## Geometry of sports ballsWritten by Paul BourkeJanuary 2017
In what follows various mathematical and graphical models will be presented for the surface features of some popular sports balls. In each case the equirectangular texture will be provided along with a couple of rendered images. The intent here is not to present high quality and realistic renderings, but rather to capture the underlying geometric essense. It should be noted that in many cases there is not one design for the balls, there is variation over the years as manufacturing has evolved but also variations due to different manufactures, in some case variations arising due to patents. Tennis ball (Softball, Baseball, Basketball, Tee ball, Hurling)
The shape of the seam on a tennis ball, like some other ball seams, arises from the initial goal of producing one 2D shape that can be cut out of a sheet of material and then stitched together in pairs. This is an example of dform surfaces. One proposed equation for the the seam is the following parameric equation y = sin(pi/2 - (pi/2 - A) cos(T)) sin(T/2 + A sin(2T)) z = cos(pi/2 - (pi/2 - A) cos(T)) Where T ranges from 0 to 2pi and parameter A in the following is 0.44.
Another construction involves constructive solid geometry, namely the subtraction of a cylinder and box from a sphere.
Equirectangular projection of one half of ball, the other half is the same form rotated 180 degrees about the vertical axis and then 90 degrees along the gap axis.
The relative size of the lobes is controlled by the relative size of the subtracted cylinder and box.
Another formation with good control over the seam spacing is to take the intersection of
an Ennepers minimal
surface with a sphere, the radius of the sphere controlling how close the
lobes are. ^{3} / 3 + u v^{2}y = v - v ^{3} / 3 + v u^{2}z = u ^{2} - v^{2}Where, in the figure below, -2 <= u <= 2, -2 <= v <= 2
The main curved seam on a basketball can be created using the same techniques as for the tennis ball, it has an additional two seams at 90 degrees to each other.
This is a rather dull case since a cricket ball only has a single seam between two hemispheres, dark line in the equirectangular below. The other curves represent the stitching which runs along lines of latitude on either side of the seam.
A soccer ball seam is the projection of a truncated icosahedron (see item 22 and 25) onto the surface of a sphere. There are 60 vertices, 90 seam edges, 20 regular hexagonal faces and 12 regular pentagonal faces. The truncated icosahedron is constructed by starting with an icosahedron with the 12 vertices truncated 1/3 of the distance. This creates 12 additional pentagon faces, and turns the original 20 triangular faces into regular hexagons.
The map can be rotated to place the two hexagons at the poles making the equirectangular projection easier to understand.
The roughness of a golf ball, introduced in the late 1800's, made for a more consistent flight path than smooth surfaces. The dimples, introduced in the early 1900's, create a turbulent layer around the ball creating a low pressure leading edge and thus less drag allowing the ball to travel further. There were disputes in the 1980's around symmetric vs non-symmetric dimple patterns, the later provided for a sort of self correction of ball spin during flight. While there are asymmetric balls they are only allowed in non-tournament games. Most balls have between 300 and 500 dimples. The example below has 308 dimples. It is formed by progressively sampling lines of latitude in equal steps with ever increasing steps in longitude and phase offsets per latitude.
But this doesn't give the local hexagonal neighbourhood around each dimple, see the example on the right above. A more regular dimple pattern can be achieved by starting with an icosahedron and tiling each triangular face with a regular triangular array of dimples.
The next approach is to perform a minimisation process to distribute points on a sphere. This may be an arbitrary simulation or a spring system as described here. The following example has 500 dimples and unlike the previous methods where only certain densities can be supported, this approach can be applied to any number of dimples.
While the early rugby balls were shaped more like the American footballs, the current rugby ball is a prolate spheroid with a major to minor axis ratio of about 1.5. Besides a sphere stretched along one axis there are many other ways of describing this shape, one example being a superellipsoid.
The ball shape can be represented as the intersection of two off-center circles,
rotated about the mirror axis. The design is intended to reduce drag when thrown
with a rotation along the long axis. ^{2} + y^{2} = R^{2}and (x - r) ^{2} + y^{2} = R^{2}
where the range for x is 0 to R-r
and the range for y is -sqrt(R
The seams are the same as for the rugby football, namely, 4 lines of longitude spaced every 90 degrees about the long axis of the ball. As with most ball seams they arise from the desire to create the ball skin from a number of equal shaped and sized pieces.
There are a few variation of the design of volley balls, the simplest to describe geometrically is presented below. One takes a cube and splits each face into three sections. The axis of the sections is the same for opposing faces of the cube, but along a different axis for each pair of opposing faces. These lines are then projected onto the surface of a sphere.
There are a wide range of designs but they are generally symmetric slices by a number of planes through a sphere, repeated on two or three axes.
There are a number of other sports balls not included here, largely because they are not interesting. Squash balls are simple spheres with only one or two small dots to indicate hardness (bounce). Similarly, ten pin bowling balls only have three finger grip holes. Pool, snooker and billiards balls may include dots (billiards) or numbers and simple bands of colour. Ping pong balls have no markings at all. Same for croquet balls, racketball, and polo (horseback). |