __R__ is the average distance from the origin to the perimeter.

That distance is equal to the radius of the circle having the same area as *A*.

Without reference to the Mean Value Theorem __R__ may just as well be defined as

*A* may be moved so that any of its interior point coincides with the origin.
Therefore the average distance from any interior point of *A* to its perimeter
is the same for all interior points..

Let V be a convex three dimensional region surrounding the origin of a spherical coordinate system (r, θ, φ). Let S be its suface. Let this surface be defined by R(θ, φ). The angle θ ranges from 0 to 2π radians and φ ranges from 0 to π radians.

V may be considered to be composed of pyramids with apex points at the origin. The base of each pyramid is a square of dimensions Rdθ by Rdφ. The volume of such a pyramid is

The total volume is therefore

V = ⅓∫

The area S of the surface is

This means that by the Mean Value Theorem

where __R__ is the average radius of the surface S. Its value is equal to the radius of the sphere with the same
volume as V.

Since V can be moved so that any of its interior points coincide with the origin it means that the average distance from any interior point of V to its surface S is the same.

Although the analysis for this case is trivial it is satisfying to see that it exactly matches the analysis in the cases of the other two dimensions.

Let *L* be a line stretching from a to b, where a<0<b. Its length L is given by

Its "surface" is the two end points. The size of this "surface" is 2.
The average distance __x__ from the origin to the "surface" points of the line is

This is just the "volume" L divided by the "size" of its surface.

For a convex region U of dimension D the average distance __R__ from an interior point to its
(D-1)-dimensional surface is

where V_{D} is the D dimesional volume of U and
S_{D-1} is the (D-1) dimesional area of the boundary
surface of U. __R__ is the same for all interior points of U.

applet-magic HOME PAGE OF Thayer Watkins, |