Horseshoe distribution

Question

Studying nonlinear dimension reduction topics I came across a "horseshoe distribution" or "horsheshoe phenomena". Figure (b) [source, p. 12] depicting horseshoe distribution is included below.

I do not have a good idea how to simulate this type of distribution.

Thanks in advance for any pointers.

Answer 1

You could simply formulate a parabolic function fo each fixed value in a particular dimension (say for the z axis) and add a random noise component to it.