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.

*How to generate uniformly random points *

on n-spheres and in n-balls

on n-spheres and in n-balls

*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.*

* Continue reading “How to generate uniformly random points on n-spheres and in n-balls”*

*Trigonometry in Pictures*

*This post shows how the core trigonometric definitions, relations and addition theorems can be simply and intuitively visualized.*

*A Formula for the Perimeter of an Ellipse*

*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.*

* Continue reading “A Formula for the Perimeter of an Ellipse”*

*Lissajous Curves*

*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.*

*A simple formula for Sequences and Series*

*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. *

* Continue reading “A simple formula for Sequences and Series”*

*Multiple Pendulums*

*This fun post illustrates the phenomenon of multiple uncoupled pendulums whose periods are all rational multiples of each other. *

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