For a
plane curve
In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic pla ...
''C'' and a given fixed point ''O'', the pedal equation of the curve is a relation between ''r'' and ''p'' where ''r'' is the distance from ''O'' to a point on ''C'' and ''p'' is the perpendicular distance from ''O'' to the
tangent line
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 ...
to ''C'' at the point. The point ''O'' is called the ''pedal point'' and the values ''r'' and ''p'' are sometimes called the ''pedal coordinates'' of a point relative to the curve and the pedal point. It is also useful to measure the distance of ''O'' to the normal
(the ''contrapedal coordinate'') even though it is not an independent quantity and it relates to
as
.
Some curves have particularly simple pedal equations and knowing the pedal equation of a curve may simplify the calculation of certain of its properties such as curvature. These coordinates are also well suited for solving certain type of force problems in
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
and
celestial mechanics
Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...
.
Equations
Cartesian coordinates
For ''C'' given in
rectangular coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
by ''f''(''x'', ''y'') = 0, and with ''O'' taken to be the origin, the pedal coordinates of the point (''x'', ''y'') are given by:
:
:
The pedal equation can be found by eliminating ''x'' and ''y'' from these equations and the equation of the curve.
The expression for ''p'' may be simplified if the equation of the curve is written in
homogeneous coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. T ...
by introducing a variable ''z'', so that the equation of the curve is ''g''(''x'', ''y'', ''z'') = 0. The value of ''p'' is then given by
:
where the result is evaluated at ''z''=1
Polar coordinates
For ''C'' given in
polar coordinates
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the or ...
by ''r'' = ''f''(θ), then
:
where
is the
polar tangential angle given by
:
The pedal equation can be found by eliminating θ from these equations.
Alternatively, from the above we can find that
:
where
is the "contrapedal" coordinate, i.e. distance to the normal. This implies that if a curve satisfies an autonomous differential equation in polar coordinates of the form:
:
its pedal equation becomes
:
Example
As an example take the logarithmic spiral with the spiral angle α:
:
Differentiating with respect to
we obtain
:
hence
:
and thus in pedal coordinates we get
:
or using the fact that
we obtain
:
This approach can be generalized to include autonomous differential equations of any order as follows: A curve ''C'' which a solution of an ''n''-th order autonomous differential equation (
) in polar coordinates
:
is the
pedal curve
A pedal (from the Latin '' pes'' ''pedis'', "foot") is a lever designed to be operated by foot and may refer to:
Computers and other equipment
* Footmouse, a foot-operated computer mouse
* In medical transcription, a pedal is used to control p ...
of a curve given in pedal coordinates by
:
where the differentiation is done with respect to
.
Force problems
Solutions to some force problems of classical mechanics can be surprisingly easily obtained in pedal coordinates.
Consider a dynamical system:
:
describing an evolution of a test particle (with position
and velocity
) in the plane in the presence of central
and Lorentz like
potential. The quantities:
:
are conserved in this system.
Then the curve traced by
is given in pedal coordinates by
:
with the pedal point at the origin. This fact was discovered by P. Blaschke in 2017.
Example
As an example consider the so-called
Kepler problem
In classical mechanics, the Kepler problem is a special case of the two-body problem, in which the two bodies interact by a central force ''F'' that varies in strength as the inverse square of the distance ''r'' between them. The force may be ei ...
, i.e. central force problem, where the force varies inversely as a square of the distance:
:
we can arrive at the solution immediately in pedal coordinates
:
,
where
corresponds to the particle's angular momentum and
to its energy. Thus we have obtained the equation of a conic section in pedal coordinates.
Inversely, for a given curve ''C'', we can easily deduce what forces do we have to impose on a test particle to move along it.
Pedal equations for specific curves
Sinusoidal spirals
For a
sinusoidal spiral
In algebraic geometry, the sinusoidal spirals are a family of curves defined by the equation in polar coordinates
:r^n = a^n \cos(n \theta)\,
where is a nonzero constant and is a rational number other than 0. With a rotation about the origin, ...
written in the form
:
the polar tangential angle is
:
which produces the pedal equation
:
The pedal equation for a number of familiar curves can be obtained setting ''n'' to specific values:
Spirals
A spiral shaped curve of the form
:
satisfies the equation
:
and thus can be easily converted into pedal coordinates as
:
Special cases include:
Epi- and hypocycloids
For an epi- or hypocycloid given by parametric equations
:
:
the pedal equation with respect to the origin is
:
or
:
with
:
Special cases obtained by setting ''b''= for specific values of ''n'' include:
Other curves
Other pedal equations are:,
[Yates p. 169, Edwards p. 163, Blaschke sec. 2.1]
See also
*
Pedal curve
A pedal (from the Latin '' pes'' ''pedis'', "foot") is a lever designed to be operated by foot and may refer to:
Computers and other equipment
* Footmouse, a foot-operated computer mouse
* In medical transcription, a pedal is used to control p ...
References
*
*
*
External links
*
{{DEFAULTSORT:Pedal Coordinates
Coordinate systems