Curve Of Pursuit on:
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The pursuit curve was first studied by
The path followed by a single pursuer, following a pursuee that moves at constant speed on a line, is a
Typical drawings of curves of pursuit have each point acting as both pursuer and pursuee, inside a
Mathworld
with a slightly narrower definition that , ''L''′(''t''), and , ''F''′(''t''), are constant
{{Differential transforms of plane curves Curves Pursuit–evasion
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 curve of pursuit is a curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
constructed by analogy to having a point
Point or points may refer to:
Places
* Point, Lewis, a peninsula in the Outer Hebrides, Scotland
* Point, Texas, a city in Rains County, Texas, United States
* Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland
* 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 in time, the pursuee is always on the pursuer's tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
. That is, given , the pursuer (follower), and , the pursued (leader), for every with there is an such that
:
History
The pursuit curve was first studied 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. In an article on navigation
Navigation is a field of study that focuses on the process of monitoring and controlling the movement of a craft or vehicle from one place to another.Bowditch, 2003:799. The field of navigation includes four general categories: land navigation, ...
, Bouguer defined a curve of pursuit to explore the way in which one ship might maneuver while pursuing another.
Leonardo da Vinci
Leonardo di ser Piero da Vinci (15 April 14522 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, Drawing, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially res ...
has occasionally been credited with first exploring curves of pursuit. However Paul J. Nahin
Paul J. Nahin (born November 26, 1940 in Orange County, California) is an American electrical engineer and author who has written 20 books on topics in physics and mathematics, including biographies of Oliver Heaviside, George Boole, and Claude Sh ...
, having traced such accounts as far back as the late 19th century, indicates that these anecdotes are unfounded.
Single pursuer
The path followed by a single pursuer, following a pursuee that moves at constant speed on a line, is a radiodrome In geometry, a radiodrome is the pursuit curve followed by a point that is pursuing another linearly-moving point. The term is derived from the Greek words and . The classic (and best-known) form of a radiodrome is known as the "dog curve"; this i ...
.
It is a solution of the differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
.
Multiple pursuers
Typical drawings of curves of pursuit have each point acting as both pursuer and pursuee, inside a polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...
, and having each pursuer pursue the adjacent point on the polygon. An example of this is the 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 ...
.
See also
*Logarithmic spiral
A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige Linie"). Mor ...
*Tractrix
In geometry, a tractrix (; plural: tractrices) is the curve along which an object moves, under the influence of friction, when pulled on a horizontal plane by a line segment attached to a pulling point (the ''tractor'') that moves at a right angl ...
* Circles of Apollonius#Apollonius pursuit problem
*Pursuit–evasion
Pursuit–evasion (variants of which are referred to as cops and robbers and graph searching) is a family of problems in mathematics and computer science in which one group attempts to track down members of another group in an environment. Early ...
References
External links
Mathworld
with a slightly narrower definition that , ''L''′(''t''), and , ''F''′(''t''), are constant
{{Differential transforms of plane curves Curves Pursuit–evasion