Loci, 2008
Generalized Baseball Curves: Three Symmetries and You're In! Allison, Diaz, and Miller

4. Geodesic curvature and other ideas from differential geometry

We'd now like to explore the properties of generalized baseball curves, but we first need to develop some basic ideas from differential geometry. In particular, we will need a basic understanding of the geodesic curvature of a curve on a surface. Because this isn't the main focus of our article, we will give just enough detail for the following discussion to make sense; the interested reader can find more details in any standard differential geometry reference, such as Millman and Parker [1977], Gray [1998], or Henderson [1998]. Brief summaries of some of these ideas can also be found at Wikipedia and MathWorld.

We will assume that any curve α : IR3 that we discuss is at least C3 and regular; that is, that dα / ds ≠ 0 for any s in I . If P(s) is a curve in three-dimensional space (or more generally, in Rn ) that is parameterized by arclength s , then we can define the unit tangent vector at any point along the curve by T(s) = dP / ds . This derivative gives a unit vector because the curve is parameterized by arclength, which means that we are essentially moving along the curve with constant unit speed. We can then define the curvature vector κ(s) at a given point along the curve to be the derivative of this unit tangent vector; that is, κ(s) = dT / ds .

What does the curvature of a curve at a point tell us? Because it is the derivative of the unit tangent vector, it is a measure of how fast the curve is turning at the given point.

However, for a curve winding and twisting around in R3 , curvature is not sufficient to characterize all of the turning that a curve can do. Another quantity, the torsion, measures the tendency of the curve to twist out of the plane that the curve comes closest to being in near the point at which we are measuring the curving. A central result of differential geometry is the Fundamental Theorem of Space Curves, which essentially says that, up to position in space, a regular curve with nonzero κ is completely determined by its curvature κ and torsion.

Curvature and torsion provide a nice way to talk about how much a curve is turning in three-dimensional space. However, in order to study curves that are constrained to stay on a two-dimensional surface like a sphere, it will be useful to refine these ideas. Consider, for example, a great circle on a sphere. Extrinsically, the great circle has a constant non-zero curvature. However, intrinsically the great circle looks like a straight line on the sphere: it has all of the intrinsic symmetries of a straight line, and a two-dimensional creature that walked along the curve would move the right-hand and left-hand parts of its body the same amount. In other words, all of the turning in the curve comes from the curving of the surface, rather than from the turning of the creature's body. Thus, intrinsically, we would like to say that the curvature of this curve should be zero.

We can capture this notion by defining the intrinsic or geodesic curvature vector at a point P of a curve that lies in a orientable surface S to be the projection of the curvature vector of the curve at P onto the plane tangent to S at P . We will sometimes also call the curvature vector of a curve the extrinsic curvature vector, to distinguish it from the geodesic curvature vector. We can furthermore define a scalar curvature k(s) that gives the length of the geodesic curvature vector. It is conventional to simply refer to the scalar geodesic curvature as the geodesic curvature. In the rest of this paper we will follow that convention and use k(s) for the (scalar) geodesic curvature. We assign the geodesic curvature of the curve at a point a positive or negative sign depending on whether the geodesic curvature vector lies to the right or the left of the curve within the surface relative to its orientation as determined by a choice of a normal vector to the surface.

As an example, again consider the case of a great circle on a sphere. Its extrinsic curvature vector always points in towards the center of the sphere; so at any point on the great circle, the extrinsic curvature vector is orthogonal to the tangent plane. Thus, the geodesic curvature of a great circle is always zero, as we'd like it to be. It should be pointed out that it isn't obvious from the definition that we have given of geodesic curvature that it should be an intrinsic quantity, since it relies on the definition of the extrinsic curvature of a curve, and this will change depending on how the surface in which the curve lies is embedded in R3 . However, a standard result of differential geometry shows that the geodesic curvature of a curve on a surface is independent of how the surface is embedded in the larger space, and thus the geodesic curvature is, in fact, an intrinsic quantity. For proof, see Millman and Parker [1977], p.106.

One nice way of understanding geodesic curvature is through the following result, which generalizes the Ribbon Test defined by Henderson [1998], Problem 3.4.

Theorem 3 [The Generalized Ribbon Test]. Let s(t) be a curve on some orientable surface, and let r(t) be a planar curve such that it is possible to cut out a ribbon around r(t) in the plane and lay it flat on the surface so that the moved image of each point r(t) lies on s(t) . (If we denote this moved image by m(r(t)) , we're saying that m(r(t)) = s(t) . By lying flat, we mean that the ribbon is tangent to the surface at each point along the curve.) Then the geodesic curvature of s at the point s(t0) is the same as the (signed) extrinsic curvature of r(t0) in the original plane.

Proof. Since the surface and the ribbon are tangent to one another at the point s(t0) , they have the same tangent plane. Thus, since s(t) = m(r(t)) and the curve can be considered to lie in either the surface or the ribbon, the curve has the same geodesic curvature whether it is measured as a curve on the surface or the ribbon. Since the geodesic curvature of a curve on a surface is independent of how that surface sits in three-dimensional space, the geodesic curvature of m(r(t0)) in the curved ribbon is the same as the geodesic curvature of r(t0) in the plane. And since the plane is its own tangent plane, the geodesic curvature of r at r(t0) is the same as its extrinsic curvature. Thus, the geodesic curvature of s at s(t0) is the same as the extrinsic curvature of r at r(t0) .

This result gives us a nice way to understand geodesic curvature-it says that the geodesic curvature of a curve C on a surface is the same as the extrinsic curvature of a curve in the plane that can be cut out and laid along C .

There are some curves in some surfaces along which it is impossible to lay a ribbon; but Henderson [1998], Problem 7.6 shows that it is always possible as long as the normal curvature of the curve is never zero. Since a standard result from differential geometry says that the normal curvature of any curve on the sphere can never be zero, a ribbon can always be laid along any spherical curve.

A third way of thinking about geodesic curvature on the sphere is in terms of rolling. If we have a curve on the sphere, we can imagine laying a ribbon along it, and putting glue on the outside of the ribbon. Then, if we roll the sphere on a plane along the curve, the ribbon will stick to the plane and will trace out the path of the sphere on the plane. Thus, we can also think of the geodesic curvature of a curve on the sphere as being the extrinsic curvature of the path that the sphere takes when rolled on a plane along the curve.

Now, if we have a planar curve and we know its starting point and direction and its curvature (as a function of arclength), then we can reconstruct the whole curve, essentially by integrating the curvature function. That is to say, the curvature function of a curve determines that curve uniquely once we pick a starting point and direction for the curve. Likewise, if we know the starting point and direction of a spherical curve and we also know its geodesic curvature function, then we can again reconstruct the curve. To be precise, we can state the following:

Theorem 4 [The Fundamental Theorem of Spherical Curves]. Let (a,b) be an interval about 0 , k a C1 function on (a,b) , x0 a fixed point of S2 , and V a fixed unit-length vector tangent to S2 at the point x0 . Then there exists a unique C1 regular curve α : (a,b) → S2 such that α(0) = x0 , T(0) = V , and the geodesic curvature of α is given by k .

To prove this theorem, we would derive the following system of differential equations known as the Spherical Frenet Equations, which are given by

P ′ = 0P + 1T + 0N
T ′ = −1P + 0 T + k(s)N
N ′ = 0Pk(s)T + 0N

where P(s) is the position vector of the unit speed parametrized curve, T = dP / ds is the unit tangent vector, N = P × T is the unit normal vector to the curve, and k(s) is the geodesic curvature of P(s) . (If you are interested, here is a derivation of these equations.)

Once we know that every smooth spherical curve satisfies these equations, Theorem 4 follows almost immediately, using the existence and uniqueness theory for systems of linear differential equations.

One nice way to actually construct this desired spherical curve, however, assuming that we know how to construct the planar curve with the given curvature function, is to cut out a ribbon in the plane around this planar curve, and to then lay the ribbon on the sphere at the prescribed starting point, heading in the prescribed direction. If we have a curve that is a candidate for the seam of a baseball and we know its geodesic curvature function, we can use this method to check how close the candidate comes to fitting an actual baseball by printing out the planar curve with the same geodesic curvature, cutting it out, and seeing how closely it fits the baseball's seam when laid flat on the baseball.