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.