What's the mathematics behind 3D modelling?

Question

I'm highly interested about 3D modelling in software, and I know that it has some deep mathematics behind it too. I would like to learn what specific topics are behind it mathematically. As long as I know the primary topics behind 3D modelling are, linear algebra, topology and differential geometry (if I'm wrong please correct me). On the other hand, what specific topics in these areas are more important? Which other areas of mathematics are useful?

Lastly, I'm currently studying Topology from Munkres' books. I would like to hear what other books or resources you advice to study the advanced mathematics behind (and partially or fully related to) 3D modelling, in order to do mathematics research in these areas?

Answer 1

I am totally not into this topic, but once I came across the book "Topology for Computing" written by Afra Zomorodian. He uses morse theory, homotopy theory, group theory, topology and much more complex stuff in graphics and surface analysis. The good thing about his book is that mathematics in it is kept with perfect rigour. You should take a look at it.

Answer 2

I'm highly interested about 3D modelling in software, and I know that it has some deep mathematics behind it too.

You give no indication of what aspect of 3D modelling or what application domain you are interested in. The field is vast. Lots of software is offered, even more for engineering.

On the other hand, what specific topics in these areas are more important? Which other areas of mathematics are useful?

Imagine:

• a 3D mesh used for a computer game
• a 3D mesh used for calculating the temperature along a gas turbine blade during casting
• a 3D mesh used for calculating the stress tensor of a motor block
• a 3D mesh used for solving a wave equation of some 3D domain

The constraints on these meshes range from

• geometrical aspects
• the impact on rendering time
• numerical precision
• cost to model by tools / humans etc etc

So you will find such meshes showing up in literature from mathematics, computer science, physics, mechanical engineering, electrical engineering, castings engineering, aerodynamics etc etc.