Work

Distinct Distances on Pfaffian Curves

Year: 2026

Type: article

Abstract: We generalize Pach and de Zeeuw’s lower bound for distinct distances between points on two curves, from algebraic curves to Pfaffian curves. Pfaffian curves include those that can be defined by any combination of elementary functions, including exponential and logarithmic functions, rational and irrational powers, trigonometric functions and their inverses, integration, and more. The bound remains $$\Omega (\min \{m^{3/4}n^{3/4},m^2,n^2\})$$ Ω ( min { m 3 / 4 n 3 / 4 , m 2 , n 2 } ) , as obtained from the proximity technique of Solymosi and Zahl.

Source: Discrete & Computational Geometry

DOI: 10.1007/s00454-026-00872-4

Authors: Abhiram Natarajan, Adam Sheffer

Institutions: University of Warwick, Baruch College

Language: English

Topic: Computational Geometry and Mesh Generation

Subfield: Computer Graphics and Computer-Aided Design

Field: Computer Science

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,090

Funders: Styrelsen för Internationellt Utvecklingssamarbete, Engineering and Physical Sciences Research Council