Study notes for Ian Stroud, Boundary Representation Modelling Techniques (Springer, 2006).

The Medial Axis Transform (MAT) is a powerful conceptual tool for shape analysis, offsetting, and feature reasoning. This chapter focuses on computing and using the MAT for solids.

1) What the MAT represents

Informally, the medial axis is the set of centres of maximal inscribed balls. It captures:

  • thickness and local radius information
  • “skeleton” structure of a shape
  • critical points where topology or proximity relationships change

For CAD kernels, MAT ideas show up in:

  • offsetting and shelling
  • feature recognition (thin walls, ribs)
  • collision and clearance reasoning

2) Critical points and dual space

The book discusses:

  • how to identify critical events where the medial structure changes
  • representing relationships in a dual space to simplify computation

3) Algorithms

Two algorithmic approaches are highlighted:

  • multiple start point strategies
  • divide-and-conquer strategies

Both reflect a common theme in geometric computing: you need careful control of numeric robustness and of combinatorial explosion.

4) Negative MAT, subdivision, and offsetting

Negative MAT and subdivision relate to:

  • inward/outward offsets
  • handling concavities
  • producing practical engineering offsets without self-intersections

Chapter outline (from the book)

Major sections

  • 15.2 Mat Of A Solid Object
  • 15.3 Calculating Critical Points
  • 15.5 The Multiple Start Point Algorithm
  • 15.6 The Divide-And-Conquer Algorithm

Selected subsections

  • 15.2 MAT of a solid object
  • 15.2.1 MAT terminology
  • 15.3 Calculating critical points
  • 15.4 Dual space
  • 15.5 The multiple start point algorithm

Implementation notes

  • MAT computation is sensitive to tolerance and to surface/curve intersection accuracy.
  • For engineering use, you often want:
    • an approximate MAT that is stable
    • guaranteed bounds on error
  • Tie MAT tools to verification:
    • offsets can create self-intersections; detect and resolve them systematically.

Where MAT connects to everyday CAD

  • offsets/shells: MAT highlights where offsets pinch off and self-intersect
  • feature inference: thin walls and ribs are “medial” phenomena
  • clearance analysis: distance-to-boundary fields are MAT-adjacent concepts

Practical exercises

  • Compute an approximate medial axis for a 2D polygon and use it to estimate local thickness.