Maximal Angle Flow on the Shape Sphere of Triangles

Abstract

There has been recent work using Shape Theory to answer the longstanding and conceptually interesting problem of what is the probability that a triangle is obtuse. This is resolved by three kissing cap-circles of rightness being realized on the shape sphere; integrating up the interiors of these caps readily yields the answer to be 3/4. We now generalize this approach by viewing rightness as a particular value of maximal angle, and then covering the shape sphere with the maximal angle flow. Therein, we discover that the kissing cap-circles of rightness constitute a separatrix. The two qualitatively different regimes of behaviour thus separated both moreover carry distinct analytic pathologies: cusps versus excluded limit points. The equilateral triangles are centres in this flow, whereas the kissing points themselves -- binary collision shapes -- are more interesting and elaborate critical points. The other curves' formulae and associated area integrals are more complicated, and yet remain evaluable. As a particular example, we evaluate the probability that a triangle is Fermat-acute, meaning that its Fermat point is nontrivially located; the critical maximal angle in this case is 120 degrees.

18 pages including 9 figures

References in corpus (9)

• The Problem of Time and Quantum Cosmology in the Relational Particle Mechanics Arena
• Foundations of Relational Particle Dynamics
• The Smallest Shape Spaces. I. Shape Theory Posed, with Example of 3 Points on the Line
• The Smallest Shape Spaces. III. Triangles in 2- and 3-d
• The Smallest Shape Spaces. II. 4 Points in 1-d Suffices to have a Complex Background-Independent Theory of Inhomogeneity
• Manifolds of Projective Shapes
• Alice in Triangleland: Lewis Carroll's Pillow Problem and Variants Solved on Shape Space of Triangles
• Two new versions of Heron's Formula
• Shape (In)dependent Inequalities for Triangleland's Jacobi and Democratic-Linear Ellipticity Quantitities

Cited by in corpus (10)

• The Smallest Shape Spaces. I. Shape Theory Posed, with Example of 3 Points on the Line
• The Smallest Shape Spaces. II. 4 Points in 1-d Suffices to have a Complex Background-Independent Theory of Inhomogeneity
• The Smallest Shape Spaces. III. Triangles in 2- and 3-d
• Monopoles of Twelve Types in 3-Body Problems
• $N$-Body Problem: Minimal $N$'s for Qualitative Nontrivialities
• Topological Shape Theory
• Background Independence: ${\mathbb{S}}^1$ and $\mathbb{R}$ absolute spaces differ greatly in Shape-and-Scale Theory
• Rubber Relationalism: Smallest Graph-Theoretically Nontrivial Leibniz Spaces
• Quadrilaterals in Shape Theory. II. Alternative Derivations of Shape Space: Successes and Limitations
• Isotropy Groups and Kinematic Orbits for 1 and 2-$d$ $N$-Body Problems