Mapping the Fibonacci lattice (aka Golden Spiral, aka Fibonacci Sphere) onto the surface of a sphere is an extremely fast and effective approximate method to evenly distribute points on a sphere. I show how small modifications to the canonical implementation can result in notable improvements for nearest-neighbor measures.
For many Monte Carlo methods, such as those in graphical computing, it is critical to uniformly sample from $d$-dimensional spheres and balls. This post describes over twenty different methods to uniformly random sample from the (surface of) a $d$-dimensional sphere or the (interior of) a $d$-dimensional ball.
This post shows how the core trigonometric definitions, relations and addition theorems can be simply and intuitively visualized.
Unlike for circles, there isn’t a simple exact closed formula for the perimeter of an ellipse. We compare several well-known approximations, and conclude that a formula discovered by Ramanujan is our favourite, due to its simplicity and extreme accuracy.
This post illustrates and explains the beautiful Lissajous Curves – trajectories of points whose coordinates follow sinusoidal movements. A simple but timeless classic curve that has numerous applications as well as artistic elegance.
If you are given the initial terms of a sequence, then here is an insanely simple method to derive a general formula for the first N terms, as well as the sum of the first N terms of the sequence.
This fun post illustrates the phenomenon of multiple uncoupled pendulums whose periods are all rational multiples of each other.