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In 
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a radiodrome is the pursuit curve
In geometry, a curve of pursuit is a curve constructed by analogy to having a point or points representing pursuers and pursuees; the curve of pursuit is the curve traced by the pursuers.
With the paths of the pursuer and pursuee parameterized ...
followed by a point that is pursuing another linearly-moving point. The term is derived from the Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group.
*Greek language, a branch of the Indo-European language family.
**Proto-Greek language, the assumed last common ancestor ...
words and . The classic (and best-known) form of a radiodrome is known as the "dog curve"; this is the path a dog follows when it swims across a stream with a current after something it has spotted on the other side. Because the dog drifts with the current, it will have to change its heading; it will also have to swim further than if it had taken the optimal heading. This case was described by Pierre Bouguer
Pierre Bouguer () (16 February 1698, Croisic – 15 August 1758, Paris) was a French mathematician, geophysicist, geodesist, and astronomer. He is also known as "the father of naval architecture".
Career
Bouguer's father, Jean Bouguer, one ...
in 1732.
A radiodrome may alternatively be described as the path a dog follows when chasing a hare, assuming that the hare runs in a straight line at a constant velocity.
Mathematical analysis
Introduce a coordinate system with origin at the position of the dog at time zero and with ''y''-axis in the direction the hare is running with the constant speed . The position of the hare at time zero is with and at time it is The dog runs with the constant speed towards the instantaneous position of the hare. The differential equation corresponding to the movement of the dog, , is consequently It is possible to obtain a closed-form analytic expression for the motion of the dog. From () and (), it follows that Multiplying both sides with and taking the derivative with respect to , using that one gets or From this relation, it follows that where is the constant of integration determined by the initial value of ' at time zero, , i.e., From () and (), it follows after some computation that Furthermore, since , it follows from () and () that If, now, , relation () integrates to where is the constant of integration. Since again , it's The equations (), () and (), then, together imply If , relation () gives, instead, Using once again, it follows that The equations (), () and (), then, together imply that If , it follows from () that If , one has from () and () that , which means that the hare will never be caught, whenever the chase starts.See also
*Mice problem
In mathematics, the mice problem is a continuous pursuit–evasion problem in which a number of mice (or insects, dogs, missiles, etc.) are considered to be placed at the corners of a regular polygon. In the classic setup, each then begins to m ...
References
*. *{{Citation, first=Francisco , last=Gomes Teixera, title= Traité des Courbes Spéciales Remarquables , editor=Imprensa da universidade , place=Coimbra , year=1909 , volume=2 , pages=255, url =http://quod.lib.umich.edu/u/umhistmath/aat2332.0005.001/261?view=pdf Plane curves Differential equations Analytic geometry Pursuit–evasion