More circles on a sphere of cubes скачать в хорошем качестве

More circles on a sphere of cubes

This is a followup video to    • Where do these circles come from?  . The original Shadertoy demo: https://www.shadertoy.com/view/7ds3zB The closest parallel planes: Suppose that v is an integer vector with no common factors between its coordinates. We want to find the integer vector u so that the component of u in the direction of v is as small as possible but still strictly positive. This component can be written using the dot product as u·v/|v|. Since v has no common factors between its coordinates, we can find an integer vector u so that u·v = 1. Thus the distance between the plane perpendicular to v and based at the origin and the parallel plane based at the end of u is 1/|v|. There is no closer plane, since u.v must be an integer. Correction: the ripple frequency is proportional to the length of v, not inversely proportional. (Longer vectors have higher frequency ripples.)