The Medial Axis of a Multi-Layered Environment and its Application as a Navigation Mesh

TL;DR

This paper extends the medial axis and ECM navigation mesh to 3D multi-layered environments, enabling efficient real-time path planning for virtual characters constrained to walkable surfaces.

Contribution

It provides formal definitions, size bounds, and an improved algorithm for constructing the medial axis in 3D multi-layered environments, facilitating efficient navigation mesh computation.

Findings

Medial axis size is O(n) for n boundary vertices.

Constructs medial axis in O(n log n log k) time.

Enables real-time path planning for large virtual crowds.

Abstract

Path planning for walking characters in complicated virtual environments is a fundamental task in simulations and games. A navigation mesh is a data structure that allows efficient path planning. The Explicit Corridor Map (ECM) is a navigation mesh based on the medial axis. It enables path planning for disk-shaped characters of any radius. In this paper, we formally extend the medial axis (and therefore the ECM) to 3D environments in which characters are constrained to walkable surfaces. Typical examples of such environments are multi-storey buildings, train stations, and sports stadiums. We give improved definitions of a walkable environment (WE: a description of walkable surfaces in 3D) and a multi-layered environment (MLE: a subdivision of a WE into connected layers). We define the medial axis of such environments based on projected distances on the ground plane. For an MLE with $n$ boundary vertices and $k$ connections, we show that the medial axis has size $O(n)$, and we present an improved algorithm that constructs the medial axis in $O(n \log n \log k)$ time. The medial axis can be annotated with nearest-obstacle information to obtain the ECM navigation mesh. Our implementations show that the ECM can be computed efficiently for large 2D and multi-layered environments, and that it can be used to compute paths within milliseconds. This enables simulations of large virtual crowds of heterogeneous characters in real-time.