GALLERY OF BISHOP CURVES AND OTHER SPHERICAL CURVES

This Gallery is a part of the project on Bishop Curves and Other Spherical Curves, conducted by

Clark Kimberling
University of Evansville

and

Peter Moses
Moparmatic Co.
Redditch, Worcestershire, UK

The project includes forthcoming research articles.

This gallery is a sequel to Gallery of Spaces Curves Made from Circles, which includes a "baseball curve" called the cubic quadrarc, or CQ. After the CQ was presented at a geometry seminar in March 2010 at the University of Illinois, Professor Richard Bishop observed that, although a tangent vector exists at each point on the CQ, this is not true for a normal vector. In other words, the CQ, although 1-smooth, is not 2-smooth. Professor Bishop suggested some curves which we believe to be new to the literature: baseball-like curves which are infinitely smooth; indeed, they are analytically smooth algebraic curves.

A Bishop curve is the intersection of two elliptic cylinders as indicated here:

Varying the cylinders results in an infinite family of Bishop curves, like these:

The smoothness of a Bishop curve ensures that the Frenet frame (the pairwise orthogonal unit vectors T, N, B in elementary calculus) exists at every point. Here is an animated Frenet frame:

At each point P on a 2-smooth curve C there is a "closest circle." As P moves on C, this osculating circle, as it is known, rolls along with continuously changing radius, like this:

Putting the Frenet frame and osculating circles together:

Here is a close-up of a Bishop curve as the intersection of an elliptic cylinder and a sphere:

These graphics were created by PM using MM. This text was written by CK, whose favorite graphic is this next one. It shows many Bishop curves in a continuous manner.

Each Bishop curve is an intersection of an elliptic cylinder and a sphere. What other cylinder-and-sphere intersections are interesting? Here is an intersection of a parabolic cylinder and a sphere, which is also the intersection of a circular cylinder and the sphere.

Next, the intersection of a hyperbolic cylinder and a sphere:

We are interested in the mean distance from a random point on a sphere to curves on the sphere. Here is a picture of an arc of a Bishop curve and a scattering of "random points" with distances (arclengths) to the curve:

Among the curves we wish to compare with Bishop curves are "spherical sine curves", shown on the left, with Bishop curves, on the right:

A curve T is called an orthogonal trajectory of a family F of curves if at each point of intersection of T with a curve in F the angle is 90 degrees. Here is a quarter-sphere representation of Bishop curves and their orthogonal trajectories:

Here is an animated representation of Bishop curves and their orthogonal trajectories. (The othogonal trajectories have amenable parametric equations!)

We introduce a large family of spherical spirals. Included are two subfamilies: the (z,W)-spirals have constant rate of change of height z, whereas the (φ,W)-spirals have constant rate of change of the angle φ (the third coordinate in spherical coordinates, which is the angle made with the positive z-axiz). For both types, W is the winding number; i.e., the number of times the orthogonal projection of the spiral winds around the origin.

Next, a few more representations of orthogonal trajectories of Bishop curves:

Graphics files numbered 1-13 first appeared here on August 16, 2010.
Files numbered 14-24 first appeared here on September 18, 2010.
Files numberd 25-2 first appeared here before November 12, 2010.