In mathematics, an orthogonal trajectory is a curve which intersects any curve of a given pencil 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 equations. The standard method establishes a first order ordinary differential equation and solves it by separation of variables. 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) and appear in physics as electric fields and their equipotential curves.
If the trajectory intersects the given curves by an arbitrary (but fixed) angle, one gets an isogonal trajectory.
Generally, one assumes that the pencil of curves is given implicitly by an equation
(0)
: F(x,y,c)=0,
: x2+y2-c=0 ,
y=cx2 \leftrightarrow y-cx2=0 ,
c
y=f(x,c)
y-f(x,c)=0
x
(1)
: Fx(x,y,c)+Fy(x,y,c) y'=0,
: 2x+2yy'=0 ,
: y'-2cx=0 .
c
(2)
: y'=f(x,y),
: y'=- | x |
y |
,
: y'=2 | y |
x |
,
(x,y)
(3)
: y'=-
1 | |
f(x,y) |
,
: y'=y/x ,
: y'=- | x |
2y |
.
y=mx, m\in\R
x2+2y2=d, d>0 .
If the pencil of curves is represented implicitly in polar coordinates by
(0p)
: F(r,\varphi,c)=0
(1p)
: Fr(r,\varphi,c)+F\varphi(r,\varphi,c) \varphi'=0,
(2p)
: \varphi'=f(r,\varphi)
(3p)
: \varphi'=- | 1 |
{\color{red |
r2}f(r,\varphi)} .
(0p)
: F(r,\varphi,c)=r-c(1+\cos\varphi)=0, c>0 .
(1p)
: Fr(r,\varphi,c)+F\varphi(r,\varphi,c) \varphi'=1+c\sin\varphi \varphi'=0,
c
(2p)
: \varphi'=- | 1+\cos\varphi |
r\sin\varphi |
(3p)
: \varphi'= | \sin\varphi |
r(1+\cos\varphi) |
r=d(1-\cos\varphi) , d>0 ,
A curve, which intersects any curve of a given pencil of (planar) curves by a fixed angle
\alpha
Between the slope
η'
y'
(x,y)
η'= | y'+\tan(\alpha) |
1-y'\tan(\alpha) |
.
\tan(\alpha+\beta)
\alpha → 90\circ
For the determination of the isogonal trajectory one has to adjust the 3. step of the instruction above:
: y'=
f(x,y)+\tan(\alpha) | |
1-f(x,y) \tan(\alpha) |
.
For the 1. example (concentric circles) and the angle
\alpha=45\circ
(3i)
: y'=
-x/y+1 | |
1+x/y |
.
z=y/x
\arctan | y | + |
x |
1 | |
2 |
ln(x2+y2)=C .
C-\varphi=ln(r) ,
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.