hopf | 🧣 A graphical program for exploring the Hopf fibration | 3D Animation library

hopf is a C++ library typically used in User Interface, 3D Animation applications. hopf has no bugs, it has no vulnerabilities and it has low support. You can download it from GitHub.

The Hopf fibration is a mapping from S3 to S2 discovered by Heinz Hopf in 1931. Confusingly (at least for me), S3 actually refers to the sphere (or hypersphere) in 4D space. Similarly, S2 refers to the sphere that we are most familiar with in 3D space. To quote Wikipedia: "Hopf found a many-to-one continuous function (or 'map') from the 3-sphere onto the 2-sphere such that each distinct point of the 2-sphere is mapped to from a distinct great circle of the 3-sphere.". The first thing to understand is that a great circle is essentially a "slice" of a sphere that passes through its center. It is difficult to visualize what a great circle on the surface of a 4-dimensional sphere looks like, but luckily, we don't have to. Each of these great circles in the domain of the mapping function forms a "fiber" that we can project down to 3-space. Doing so results in a beautiful structure of nested shapes that appear to be intricately woven together. To be more specific, the mapping treats each point on the surface of S3 as a unit quaternion r = a + b*i + c*j + d*k (a unit quaternion is one whose norm is 1). Next, we pick a "principal point" on S2: in the literature, this is usually a unit vector along one of the standard basis vectors, i.e. <1, 0, 0>. The fiber for any point p on S2 consists of all of the unit quaternions that send the principal point p0 to p, via a rotation in 3-space. For example, let p be the principal point such that p = p0. The fiber corresponding to p is the set of all unit quaternions for any scalar value t. Each of these quaternions, when applied to p0, result in p0 itself. So the question becomes: given any point p on S2, how do we find the set of all quaternions that rotate the principal point to p (i.e. the fiber on S3 that corresponds to p)?.