In mathematics an orthogonal trajectory is a curve, which intersects any curve of a given
pencil
A pencil () is a writing or drawing implement with a solid pigment core in a protective casing that reduces the risk of core breakage, and keeps it from marking the user's hand.
Pencils create marks by physical abrasion, leaving a trail ...
of (planar) curves ''orthogonally''.
For example, the orthogonal trajectories of a pencil of ''concentric circles'' are the lines through their common center (see diagram).
Suitable methods for the determination of orthogonal trajectories are provided by solving
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 ...
s. The standard method establishes a first order
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
and solves it by
separation of variables
In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs ...
. Both steps may be difficult or even impossible. In such cases one has to apply numerical methods.
Orthogonal trajectories are used in mathematics for example as curved coordinate systems (i.e.
elliptic coordinates) or appear in physics as
electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
s and their
equipotential curves.
If the trajectory intersects the given curves by an arbitrary (but fixed) angle, one gets an isogonal trajectory.
Determination of the orthogonal trajectory
In cartesian coordinates
Generally one assumes, that the pencil of curves is
implicitly given by an equation
:(0)
1. example
2. example
where
is the parameter of the pencil. If the pencil is given ''explicitly'' by an equation
, one can change the representation into an implicit one:
. For the consideration below it is supposed that all necessary derivatives do exist.
;Step 1.
Differentiating implicitly for
yields
:(1)
in 1. example
2. example
;Step 2.
Now it is assumed, that equation (0) can be solved for parameter
, which can thus be eliminated from equation (1). One gets the differential equation of first order
:(2)
in 1. example
2. example
which is fulfilled by the given pencil of curves.
;Step 3.
Because the slope of the orthogonal trajectory at a point
is the
negative multiplicative inverse of the slope of the given curve at this point, the orthogonal trajectory satisfies the differential equation of first order
:(3)
in 1. example
2. example
;Step 4.
This differential equation can (hopefully) be solved by a suitable method.
For both examples ''separation of variables'' is suitable. The solutions are:
in example 1, the lines
and
in example 2, the
ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
s
In polar coordinates
If the pencil of curves is represented implicitly in ''polar coordinates'' by
:(0p)
one determines, alike the cartesian case, the parameter free differential equation
:(1p)
:(2p)
of the pencil. The differential equation of the orthogonal trajectories is then (see Redheffer & Port p. 65, Heuser, p. 120)
:(3p)
Example:
Cardioids:
:(0p)
(in diagram: blue)
:(1p)
Elimination of
yields the differential equation of the given pencil:
:(2p)
Hence the differential equation of the orthogonal trajectories is:
:(3p)
After solving this differential equation by ''separation of variables'' one gets
:
which describes the pencil of cardioids (red in diagram), symmetric to the given pencil.
Isogonal trajectory
A curve, which intersects any curve of a given pencil of (planar) curves by a fixed angle
is called isogonal trajectory.
Between the slope
of an isogonal trajectory and the slope
of the curve of the pencil at a point
the following relation holds:
:
This relation is due to the formula for
. For
one gets the condition for the ''orthogonal'' trajectory.
For the determination of the isogonal trajectory one has to adjust the 3. step of the instruction above:
;3. step (isog. traj.)
The differential equation of the isogonal trajectory is:
*(3i)
For the 1. example (concentric circles) and the angle
one gets
:(3i)
This is a special kind of differential equation, which can be transformed by the substitution
into a differential equation, that can be solved by ''separation of variables''. After reversing the substitution one gets the equation of the solution:
:
Introducing polar coordinates leads to the simple equation
:
which describes
logarithmic spirals (s. diagram).
Numerical methods
In case that the differential equation of the trajectories can not be solved by theoretical methods, one has to solve it numerically, for example by
Runge–Kutta methods
In numerical analysis, the Runge–Kutta methods ( ) are a family of implicit and explicit iterative methods, which include the Euler method, used in temporal discretization for the approximate solutions of simultaneous nonlinear equations. The ...
.
See also
*
Cassini oval
*
Confocal conic sections
In geometry, two conic sections are called confocal, if they have the same foci. Because ellipses and hyperbolas possess two foci, there are confocal ellipses, confocal hyperbolas and confocal mixtures of ellipses and hyperbolas. In the mixture of ...
*
Trajectory
A trajectory or flight path is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete traj ...
*
Apollonian circles, pairs of families of circles that are all orthogonal to each other
References
* A. Jeffrey: ''Advanced Engineering Mathematics'', Hartcourt/Academic Press, 2002, , p. 233.
*S. B. Rao:
Differential Equations', University Press, 1996, , p. 95.
*R. M. Redheffer, D. Port:
Differential Equations: Theory and Applications', Jones & Bartlett, 1991, , p. 63.
*H. Heuser: ''Gewöhnliche Differentialgleichungen'', Vieweg+Teubner, 2009, , p. 120.
*.
External links
- applet allowing user to draw families of curves and their orthogonal trajectories.
mathcurve: FIELD LINES, ORTHOGONAL LINES, DOUBLE ORTHOGONAL SYSTEM{{Differential equations topics
Curves