[Paper Review] Oblique Circular Cones and Cylinders

Steven R. Finch|arXiv (Cornell University)|Dec 24, 2012

Point processes and geometric inequalities8 references3 citations

TL;DR

This paper derives exact formulas for the surface area, mean width, and mean curvature of oblique circular cones and cylinders in ℝ³, introducing novel expressions for mean width that are new in the literature. The key contribution is solving an optimization problem to find the minimum ratio of smaller to larger measures (surface area or mean width) when splitting an oblique cone along its axial plane, yielding a minimum ratio of approximately 0.8431 for mean width and 0.5946 for surface area under different assumptions.

ABSTRACT

Surface area and mean width of a cylinder (the convex hull of two parallel disks) in R^3 are computed. It is more difficult to obtain analogous results for a cone (the convex hull of a disk D and a point p). Oblique formulas for mean width, as well as those for mean curvature, are new. Let L denote the unique diameter of D whose endpoints are equidistant from p. We conclude with a question involving the plane that bisects the cone and contains {p,L}, as p varies. What is the minimum ratio of the smaller measure to the larger?

Motivation & Objective

• To derive exact analytical expressions for surface area and mean width of oblique circular cones and cylinders in ℝ³.
• To address the lack of prior closed-form results for mean width in the oblique case, particularly for cones.
• To investigate the optimization of the ratio between measures (surface area or mean width) of the two halves when an oblique cone is split along its axial plane.
• To resolve a geometric optimization problem involving the minimum ratio of smaller to larger measures as the apex point varies.
• To clarify the asymmetry in how surface area and mean width are treated when splitting a cone into two half-cones.

Proposed method

• Parametric representation of the lateral surface of oblique cylinders and cones using parameters u and v, with partial derivatives used to compute the first fundamental form.
• Computation of surface area via integration of the square root of the determinant of the first fundamental form, involving complete elliptic integrals of the second kind.
• Derivation of mean width using the indirect approach: integration of mean curvature over the surface and summation of exterior dihedral angles along edges.
• Use of normal vector fields and second fundamental form components (L, M, N) to compute mean curvature H.
• Asymptotic analysis as b → 0⁺ to study limiting behavior of integrals involving arccosine and square roots.
• Numerical solution of transcendental equations derived from differentiation of the objective function to find optimal parameters.

Experimental results

Research questions

• RQ1 What is the exact expression for the mean width of an oblique circular cone, and how does it differ from that of a right cone?
• RQ2 What is the minimum value of the ratio of the smaller to the larger surface area when an oblique cone is split along its axial plane?
• RQ3 What is the minimum value of the ratio of the smaller to the larger mean width for the two half-cones resulting from axial splitting of an oblique cone?
• RQ4 How does the treatment of overlapping triangular faces affect the optimization of surface area ratios in the half-cone decomposition?
• RQ5 What is the optimal apex position (parameterized by a) that minimizes the ratio of smaller to larger measures under different assumptions?

Key findings

• The paper derives a new closed-form expression for the mean width of an oblique circular cone, which is the first such result in the literature.
• The minimum ratio of the smaller to larger mean width for the two half-cones is approximately 0.8431, achieved when the apex parameter a ≈ 1.3638.
• For surface area, under a revised model accounting for overlapping triangular faces, the minimum ratio is approximately 0.5946, occurring at a ≈ 1.2438.
• The optimization problem for mean width leads to a transcendental equation: √(a²−1)(2 + π√(a²+1))/(2a²) = 1 + arccsc(a), solved numerically.
• The surface area ratio optimization under the same model yields the identical equation √(a²−1)π/(2a) = arccsc(a), confirming consistency in the optimal parameter.
• The analysis reveals a fundamental asymmetry: mean width is additive across half-cones (MW₁ + MW₂ > original MW), while surface area is constrained by fixed material (AR₁ + AR₂ = original AR).