geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a radiodrome is a specific type of

pursuit curve In geometry, a curve of pursuit is a curve constructed by analogy to having a point (geometry), point or points representing pursuers and pursuees; the curve of pursuit is the curve traced by the pursuers. Definition With the paths of the purs ...

: the path traced by a point that continuously moves toward a target traveling in a straight line at constant speed. The term comes from the Latin ''radius'' (ray or spoke) and the Greek ''dromos'' (running or racetrack), reflecting the radial nature of the motion. The most classic and widely recognized example is the so-called dog curve, which describes the path of a dog swimming across a river toward a hare moving along the opposite bank. Because of the current, the dog must constantly adjust its heading, resulting in a longer, curved trajectory. This case was first described by the French mathematician and hydrographer

Pierre Bouguer Pierre Bouguer () (16 February 1698, Le 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, ...

in 1732. Radiodromes are distinguished from other pursuit curves by the assumption that the pursuer always heads directly toward the target’s current position, while the target moves at a constant velocity along a straight path.

Mathematical analysis

Introduce a

coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are ...

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

T_x-x

and taking the derivative with respect to , using that one gets or From this relation, it follows that where is the

constant of integration In calculus, the constant of integration, often denoted by C (or c), is a constant term added to an antiderivative of a function f(x) to indicate that the indefinite integral of f(x) (i.e., the set of all antiderivatives of f(x)), on a connecte ...

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

\lim_y(x) = \infty

, which means that the hare will never be caught, whenever the chase starts.

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

Mathematical analysis

See also

References