The mean width of the oloid andintegral geometric applications of it

The mean width of the oloid and
integral geometric applications of it

Uwe Bäsel
Abstract

The oloid is the convex hull of two circles with equal radius in perpendicular planes so that the center of each circle lies on the other circle. We calculate the mean width of the oloid in two ways, first via the integral of mean curvature, and then directly. Using this result, the surface area and the volume of the parallel body are obtained. Furthermore, we derive the expectations of the mean width, the surface area and the volume of the intersections of a fixed oloid and a moving ball, as well as of a fixed and a moving oloid.
2010 Mathematics Subject Classification: 53A05, 52A15, 52A22, 60D05
Keywords: Oloid, convex hull, integral of mean curvature, mean width, Steiner formula, parallel body, intrinsic volumes, principal kinematic formula

1 Introduction

The oloid is the convex hull of two circles , with equal radius in perpendicular planes so that the center of each circle lies on the other circle (see Figures 2 and 2). Dirnböck & Stachel [4, p. 117] calculated the surface area and the volume of the oloid (see also [14], [15], and Equations (2), (6), (2) and (8) of the present paper). The surface is part of a developable surface [4], [2], [14], [15].

Figure 1: The circles and with
Figure 2: The oloid

Finch [5] calculated surface areas, volumes and mean widths of the convex hulls of three different configurations of two orthogonal disks with equal radius. The mean width of every convex hull is determined twice: 1) using the integral of the mean curvature and the relation , 2) calculating directly.
     According to [4, pp. 105-106], the circles with can be defined by

(1)

(see Fig. 1). A parametrization of the surface [2, p. 165, Eq. (2)] is

(2)

with

(3)

To the authors knowledge, the mean width of the oloid is not aready known. In Section 3 we calculate the mean width of using the integral of mean curvature, and in Section 4 we calculate it directly. With the help of this result we derive the volume, the surface area and the mean width of the parallel body of in Section 5. Using the principal kinematic formula of integral geometry, the expectations of the mean width, the surface area and the volume of the intersections of a fixed oloid and a moving ball, as well as of a fixed and a moving oloid are calculated in Section 6.

2 Preliminaries

In the following, we work in the real vector space with its standard scalar product and its vector product for two vectors , . We denote the partial derivatives

of (see (2)) by , , and so on.
     Using (3), for the coefficients , , of the first fundamental form (see e. g. [6, pp. 87-88], translation: p. 68) we find

(4)

Now, we able to calculate the surface area of the oloid :

(5)

Mathematica evaluates this integral to

(6)

Now, we calculate the volume of , and start with

From (3) it follows that

So we have

(7)

Mathematica finds

(8)

where

(9)

is the complete elliptic integral of the first kind, and

(10)

is the complete elliptic integral of the second kind. A numerical integration integration of (2) and evaluation of (8) with Mathematica yields the decimal expansion

(see also [1]).

3 The integral of mean curvature

The surface of the oloid is piecewise continuously differentiable. We denote by the mean curvature in one point of . The circles and (see (1)) produce two edges and , respectively, that are smooth curves. Let denote the angle between the two unit normal vectors in every point of . Applying the general formula for the integral of the mean curvature (see [12, pp. 76-77, Eqs. (3.5), (3.7)]; cp. the formula for the mean width in [5, p. 3]) to gives

(11)

For the unit normal vector one finds

(12)

Since is part of a developable surface and the line segments , are part of the generators of (see [4], [2]), it is not surprising that does not depend on . The mean curvature in a point of a surface is defined by

where , are the principal curvatures, are the coefficients of the second fundamental form (see e. g. [6, p. 99], translation: p. 79), and , are given by (4). In our case we have

and

It follows that

and

Mathematica finds

(13)

where is the complete elliptic integral of the first kind (9), hence

(14)

A handmade proof for the relation in (13) may be found in Section 7.
     Now we calculate the integral of mean curvature for the edges , (see (11)). Therefore, we consider . The first unit normal vector in a point , is given by (12), the second is . So one gets

hence

We observe that the inverse function of the integrand is equal to the integrand, and hence the graph of the integrand symmetrical with respect to the line . As solution of

we find , hence

(15)

Mathematica and we, too, are not able to solve the last integral. It looks similar to Coxeter’s integral in [7, pp. 194-201]. The NIntegrate-function of Mathematica provides

(16)

hence

From (11), (14), (3), with (9) and (16), it follows that the integral of the mean curvature of is given by

For a convex body , the mean width is given by the relation (see [12, p. 78, Eq. (3.9)]). So we we have proved the following theorem.

Theorem 1.

The mean width of the oloid is

where is the complete elliptic integral of the first kind (9), and

4 Direct calculation of the mean width

Let

be a supporting plane of given in the Hesse normal form. So with , is the normal unit vector of and is the distance of from the origin. intersects the plane in the line

and the plane in the line

The equation of in Hesse normal form is

therefore, the distance of from the center of is

(see e. g. [3, p. 172]). Since is tangent to , we have

Analogously one finds that the distance of from the center of is

hence

It follows that the distance between the support plane and the origin is

Now we use spherical coordinates and as coordinates of the unit normal vector :

So we have

and can write as

Clearly, is the support function of in the direction . Hence the width of in this direction is given by

In order to calculate the mean width of we have to integrate over all directions, hence over the unit hemisphere. Let denote the surface element of the unit sphere, we have

(cp. [12, p. 78, Eq. (3.9)]), where the last equal sign follows from the fact that there are two congruent portions of in the half spaces and . Due to the symmetry of with respect to the planes and we can restrict the spherical coordiates to the intervals and , respectively, hence

(17)

where

is the solution of the equation

for . Numerical integration of (4) with Mathematica gives

5 The parallel body

For a convex body and , the set (Minkowski sum)

is the parallel body of at distance , where is the -ball of radius ,

and is the distance between the point and . The volume of the parallel body is given by the Steiner formula

(18)

where

(19)

is the volume of the -dimensional unit ball , and are the intrinsic volumes of . [11, p. 2, pp. 12-13, p. 600]
     Using the relations in [10, p. 301], where denotes the Euler characteristic, the intrinsic volumes of are

(20)

with (see (10)), (see (9)), and (see (16)). Let denote the parallel body of at distance . Due to (18), its volume is

Applying [12, p. 82, (3.17)] allows to calculate the surface area of :

Clearly, the mean width of is equal to , hence

(see also [12, p. 82, (3.17)]). The results of the following theorem follow immediately.

Theorem 2.

The integral of mean curvature, the surface area and the volume of the parallel body are given by

with

6 Intersections with an oloid

Now, we apply our results and the principal kinematic formula to derive some expectations for the intersections of the oloid and the three-dimensional ball of radius , and of two oloids .
     The principal kinematic formula (see [10, p. 301]) for a fixed convex body and a moving convex body is for given by

(21)

with the notation

group of proper (orientation-preserving) rotations [11, p. 13],
proper rotation, ,
translation vector,
Lebesgue measure on ,
unique Haar measure on with [11, p. 584],

and

where is the volume of the unit -ball (see (19)). Since the intersection of two convex sets is a convex set, we have

(22)

where is the indicator function of the event . So we see that is the measure of the set of rigid motions bringing into a hitting position with (see [11, p. 175], [8, p. 262, p. 267]).
     For , (21) gives

(23)

From (6) it follows that

(24)

is the expected volume of . Analogously, we get the expected mean width and the expected surface area:

(25)
(26)

Clearly, it is possible to reverse the roles of the fixed body and the moving body.

Example 1.

As an example we calculate the expected values (25), (26) and (24) for and , or, equivalently, for and . For the ball one easily gets

(27)

Note that these terms also follow from the general formula

<