# Curve on a basketball

The sewing pattern on a basketball is composed of two great circles and a single curve that intersects each great circle twice. Does this curve have a name? Are there any parametric descriptions of the curve?

The same question can apply to the sewing pattern on a tennis ball.

Answers about why this sewing pattern came to be used are also welcome.

• The curve on the basketball intersects each great circle twice. It came to be used because it allows one to sew two flat sheets of material together into a (very close approximation of) a ball. – Servaes Apr 19 '14 at 10:18
• @Servaes Oops, my mistake. (In hindsight, this should have been obvious from Jordan curve theorem.) – Mario Carneiro Apr 19 '14 at 12:30
• Also refer to this: math.stackexchange.com/questions/688749/… – heropup Apr 19 '14 at 21:33

Actually I thought the seams on a basketball were two great circles and two ovals that each intersect the same great circle twice. And so apparently do many people who make basketball texture maps one can find on the web. (Randall Munroe too).

It is difficult to settle the question without having a physical ball to inspect; a usual photograph doesn't show both poles of the ball which is one needs. However, here's a photo showing that at least some basketballs are constructed like you describe:

I don't think the shape of the curved seam has a name -- judging from the photos I've found its precise shape can vary from ball to ball. The most realistic-looking images look like it's made up of four about 150° arcs of great circles, joined by curved segments near the poles, but how tight the joining curves are seems to vary.

From a construction point of view, the one-curve design does make more sense, because then the seams divide the surface of the ball into eight congruent pieces, each of which can be cut out of flat material without too much distortion (at compared to dividing into eights by three great circles), and also without needing eight-way meets at the poles. With two great circles and two ovals, the pieces would have two different shapes.

Incidentally, I don't think they're actually seams with modern manufacturing methods -- just a decoration that's put on the ball for reasons of tradition, and possibly to make it easier for players to perceive its spin.

• A better image, showing more of the intersections, can be found at en.wikipedia.org/wiki/Basketball_(ball) – Barry Cipra Apr 19 '14 at 12:28
• @Barry That link also shows that there is a considerable amount of variation in the topology of the lines (there is even one ball there made from 10 segments). @${}$Henning, is there any reason to believe that there is a local minimum to be found with respect to minimizing the distortion of the segments that would produce the general shape that is observed? I suppose if it is just a contest between minimizing distortion and the desire to avoid 8-way meets or sharp turns then there wouldn't be much of a mathematical so much as economic reason to choose the particular shape that is seen. – Mario Carneiro Apr 19 '14 at 12:37
• Here's an animation of a basketball that shows the intersections quite clearly. – Mario Carneiro Apr 19 '14 at 12:51
• @Barry: No, those images don't show whether the pattern at the other side of the ball is oriented the same or with a 90° turn. – hmakholm left over Monica Apr 19 '14 at 15:44
• @Mario: Yes, I agree that the shape probably isn't determined by a mathematical minimization procedure. Apart from production cost, there are market forces at work -- even if you could produce a different seam pattern cheaper, if people wouldn't buy it because it doesn't "look like a real basketball", then it wouldn't be any good. – hmakholm left over Monica Apr 19 '14 at 15:48

I don't have a basketball here to compare. However, I can say that the seams on a baseball and on a tennis ball are similar to the intersection of Enneper's minimal surface with a sphere centered at the, well, the origin of all the symmetries of the surface. See also HERE.

Edit by Mario: Here is an animation of the basketball curve based on the idea given above. The degree to which the two bells of the curve approach each other depends on the radius of the sphere chosen to intersect the surface; this animation uses $r=\frac32$, which produces a reasonable approximation.

• I hope you don't mind my adding an image of your idea to the answer. – Mario Carneiro Apr 19 '14 at 21:07
• @MarioCarneiro, very nice, and nothing i could have arranged myself. – Will Jagy Apr 19 '14 at 21:18