Rebounds of basketball field shots

Abstract

A general three-dimensional dynamic computational model is used to estimate likely basketball rebounding regions for short-, medium-, and long-range direct and bank shots. The deterministic part of the quasi-rigid body model includes radial basketball stiffness and damping, and contains six distinct sub-models that completely encompass all reasonably possible qualitative basketball behaviors: gravitational flight with air drag, and ball-contact sub-models for ball–rim, ball–bridge, ball–board, ball–bridge–board, and ball–rim–board contact. Each contact sub-model allows slipping and non-slipping, and spinning and non-spinning motions at the ball-contact point(s). The deterministic model is driven with random initial conditions. Using independent Gaussian probability distributions for release velocity, release angle, and lateral deviation angle, simulations calculate ball trajectories of shots on a grid of three distances and seven different floor angles around the hoop on the right side of the court. The main results are the induced probability distributions over the court surface of rebounding ball locations at the height of the rim. Angled direct and bank shots have different most-likely rebounding positions, and usually have two high-probability positions, one on the same side as, and the other on the side opposite from, the shooter. The general high-probability rebound position on the shooter side is close to the hoop, even in long-range shots. On the opposite side, the most likely rebound distance from the hoop is roughly proportional to shot distance. Assumed standard deviations of release conditions affect rebound positions. A larger deviation of release velocity increases the likely rebound distance from the hoop on the shooter side. A larger deviation of lateral deviation angle moves the most likely rebound position out of the optimal shot path plane on the shooter side. Shots with larger deviation in release angle have a tendency to rebound to the shooter side.