[Paper Review] Convex Hull of Two Orthogonal Disks

Steven R. Finch|arXiv (Cornell University)|Nov 19, 2012

Point processes and geometric inequalities9 references3 citations

TL;DR

This paper computes exact closed-form expressions for the volume, surface area, and mean width of the convex hulls formed by two orthogonal disks in R³ under three configurations: concentric, touching at a point with offset centers, and maximally separated. Using direct integration and an indirect method linking mean width to integrated mean curvature, the study derives exact results for volume and surface area, with mean width expressed via elliptic integrals in complex forms for non-trivial cases.

ABSTRACT

Three configurations of two perpendicular disks in R^3 are examined, the first in which the disks share centers and the other two in which the disks touch at precisely one point. Volume, surface area and mean width calculations dominate the discussion. Integrated mean curvature also appears as an indirect way to compute mean width.

Motivation & Objective

• To determine exact closed-form expressions for the volume, surface area, and mean width of convex hulls formed by two orthogonal disks in R³.
• To investigate three distinct configurations: concentric disks, disks touching at a single point with offset centers, and maximally separated disks.
• To validate the indirect method of computing mean width via integrated mean curvature for non-polyhedral, smooth-singular bodies.
• To resolve open questions about mean width expressions for the 'two-circle roller' and 'oloid' configurations using advanced elliptic integrals.

Proposed method

• Employs direct integration over planar projections to compute volume and surface area using parametric representations of the convex hull boundary.
• Applies the indirect approach linking mean width to integrated mean curvature, using the formula: $ MW = \frac{1}{2\pi}\int_{\partial\Omega} H\,dS + \frac{1}{4\pi}\sum_{j=1}^{n}\int_{\varepsilon_j} \alpha_j\,ds $, where $ H $ is mean curvature and $ \alpha_j $ is the exterior dihedral angle.
• Uses partial derivatives of height functions to compute mean curvature $ H $ on smooth surface patches.
• Evaluates integrals involving $ \sqrt{1-x^2} $, $ \sqrt{2-x^2} $, and trigonometric substitutions to derive closed forms.
• Applies transformations to elliptic integrals of the first, second, and third kinds to express surface area and mean width in exact form.
• Validates results numerically and compares with known values (e.g., $ VL_1 = 8/3 $, $ VL_2 = \pi $) to confirm correctness.

Experimental results

Research questions

• RQ1 What are the exact closed-form expressions for the volume, surface area, and mean width of the convex hull of two orthogonal disks in R³?
• RQ2 Can the indirect method of computing mean width via integrated mean curvature be rigorously applied to non-polyhedral, piecewise-smooth convex bodies?
• RQ3 How do the geometric invariants—volume, surface area, and mean width—vary with the relative position (centered, touching, or maximally separated) of two orthogonal disks?
• RQ4 What is the exact expression for the mean width of the 'two-circle roller' and 'oloid' configurations, and how do they relate to elliptic integrals?
• RQ5 Can symbolic computation tools resolve the algebraic complexity of mean width expressions involving complex arguments in elliptic integrals?

Key findings

• For concentric orthogonal disks, the volume is exactly $ \frac{8}{3} $, surface area is $ 2(2 + \pi) \approx 10.283 $, and mean width is $ \frac{1}{\pi} \left( 2\sqrt{2} \ln(\sqrt{2} - 1) - 4 \text{Li}_2(-1 + \sqrt{2}) + 4 \right) $.
• For the offset configuration (Example 2), the volume is $ \pi $, surface area is approximately $ 13.92 $, and mean width is approximately $ 2.277 $, with exact expressions involving elliptic integrals.
• For the maximally separated configuration (Example 3), the volume is approximately $ 3.627 $, surface area $ \approx 15.97 $, and mean width $ \approx 2.645 $, with surface area expressed as $ \frac{4}{3}\left[9E(1/9) - 8K(1/9) + 8\Pi(-1/3, 1/9)\right] $.
• The indirect method for computing mean width via integrated mean curvature is confirmed valid for non-polyhedral, piecewise-smooth convex hulls, supporting its use in materials science and astrophysics.
• The 'oloid' configuration yields volume $ \approx 3.052 $ and surface area $ 4\pi $, with exact expressions in terms of incomplete elliptic integrals $ E $ and $ F $.
• The 'two-circle roller' has volume $ \frac{8}{3\sqrt{2}}\gamma $ and surface area $ 8\gamma $, where $ \gamma $ is a complex integral expression involving $ E $, $ F $, and $ \Pi $, with $ \gamma \approx 1.6409 $.