# Generalizations of Ripley's K-function with Application to Space Curves

## TL;DR

This paper develops new, flexible variants of Ripley's K-function tailored for analyzing the spatial structure of curve pieces and surface patches, extending its application beyond point sets.

## Contribution

It introduces generalizations of Ripley's K-function for curves and surfaces, providing a computational approach and theoretical comparison with existing methods.

## Key findings

- New variants of Ripley's K-function effectively analyze curve and surface structures.
- The proposed methods outperform previous approaches in distinguishing different spatial configurations.
- Theoretical analysis confirms the flexibility and applicability of the new generalizations.

## Abstract

The intensity function and Ripley's K-function have been used extensively in the literature to describe the first and second moment structure of spatial point sets. This has many applications including describing the statistical structure of synaptic vesicles. Some attempts have been made to extend Ripley's K-function to curve pieces. Such an extension can be used to describe the statistical structure of muscle fibers and brain fiber tracks. In this paper, we take a computational perspective and construct new and very general variants of Ripley's K-function for curves pieces, surface patches etc. We discuss the method from [Chiu, Stoyan, Kendall, & Mecke 2013] and compare it with our generalizations theoretically, and we give examples demonstrating the difference in their ability to separate sets of curve pieces.