What mathematics would I need to know in order to build a 2.5D raycasting engine

Question

I realize this is a fairly common-type question, but I've been studying Maths in my spare time, fairly slowly at that, and I really want to develop a raycasting engine like the original DOOM.

I know more-or-less the subjects / categories of Maths needed for modern 3D games, such as Linear Algebra, but I don't they all would be applicable for a raycasting engine.

Assume I am starting from the ground-up in Maths, what would I need to know? I assume Algebra, Trigonometry and Geometry would be the most important. I'm wondering if I could pull it off just with that?

Answer 1

A Wolfenstein-style raycaster is really simple. You basically shoot beams horizontally in an arc from one side of the player view to the other edge. When a beam encounters a wall, you get the wall slices height on screen from the distance (e.g. scale_factor / distance). E.g. for this height calculation you don't need to understand anything about matrices etc, just think that you want a smaller height on screen when the distance from viewer is greater.

The "trickiest" part is "shooting" the beam (i.e. determining direction and traversing it in small steps), but that's trivial for any 2d game programmer - just use sin and cos as you would have done for the player movement.

It gets slightly more complicated when adding stuff like textured walls (still easy) and sprite enemies, and you can always explain the thing with difficult mathematics using projection planes and whatnot, but the basic idea is just traversing 2d beams on a 2d grid - if you're language/programming environment has an easy way to output graphics, you can get a basic raycaster running in less than 50 lines of code without any matrices or even vector classes.

Answer 2

Well, if you stick to something very simple then you can do it using only basic linear algebra. I would list the absolute necessities as understanding a) orthogonal coordinate systems; b) matrix multiplication and c) how and why the two are related.

And if you want to actually construct some arbitrary rotation matrices then you'll at least need to understand the basic trigonometric functions and their relation to the circle.

The wikipedia page for Transformation matrix summarizes pretty much everything, and the links from that page has even more information (see 3D projection in particular).

There's also some amount of non-mathematical understanding required of course. If you want to put something together on your own then figuring out how and why to do something might require quite a lot of logical thinking, even when the math in question would turn out to be trivial (which might turn out to be most of the time).

However, I'd say that the most important skill isn't math but the ability to search for information. There is a lot of material in books and on the internet. If there is something you don't understand when reading then you can be quite certain that there are articles and books (or Stack Exchange for that matter) available to explain.

[Edit] Right... It seems I've misunderstood the question, and overshot terribly. See above for a better aimed answer :)