Clark Kimberling

University of Evansville

and

Peter Moses

Moparmatic Co.

Redditch, Worcestershire, UK

The project includes forthcoming research articles.

To view a curve, click its name - and then use your browser's **BACK** to return to this page.

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:

5. **Animated Frenet Frame and Osculating Circle**

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.

8. **Another Algebraic Curve: Parabolic**

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

9. **Another Algebraic Curve: Hyperbolic**

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:

10. **How Far, on Average, from Random Point to 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:

11. **Two Families of Curves: Sines and Bishop**

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!)

13. **Orthogonal Trajectories, Animated **

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.

14. **Both kinds of spirals, with W = 10.**

15. **Orthogonal trajectories of seven (z, 2)-spirals, with W = 2, first view**

16. **Orthogonal trajectories of seven (z, 2)-spirals, with W = 2, second view.**

17. **Orthogonal trajectories of seven (z, 2)-spirals, with W = 2, third view.**

18. **Animated orthogonal trajectories of seven (z, 1/2)-spirals**

19. **Orthogonal trajectories of three (φ, 1/2)-spirals**

20. **Orthogonal trajectories of three (φ, 2)-spirals**

21. **Animated orthogonal trajectories of seven (φ, 1)-spirals**

22. **Animated center of osculating circle of (z, 5)-spiral**

23. **Animated osculating circle of (z, 5)-spiral**

24. **Animated osculating circle of (φ, 5)-spiral**

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

26. **Animated Cylinder Forming Bishop Orthogonal Trajectories**

27. **Two Cylinders from Item 26**

28. **Animated Hyperbolic Cylinder Forming Bishop Orthogonal Trajectories**

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.