About this Topic - Mathematical Tools for Shape Analysis and Description - page 4

Figure 1.3: What do similar and different mean?

ese considerations suggest a more operational definition, where the shape is a property
of both a set of objects and a particular method of observation, or analysis. Starting from this
assumption, the mathematics discussed in this topic is related to approaches which can be used
to define the universe of objects and the methods of observation. Spaces, metrics, invariants and
transformations are presented at the theoretical and application level, to explain how to derive
shape information from objects stored in the form of geometric models. e topic cannot cover
the wealth of mathematical knowledge pertaining to 3D shape analysis, but it attempts to group
recent, and rather complex mathematics, which is necessary to understand and apply techniques
at the state-of-the-art. With this topic readers will familiarize themselves with basic concepts
of geometry and topology, then proceed to more advanced concepts in differential geometry and
topology, up to algebraic topology.

1.3 WHAT THIS TOPIC IS AND WHAT IT IS NOT

e topic is meant to be:

• A guide to the mathematics needed for 3D shape analysis, ranging from concepts in differ-
ential geometry to notions of algebraic topology;

• An introductory reading to problems, solutions, and applications of mathematics for 3D
shape analysis;

• A window into the literature on 3D shape analysis dealing with spaces, metrics, invariants
and transformations;