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.

# Evenly Distributing Points in a Triangle.

Most two dimensional quasirandom methods focus on sampling over a unit square. However, sampling evenly over the triangle is also very important in computer graphics. Therefore, I describe a simple and direct construction method for a point sequence to evenly cover an arbitrary shaped triangle.

# 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.

# The Unreasonable Effectiveness

of Quasirandom SequencesI present a new low discrepancy quasirandom sequence that offers substantial improvement over current state-of-the art sequences eg Sobol, Niederreiter,…

# Going beyond the Golden Ratio.

I show that for the same reason that the golden ratio, $\phi=1.6180334..$, can be considered the most irrational number, that $1+\sqrt{2}$ can be considered the 2nd most irrational number, and indeed why $(9+\sqrt{221})/10$ can be considered the 3rd most irrational number.

# Trigonometry in Pictures

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

# A simple method to construct isotropic quasirandom

blue noise point sequencesI describe a simple method for constructing a sequence of points, that is based on a low discrepancy quasirandom sequence but exhibits enhanced isotropic blue noise properties. This results in fast convergence rates with minimal aliasing artifacts.

# Evenly distributing points on a sphere

How to distribute points on the surface of a sphere as evenly as possibly is an incredibly important problem in maths, science and computing, and mapping the Fibonacci lattice onto the surface of a sphere via equal-area projection is an extremely fast and effective approximate method to achieve this. I show that with only minor modifications it can be made even better.

# The Theil-Sen and Siegel non-parametric estimators for linear regression

The Theil-Sen and Siegel estimators are non-parametric distribution-free methods used to fit a line to data, in ways that are very robust to large levels of noise and outliers. We briefly illustrate how the lesser-known Siegel estimator is typically better than the more commonly used Theil-Sen estimator.

# 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.

# The Fisher-Yates Algorithm

Creating unbiased random permutations of lists is often crucial to sampling. The Fisher-Yates shuffle is the definitive method to shuffle a sequence of items. Popularised by Knuth, it is unbiased, has optimal linear time efficiency; uses constant space; and is incremental.

# A probabilistic approach

to fractional factorial designI describe a probabilistic alternative to fractional factorial design based on the Sobol’ low discrepancy quasirandom sequence. This method is robust to aliasing (confounders), is often simpler to implement than traditional fractional factorial sample designs, and produces more accurate results than simple random sampling.

# Multiple Pendulums

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

# Efficient methods to estimate accuracy and variance

for quasi-monte carlo integration.In this brief post, I describes various methods to estimate uncertainty levels associated with numerical approximations of integrals based on quasirandom Monte Carlo (QMC) methods.