The geometry of basketball

In the 1990s, I heard an early version of Jim Faller's "The Physics of Basketball" talk. I wish he would post a version of the talk on the web because it really changed the way I see ball sports.

Q: When does a basketball go through the hoop?
A: When the ball is smaller than the hoop.

That can sound nonsensical. A NBA regulation ball is about 9" in diameter and the hoop is 18" in diameter. If the ball is always smaller than the hoop, why doesn't it go in every time?

The hoop does NOT appear circular to the ball as it travels through the hoop.

I drew the schematic for a ball passing through a hoop perpendicular to the plane of the hoop, 90°; and 60°, 45°, 30°, 15° to illustrate.

The ball is easily smaller than the hoop when directed straight downward--like when slam dunking. But, if you are shooting a basket from the court, the size of the hoop depends on the angle of approach of the ball. At around 30°, the ball and the rim are roughly the same size.

This simplification ignores spin, velocity, the backboard and the energy dissipation capacity of the rim. That's why I titled this post, "the geometry of basketball" instead of "the physics of basketball".

But, as soon as Jim mentioned the size thing, I saw a picture of the rim as a separatrix between the scoring and non-scoring regions in my mind. And then I understood the importance of standardizing the materials and flex of the rims and the backboard, and why a player should shoot from as high a vantage point as they can (and still do it accurately).

Then I lost track of the next 10 minutes of the talk as I started applying this to volleyball approaches. My HS volleyball coach was soooo right about approaching from outside the court. Leave a comment if you want me to draw diagrams of how your VB spiking approach determines your collisional cross-section with the set.

Aside:
This post was motivated by Lin-mania. As I mentioned here:

In high school, I worked harder and longer on competitive sports than academics. This was a poor use of my time as Caltech was the only school interested in me athletically. However, I doubted their sincerity; I thought they were secretly interested in my math skills. I ended up at a division I school where the coaches told me not to even bother trying out.

I'm a big believer in practice. Yes, some people are more intrinsically gifted than others. But, all people can improve with practice. For the most part, school wasn't challenging for me until college (with the exception of a few HS classes). So I found something that was hard for me, sports, and worked very, very hard at it until I became good.

Over time, people started calling me a "natural" athlete. Nothing could be further from the truth. It took practice, good coaching, and development of awareness.

I don't think it had anything to do with my Taiwanese-American heritage, or the Japanese and African-American ethnicities of my coaches. They both taught me the importance of drills, conditioning and analyzing the game to see what was effective and what was not. Those were good lessons when the schoolwork became tougher.