Return to Answer

Is it possible to uniformly draw points over a D−2 sphere, given that one has an algorithm to draw over the D−1 sphere in D-dimensional space?

You can consider an n-sphere as a sum of (n-1)-spheres.. An n-sphere can be constructed by stacking scaled (n-1)-spheres.

And you can turn around that process.

The n-sphere has n+1 coordinates and if you take n of the coordinates and scale them to get the appropriate length of the desired radius of the sphere, then you have the n coordinates of a n-1 sphere.

Further, suppose now that I have a uniform random sampling algorithm over an arbitrary Riemannian manifold. Would the same principles necessarily apply?

The case of spheres has a simple solution. But for other surfaces you will need to take another approach to get the slice (like rejection sampling and taking only points in a small range of distance to a plane). Also the slice will not have a uniform distribution.

E.g take a slice of carambola cut at an angle. If you trace that slice with a measuring tape and compute for each amount of length that you go around the amount of surface that you have, then there will be a difference. The same is true when you cut a cucumber at an angle (and this you could model like cutting an ellipsoid at an angle).

Is it possible to uniformly draw points over a D−2 sphere, given that one has an algorithm to draw over the D−1 sphere in D-dimensional space?

You can consider an n-sphere as a sum of (n-1)-spheres.. An n-sphere can be constructed by stacking scaled (n-1)-spheres.

And you can turn around that process.

The n-sphere has n+1 coordinates and if you take n of the coordinates and scale them to get the appropriate length of the desired radius of the sphere, then you have the n coordinates of a n-1 sphere.

Further, suppose now that I have a uniform random sampling algorithm over an arbitrary Riemannian manifold. Would the same principles necessarily apply?

The case of spheres has a simple solution. But for other surfaces/spaces you will need to take another approach to get the slice (like rejection sampling and taking only points in a small range of distance to a plane). Also the slice will not have a uniform distribution.

E.g take a slice of carambola. If you trace that slice with a measuring tape and compute for each amount of length that you go around the amount of surface that you have, then there will be a difference, the points have more surface. The same is true when you cut a cucumber at an angle (and this you could model like cutting an ellipsoid at an angle).

So a D-2 dimensional hyperspace slice from a uniform distribution on a D-1 hyperspace may not itself have a uniform distribution. If your question is about "how to resolve this?" or "how to get a slice with a uniform distribution?" then you could apply the rejection sampling method that is an answer to the question How to sample uniformly from the surface of a hyper-ellipsoid (constant Mahalanobis distance)?

You can consider an n-sphere as a sum of (n-1)-spheres.. An n-sphere can be constructed by stacking scaled (n-1)-spheres.

And you can turn around that process.

The n-sphere has n+1 coordinates and if you take n of the coordinates and scale them to get the appropriate length of the desired radius of the sphere, then you have the n coordinates of a n-1 sphere.

Is it possible to uniformly draw points over a D−2 sphere, given that one has an algorithm to draw over the D−1 sphere in D-dimensional space?

You can consider an n-sphere as a sum of (n-1)-spheres.. An n-sphere can be constructed by stacking scaled (n-1)-spheres.

And you can turn around that process.

The n-sphere has n+1 coordinates and if you take n of the coordinates and scale them to get the appropriate length of the desired radius of the sphere, then you have the n coordinates of a n-1 sphere.

Further, suppose now that I have a uniform random sampling algorithm over an arbitrary Riemannian manifold. Would the same principles necessarily apply?

The case of spheres has a simple solution. But for other surfaces you will need to take another approach to get the slice (like rejection sampling and taking only points in a small range of distance to a plane). Also the slice will not have a uniform distribution.

E.g take a slice of carambola cut at an angle. If you trace that slice with a measuring tape and compute for each amount of length that you go around the amount of surface that you have, then there will be a difference. The same is true when you cut a cucumber at an angle (and this you could model like cutting an ellipsoid at an angle).

You can consider an n-sphere as a sum of (n-1)-spheres.. An n-sphere can be constructed by (n-1)-spheres.

And you can turn around that process.

The n-sphere has n+1 coordinates and of you take n of the coordinates and scale them to get the appropriate length of the vector, then you have the n coordinates of a n-1 sphere.

You can consider an n-sphere as a sum of (n-1)-spheres.. An n-sphere can be constructed by stacking scaled (n-1)-spheres.

And you can turn around that process.

The n-sphere has n+1 coordinates and if you take n of the coordinates and scale them to get the appropriate length of the desired radius of the sphere, then you have the n coordinates of a n-1 sphere.