Orthogonal Trajectories
   HOME

TheInfoList



OR:

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 In geometry, the elliptic coordinate system is a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal ellipses and hyperbolae. The two foci F_ and F_ are generally taken to be fixed at -a and +a, respectively ...
) 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 curve A contour line (also isoline, isopleth, or isarithm) of a function of two variables is a curve along which the function has a constant value, so that the curve joins points of equal value. It is a plane section of the three-dimensional grap ...
s. 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) :\ F(x,y,c)=0,\qquad 1. example :\ x^2+y^2-c=0\ ,\qquad 2. example y=cx^2 \ \leftrightarrow \ y-cx^2=0 \ , where c is the parameter of the pencil. If the pencil is given ''explicitly'' by an equation y=f(x,c), one can change the representation into an implicit one: y-f(x,c)=0. For the consideration below it is supposed that all necessary derivatives do exist. ;Step 1. Differentiating implicitly for x yields :(1) :\ F_x(x,y,c)+F_y(x,y,c)\;y'=0,\qquad in 1. example :\ 2x+2yy'=0\ ,\qquad 2. example :\ y'-2cx=0\ . ;Step 2. Now it is assumed, that equation (0) can be solved for parameter c, which can thus be eliminated from equation (1). One gets the differential equation of first order :(2) :\ y'=f(x,y), \qquad in 1. example :\ y'=-\frac\ ,\qquad 2. example :\ y'=2\frac \ , which is fulfilled by the given pencil of curves. ;Step 3. Because the slope of the orthogonal trajectory at a point (x,y) 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) : \ y'=-\frac\ , \qquad in 1. example :\ y'=y/x\ , \qquad 2. example :\ y'=-\frac \ . ;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 y=mx,\ m\in \R 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 x^2+2y^2=d,\ d>0\ .


In polar coordinates

If the pencil of curves is represented implicitly in ''polar coordinates'' by :(0p) :\ F(r,\varphi,c)=0 one determines, alike the cartesian case, the parameter free differential equation :(1p) :\ F_r(r,\varphi,c)+F_\varphi(r,\varphi,c)\;\varphi'=0,\qquad :(2p) :\ \varphi'=f(r,\varphi) of the pencil. The differential equation of the orthogonal trajectories is then (see Redheffer & Port p. 65, Heuser, p. 120) :(3p) :\ \varphi'=-\frac\ . Example:
Cardioid In geometry, a cardioid () is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It can also be defined as an epicycloid having a single cusp. It is also a type of sinusoidal spi ...
s: :(0p) :\ F(r,\varphi,c)=r-c(1+\cos\varphi)=0,\ c>0\ .\ (in diagram: blue) :(1p) :\ F_r(r,\varphi,c)+F_\varphi(r,\varphi,c)\;\varphi'=1+c\sin\varphi\;\varphi'=0,\qquad Elimination of c yields the differential equation of the given pencil: :(2p) :\ \varphi'=-\frac Hence the differential equation of the orthogonal trajectories is: :(3p) :\ \varphi'=\frac After solving this differential equation by ''separation of variables'' one gets :r=d(1-\cos\varphi)\ ,\ d>0 \ , 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 \alpha is called isogonal trajectory. Between the slope \eta' of an isogonal trajectory and the slope y' of the curve of the pencil at a point (x,y) the following relation holds: : \eta'=\frac\ . This relation is due to the formula for \tan(\alpha+\beta). For \alpha \rightarrow 90^\circ 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) : \ y'=\frac\ . For the 1. example (concentric circles) and the angle \alpha=45^\circ one gets
:(3i) : \ y'=\frac\ . This is a special kind of differential equation, which can be transformed by the substitution z=y/x into a differential equation, that can be solved by ''separation of variables''. After reversing the substitution one gets the equation of the solution: :\arctan\frac+\frac\ln(x^2+y^2)=C\ . Introducing polar coordinates leads to the simple equation :C-\varphi=\ln(r)\ , which describes
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 ...
s (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 In geometry, a Cassini oval is a quartic plane curve defined as the locus (mathematics), locus of points in the plane (geometry), plane such that the Product_(mathematics), product of the distances to two fixed points (Focus (geometry), foci) is ...
*
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 In geometry, Apollonian circles are two families (pencils) of circles such that every circle in the first family intersects every circle in the second family orthogonally, and vice versa. These circles form the basis for bipolar coordinates. Th ...
, 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