The mean width of the oloid and
integral geometric applications of it
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
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 , , and Equations (2), (6), (2) and (8) of the present paper). The surface is part of a developable surface , , , .
Finch  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
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.
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
Now, we able to calculate the surface area of the oloid :
Mathematica evaluates this integral to
Now, we calculate the volume of , and start with
From (3) it follows that
So we have
is the complete elliptic integral of the first kind, and
(see also ).
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
For the unit normal vector one finds
Since is part of a developable surface and the line segments , are part of the generators of (see , ), it is not surprising that does not depend on . The mean curvature in a point of a surface is defined by
It follows that
where is the complete elliptic integral of the first kind (9), hence
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
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
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
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.
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
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
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
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
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
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.
The integral of mean curvature, the surface area and the volume of the parallel body are given by
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
with the notation
|group of proper (orientation-preserving) rotations [11, p. 13],|
|proper rotation, ,|
|Lebesgue measure on ,|
|unique Haar measure on with [11, p. 584],|
where is the volume of the unit -ball (see (19)). Since the intersection of two convex sets is a convex set, we have
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
From (6) it follows that
is the expected volume of . Analogously, we get the expected mean width and the expected surface area:
Clearly, it is possible to reverse the roles of the fixed body and the moving body.