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Fibonacci Sphere

extremelearning.com.au

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Re: Fibonacci Sphere

#31

Post by nmaleki »

extremelearning wrote: post #2

Author here. Happy to try to answer any questions! ;)

Can you explain the notation [0,1)^2 unit square, does the 2 represent the spatial dimensionality? So,[0,1)^3 is the unit cube? Why is 0 inclusive, but the 1 is exclusive?

"The first is that this mapping is area-preserving, not distance-preserving." Which area is being preserved?

Is there a volume preserving choice function?

What are points t0 and t3, are those the location of the singularity points? What is the definition of those "singularity points"? Is it that seeming void in the center of the fibonacci spiral? And that void doesn't exist within the unit square case?

I especially enjoyed footnote #1.

Re: Fibonacci Sphere

#32

Post by jmiskovic »

In past weeks I've done some work on procedural meshes and filling out the space with various layout strategies. The posted article was a good introduction, and then I found this very approachable paper with great overview of different algorithms: "Point Picking and Distributing on the Disc and Sphere".

https://apps.dtic.mil/dtic/tr/fulltext/u2/a626479.pdf

Re: Fibonacci Sphere

#33

Post by extremelearning »

nmaleki wrote: post #31
extremelearning wrote: post #2

Author here. Happy to try to answer any questions! ;)

Can you explain the notation [0,1)^2 unit square, does the 2 represent the spatial dimensionality? So,[0,1)^3 is the unit cube? Why is 0 inclusive, but the 1 is exclusive? "The first is that this mapping is area-preserving, not distance-preserving." Which area is being preserved? Is there a volume preserving choice function? What are points t0 and t3, are those the location of the singularity points? What is the defi…

1. yes, the index represents the dimensionality

So[0,1)^1 is a line interval, [0,1)^2 is a unit square and [0,1)^3 is the unit cube, and [0,1]^d is a d-dimensional cube.

2.Only one boundary can be included

It includes 0 but not 1 because it can only the context is usually that practitioners want a region where one edge will wrap to the opposite edge. Thus they treat [0,1)^2 as if it is actually a 2-dimensional torus.

thus the the 2 boundaries acutally map to the same point, so you can only include one of them. In our case as we are using x %1 = fractional part of x, the fractional part could be 0, if x=3.0, but it could never be exactly 1.

3) the mapping from the circle to the surface of the sphere is described here https://en.wikipedia.org/wiki/Lambert_azimuthal_equal-area_p...

the entire top edge of the square maps to the north pole, and the entire bottom edge maps to the south pole.

4.) t0 is the first point, t3 is the 4-th point.

Hope that helps!

Re: Fibonacci Sphere

#36

Post by sls »

avmich wrote: post #28
sls wrote: post #23

Earlier quoted context omitted.

https://web.archive.org/web/20000506214514/http://www.telepo...

I know what dodecahedron is, I wanted to see the corresponding (by the number of vertices) maximally-separated polyhedron.

You cited a dead link. What I posted is the Internet Archive record of what was originally at that link.

Re: Fibonacci Sphere

#37

Post by bruce343434 »

10000truths wrote: post #21
bruce343434 wrote: post #12

Earlier quoted context omitted.

What you're saying sounds promising but I have no idea how to implement it - any articles about it that contain algorithms?

Here's some Python pseudocode to get you started: import math def fibonacci_sphere_point(idx, num_points): i = idx + 0.5 phi = math.acos(1 - 2 * i / num_points) golden_ratio = (1 + 5 ** 0.5) / 2 theta = 2 * math.pi * i / golden_ratio sin_phi = math.sin(phi) cos_phi = math.cos(phi) sin_theta = math.sin(theta) cos_theta = math.cos(theta) return ( cos_theta * sin_phi, sin_theta * sin_phi, cos_phi, ) table = [fibonacci_s…

Ah, I see now. Encoding still seems expensive though. O(n)
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