In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, the tennis ball theorem states that any
smooth curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
on the surface of a sphere that divides the sphere into two equal-area subsets without touching or crossing itself must have at least four
inflection point
In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case ...
s, points at which the curve does not consistently bend to only one side of its
tangent line
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
.
The tennis ball theorem was first published under this name by
Vladimir Arnold
Vladimir Igorevich Arnold (alternative spelling Arnol'd, russian: link=no, Влади́мир И́горевич Арно́льд, 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician. While he is best known for the Kolmogorov– ...
in 1994, and is often attributed to Arnold, but a closely related result appears earlier in a 1968 paper by
Beniamino Segre
Beniamino Segre (16 February 1903 – 2 October 1977) was an Italian mathematician who is remembered today as a major contributor to algebraic geometry and one of the founders of finite geometry.
Life and career
He was born and studied in Turin ...
, and the tennis ball theorem itself is a special case of a theorem in a 1977 paper by Joel L. Weiner. The name of the theorem comes from the standard shape of a
tennis ball
A tennis ball is a ball designed for the sport of tennis. Tennis balls are fluorescent yellow in organised competitions, but in recreational play can be virtually any color. Tennis balls are covered in a fibrous felt which modifies their aerodyna ...
, whose seam forms a curve that meets the conditions of the theorem; the same kind of curve is also used for the seams on
baseballs.
The tennis ball theorem can be generalized to any curve that is not contained in a closed hemisphere. A centrally symmetric curve on the sphere must have at least six inflection points. The theorem is analogous to the
four-vertex theorem
The four-vertex theorem of geometry states that the curvature along a simple, closed, smooth plane curve has at least four local extrema (specifically, at least two local maxima and at least two local minima). The name of the theorem derives fro ...
according to which any smooth closed
plane curve
In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic ...
has at least four points of extreme curvature.
Statement
Precisely, an inflection point of a
doubly continuously differentiable () curve on the surface of a sphere is a point
with the following property: let
be the connected component containing
of the intersection of the curve with its tangent great circle at
. (For most curves
will just be
itself, but it could also be an arc of the great circle.) Then, for
to be an inflection point, every
neighborhood of
must contain points of the curve that belong to both of the hemispheres separated by this great circle.
The theorem states that every
curve that partitions the sphere into two equal-area components has at least four inflection points in this sense.
Examples
The tennis ball and baseball seams can be modeled mathematically by a curve made of four semicircular arcs, with exactly four inflection points where pairs of these arcs meet.
A
great circle also bisects the sphere's surface, and has infinitely many inflection points, one at each point of the curve. However, the condition that the curve divide the sphere's surface area equally is a necessary part of the theorem. Other curves that do not divide the area equally, such as circles that are not great circles, may have no inflection points at all.
Proof by curve shortening
One proof of the tennis ball theorem uses the
curve-shortening flow
In mathematics, the curve-shortening flow is a process that modifies a smooth curve in the Euclidean plane by moving its points perpendicularly to the curve at a speed proportional to the curvature. The curve-shortening flow is an example of a g ...
, a process for continuously moving the points of the curve towards their local
centers of curvature. Applying this flow to the given curve can be shown to preserve the smoothness and area-bisecting property of the curve. Additionally, as the curve flows, its number of inflection points never increases. This flow eventually causes the curve to transform into a
great circle, and
the convergence to this circle can be approximated by a
Fourier series. Because curve-shortening does not change any other great circle, the first term in this series is zero, and combining this with a theorem of
Sturm on the number of zeros of Fourier series shows that, as the curve nears this great circle, it has at least four inflection points. Therefore, the original curve also has at least four inflection points.
Related theorems
A generalization of the tennis ball theorem applies to any simple smooth curve on the sphere that is not contained in a closed hemisphere. As in the original tennis ball theorem, such curves must have at least four inflection points. If a curve on the sphere is
centrally symmetric
In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...
, it must have at least six inflection points.
A closely related theorem of also concerns simple closed spherical curves, on spheres embedded into three-dimensional space. If, for such a curve,
is any point of the three-dimensional
convex hull of a smooth curve on the sphere that is not a vertex of the curve, then at least four points of the curve have
osculating planes passing through
. In particular, for a curve not contained in a hemisphere, this theorem can be applied with
at the center of the sphere. Every inflection point of a spherical curve has an osculating plane that passes through the center of the sphere, but this might also be true of some other points.
This theorem is analogous to the
four-vertex theorem
The four-vertex theorem of geometry states that the curvature along a simple, closed, smooth plane curve has at least four local extrema (specifically, at least two local maxima and at least two local minima). The name of the theorem derives fro ...
, that every smooth
simple closed curve
In topology, the Jordan curve theorem asserts that every ''Jordan curve'' (a plane simple closed curve) divides the plane into an " interior" region bounded by the curve and an "exterior" region containing all of the nearby and far away exterior ...
in the plane has four
vertices (extreme points of curvature). It is also analogous to a theorem of
August Ferdinand Möbius
August Ferdinand Möbius (, ; ; 17 November 1790 – 26 September 1868) was a German mathematician and theoretical astronomer.
Early life and education
Möbius was born in Schulpforta, Electorate of Saxony, and was descended on hi ...
that every non-contractible smooth curve in the
projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that d ...
has at least three inflection points.
References
External links
*{{MathWorld, id=TennisBallTheorem, title=Tennis Ball Theorem, mode=cs2
Theorems in differential geometry
Theorems about curves
Spherical geometry
Spherical curves