![]() ![]() My name is Dr Martin Roberts, and I’m a freelance Principal Data Science consultant, who loves working at the intersection of maths and computing. Vanderhaeghe Spherical Fibonacci Mapping.Saff: A Comparison of poopular point confugrations on $S^2$“:.$$ t_i = (x_i,y_i) = \left( \frac$ by up to 0.2%, so for most applications it is probably not worth implementing. ![]() The usual modern definition of the Fibonacci lattice (see figure 2, top row), which evenly distributes $n$ points inside a unit square $[0, 1)^2$ is: This method was apparently inspired by phyllotaxis (which describes the seed distribution on the head of a sunflower or a pine cone).įurthermore, unlike many of the other iterative or randomised methods such a simulated annealing, the Fibonacci spiral is one of the few direct construction methods that works for arbitrary $n$. Of these near-optimal solutions, one of the most common simple methods is one based on the Fibonacci lattice or the Golden Spiral. Therefore, in nearly all situations, we can merely hope to find near-optimal solutions to this problem. Unfortunately, except for a small handful of cases, it still has not been exactly solved. It is of huge signficance in many areas of mathematics, physics, chemistry and computing – including numerical analysis, approximation theory, crystallography, viral morphology, electrostatics, coding theory, and computer graphics, to name just a few. The problem of how to evenly distribute points on a sphere has a very long history. This appeared on the front page of HackerNews in August 2021, and the discussion can be found here. This is an updated version of my most popular post “Evenly distributing points on a sphere”. A simple modification to the canonical Fibonacci lattice can result in an improvement of up to 8.3% in packing distance (maximum nearest neighboring distance). I show how small modifications to the canonical implementation can result in notable improvements for nearest-neighbor measures. 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. ![]()
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