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 out the source code quite a bit and used lengthy variable names in order to make the source code easier to understand.

This project is an application of mathematical concepts I had recently learned in school. Specifically, I wanted to reimplement all of the necessary math functions needed for rendering a parameterized torus without using any libraries. Instead, I would rely on basic floating point arithmetic to derive numerical approximations for more complicated functions.

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 using modular arithmetic. In order to compute this power series, I need exponentiation and the factorial function. However, neither of these operations are primitive arithmetic operators in the C programming language. For exponentiation of floats to non-negative integers, I used iterated multiplication. 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 (vector) 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 determine luminosity. A “Z” buffer is used to prevent clipping, as the ray will often intersect multiple surfaces on the moving torus.