An irregular ball rolling on a plane, if know the path on ball surface, how to find the path on the plane?

Question

An irregular ball has its local radius or curvature different at any surface point. It has pure rolling movement on the plane P.

Several questions here:

1. If I know the path curve from A to B on the ball, how to know the path on the plane, and vice versa?

2. If I know the path on the ball is the geodesic from A to B , does it have a simpler solution on the plane? Or the other way, if the path on plane is straight, how is the path on the irregular ball?

3. Is there a situation(for example how the ball have to be, or probably the plane have to be some kind of special curved plane,etc), the problem becomes path independent, that is to say, if I know the destination point B (and start point A) on the ball, I then know the point B on the (special) plane and vice versa.

4. If only specify on the ball the start point A and a tangent direction $\hat{t}_A$ at A, the destination B and a tangent direction $\hat{t}_B$ at point B, meanwhile on the plane the start point A' and a tangent direction $\hat{t}_A'$ at A', the destination B' and a tangent direction $\hat{t}_B'$ at point B', is it possible to find a path on the ball (and the correspondent path on the plane), so that point A and A', point B and B', direction $\hat{t}_A$ and $\hat{t}_A'$,direction $\hat{t}_B$ and $\hat{t}_B'$ coincide, respectively?

Thanks!

Answer 1

The geodesic curvature $\kappa_g$ ( which is the common isometric guide) of the curve in tangent plane of ball equals the geodesic curvature of the trace in the plane.

We attempt to address the question from easier curvature cases to tougher ones, depending on number of constants needed to create the surface.

With this constraint the planar curve of known initial position and inclination to reference can be determined by integration once the curved line on ball is known.

And conversely,the entire space curve $R^3$ of known parameters viz., initial position, inclination, initial normal curvature and geodesic torsion on ball can be determined by integration once the ball geometry and ground trace curve is given.

It will be interesting to trace seams of a a base ball or rugby ball as the joint lines have non-zero geodesic curvature.

Trace of a geodesic is always a straight line whatever be the ball shape or size drawn onto the plane P.

Traces of curved lines follow intrinsic/natural equations on the plane P. That is, we isolate arc-$\kappa_g $ relation of 3D line and transfer onto plane and hold the same relation as valid to be drawn on plane P.

At first we can set the ball rolling with a small circle of a sphere (latitude $=\phi$) of a circle remaining in contact without slipping with plane $P$. We choose spheres at present as the normal curvatures are constant and more easy to formulate. In following figure latitude radius of curvature

$$R_g= R \cot \phi $$

holds along the circular arc parallel, that is also a tangential cone base line.

Later on the same can be generalized to formulate and plot a non-spherical 3D object along chosen parameter lines.

Next, images for Rhumb lines or Loxodromes are presented. Curved smooth lines are oriented at angles $ \alpha= \pi/6,\pi/4,\pi/3 $ to a sphere equator, smaller length to reach pole for smaller constant \alpha.$

The relations depict their natural intrinsic equations:

$$ R \kappa_g = \sin \alpha \, \tan ( \cos \alpha \, s/ R )$$

whose name if any exists, I do not know.

Next we come to non-spherical surfaces in 3D. We write natural/intrinsic equation of a Cornu spiral on an ellipsoid and on a hyperboloid (same curve on $ \mathbb R^2$, embedded $ \mathbb R^3 $) and roll it in the manner you indicated onto flat plane P. To effect transfer of the spiral curve onto the plane we use the same natural/intrinsic equation in all the three cases.

$$ \kappa_g = \frac{s}{a^2} $$

Three of your questions are answered. For the fourth, answer is yes. The intrinsic equation has these two scalar parameters$\kappa_g,s$ depending only upon coefficients of the first fundamental form. Accordingly Intrinsic equations are bending/isometry invariant.

we have to take boundary conditions to integrate arc/geodesic curvature natural ODE in order to fix position in Euclidean space.The Intrinsic equation is said to determine the curve only up to Euclidean motions of position and orientation. In fact the images above are made taking advantage of these two degrees of freedom.

Replying to your further questions made in comments:

1) The "ball" that you mention.. I have taken it to mean in a generic sense. That is why I had included even a hyperboloid of revolution of grossly negative Gauss curvature $K.$ Please see my answer updated in that link.

2) I have used coupled equations; one for the intrinsic curvature and other using normal curvature or whatever you want to generate a surface definition. While integrating choose either 3D or a plane, i.e., separate programs for 3D and for the plane.

3) & 4) No need to change the rigid property of the rolling ball or change flatness of the plane! The curve and ground trace form a dual pair. If one is given, then the other is known through the go-between intrinsic equation.

5) There are many. I suppose the book by author D.J. Struik is classical by now. You have a strong imagination in 3D geometry. So do not miss the book "Geometry and Imagination" by David Hilbert and Cohn-Vossen.

We can interact further after a month or so if you want, allowing you time for DG study, learn from others' replies, and of course now not prevail on the mod towards chatroom.. Good luck!