Wave fronts and caustics in the tropical plane

TL;DR

This work develops an intrinsic tropical geometry of the real affine plane by treating tropical isomorphisms as translations together with $GL_2(\mathbb{Z})$ actions, and formulates a wave-front evolution for convex domains in the tropical plane. It proves that any compact convex domain evolves into a finite polygon for arbitrarily small times and that the resulting caustic is a tropical analytic curve; it further links the caustic structure of tropical angles to continued fractions via tropical trigonometry. A Noether-type balance, a dual-fan evolution theory, and a Huygens-type propagation principle are established, with deep connections to toric surfaces and their singularities. The paper also develops a particle-trajectory viewpoint for caustics, analyzes final singularities and lengths of caustics, and provides a detailed tropical account of continued fractions and Hirzebruch–Jung resolutions. Together, these results illuminate the interplay between tropical geometry, convex geometry, and toric torus actions, with implications for tropical wave-front dynamics and the combinatorics of caustics.

Abstract

The paper studies intrinsic geometry in the tropical plane. Tropical structure in the real affine $n$-space is determined by the integer tangent vectors. Tropical isomorphisms are affine transformations preserving the integer lattice of the tangent space, they may be identified with the group $\operatorname{GL_n}(\mathbb{Z})$ extended by arbitrary real translations. This geometric structure allows one to define wave front propagation for boundaries of convex domains. Interestingly enough, an arbitrary compact convex domain in the tropical plane evolves to a finite polygon after an arbitrarily small time. The caustic of a wave front evolution is a tropical analytic curve. The paper studies geometry of the tropical wave fronts and caustics. In particular, we relate the caustic of a tropical angle to the continued fraction expression of its slope, and treat it as a tropical trigonometry notion.

Figures (12)

• Figure 1: A cone $\Sigma\subset N_{\mathbb R}$ (on the left) with its caustic, and its dual cone $\Sigma^*\subset M_{\mathbb R}$ with the intervals of $\mathcal{P}_\Sigma$ (on the right).
• Figure 2: An interim collision of a particle of weight 1 and a particle of weight $n$ in $N_{\mathbb R}$ and the dual diagram in $M_{\mathbb R}$.
• Figure 3: Sixteen types of lattice polygons (up to automorphisms of the lattice) with a single lattice point in the interior together with their tropical caustics. Each caustic consists of segments connecting the central point to the vertices. The polygons are paired by duality, the multiplicity of an edge of the caustic is the length of the corresponding side of the dual polygon. Note that some of these types are self-dual.
• Figure 4: Caustics near an endpoint of of $\Phi(t_\Phi)$ and their local dual polygons.
• Figure 5: A polygonal tropical wave front of a lattice triangle.

Theorems & Definitions (104)

• Proposition 1
• proof
• Definition 2
• Definition 3
• Definition 4
• Proposition 5
• Proposition 6
• proof
• Definition 7
• Remark 8