Rotating Torus

UPDATE: An browser based rotating torus which uses similar source code can be viewed here!

I recently read about a spinning donut, and decided to render one for myself. My donut is here. I commented it out quite a bit and used relatively lengthy variable names, so hopefully it is easy to understand.

This project is an application of mathematical concepts I had recently learned in school. Namely, I wanted to reimplement all of the necessary math functions needed for rendering a parameterized torus without using any libraries, instead relying on basic floating point arithmetic.

My parameterized torus uses three parameters: one for rotation of the torus about the X and Z axis, one for rotating the circular cross section of the torus about its current axis of revolution, and another for rotating a point around that circle.

The sine function is approximated using a Maclaurin series with eight terms on the interval \( [-\pi/2, \pi/2] \), which is then extended to approximate the function on the domain of all real numbers through modular arithmetic. In order to compute this power series, I need exponentiation and the factorial function. Neither of these are primitive arithmetic operators in the C programming language. For exponentiation of floats to non-negative integers, I used an iterative function which performs a simple repeated multiplication. However, this is a rudimentary exponentiation which ignores edge cases such as indeterminate forms. For the purposes of the Maclaurin series, \( 0^0 \) is assumed to be 1. The square root function is approximated using Newton’s method, which is executed recursively.

The motion of the entire animation is periodic, so the sine (and cosine) functions are memoized such that approximately 1000 values of those functions on the interval \( [0, 2\pi] \) are stored in an array. When rendering each point on the torus, these tables are looked up, instead of recalculating the function on each iteration.

Lighting is done through a single directional light positioned in front of the torus. Rays of light and the normals to the surface of the torus are multiplied as a dot product to determined luminosity. A Z buffer is used to prevent clipping, as often the ray will encounter multiple surfaces of the torus as it passes through.

Next up will be placing this example into the Projects section of this website.