Fetch is a game usually played with a dog

Dog

The domestic dog is a domesticated form of the gray wolf, a member of the Canidae family of the order Carnivora. The term is used for both feral and pet varieties. The dog may have been the first animal to be domesticated, and has been the most widely kept working, hunting, and companion animal in...

. An object, such as a stick or ball, is thrown a moderate distance away from the animal, and it is the animal's objective to grab and retrieve it. Many times, the owner of the animal will say "Fetch" to the animal before or after throwing the object. In rare instances, cats, especially younger cats, have been known to engage in fetch behavior.

Mathematics of Fetch

Arizona State

Arizona State University

Arizona State University is a public research university located in the Phoenix Metropolitan Area of the State of Arizona...

 psychology professor Michael McBeath has proposed a simple model to explain how dogs play Fetch. By mounting a camera on the head of a dog, he found that the dog changed its speed and direction in order to keep the frisbee's image in a constant position on its retina. This approach, called the Linear Optical Trajectory, makes the frisbee appear to move in a linear path at a constant speed. McBeath had previously noticed this interception strategy in professional baseball players pursuing fly balls.

Tim Pennings, a mathematics professor at Hope College

Hope College

Hope College is a medium-sized , private, residential liberal arts college located in downtown Holland, Michigan, a few miles from Lake Michigan. It was opened in 1851 as the Pioneer School by Dutch immigrants four years after the community was first settled...

, has found that dogs are somehow able to calculate the optimal path to a ball thrown in the water. While playing Fetch with his Welsh Corgi

Welsh Corgi

The Welsh corgi is a small type of herding dog that originated in Wales. Two distinct breeds are recognized: the Pembroke Welsh corgi and the Cardigan Welsh corgi, with the Pembroke being the more common..-Description:...

, he noticed that the dog ran along the beach for a certain distance before jumping into the water. Because the dog is faster on land, this technique minimizes the total retrieval time. He showed that the dog is able to calculate the optimal point to jump into the water with statistical significance, a problem Pennings must resort to calculus

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

to solve.

Perruchet and Gallego have demonstrated a method for calculating this optimal path using calculus in a different way. They propose that the dog optimizes its behaviour on a moment-to-moment basis, choosing at each moment the path that allows it to maximize its speed of approach to the ball. This requires the dog to be able to accurately estimate its speed on both land and water, but does not rely on the premise that the dog plans the entire route in advance.