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FIELD LINES, ORTHOGONAL LINES, DOUBLE ORTHOGONAL SYSTEM
Two families of curves are said to be orthogonal
when at every point common to a curve of each family, the tangents are
orthogonal, and one of the families is said to be composed of the orthogonal
trajectories of the other. This constitutes a double orthogonal
system of curves.
If the first family of curves is defined by:  then the orthogonal trajectories are defined by:  
Geometrical definition  f( M ) = constant  g (M) = constant with 
Cartesian implicit equation  P(x, y) = constant  Q(x, y ) = constant with 
Harmonic Cartesian implicit equation  P(x, y) = constant with P harmonic  Q(x, y ) = constant with 
Complex implicit equation  Re (f (z) ) = constant with f holomorphic (hence conformal)  Im (f (z) ) = constant 
Polar implicit equation  P(r, ) = constant  Q(r, ) = constant with 
Cartesian differential equation  y' = f(x, y)  y' = 1 / f(x, y) 
Polar differential equation  r' = f(r,)  r' =  r² / f(r, ) 
Field lines of the Cartesian field:  (f(x, y), g(x, y))  (g(x, y), f(x, y)) 
Field lines of the polar field:  (f(r, ), g(r,))  (g(r, ), f(r, )) 
If both the families are given by a unique parametric form: , with fixed u and variable v for the first family, fixed v and variable u for the second one, then the families are orthogonal iff , which is always the case when P and Q are the real and imaginary part of a holomorphic function (inverse function of the one above). 
The orthogonal trajectories of a family of lines are the involutes of the envelope of this family; therefore, they are parallel curves (see example 13 below).
Examples :

Common parametric expression
(red curves: u = constant blue curves: v = constant) 






so 
Magnetic field lines induced by a uniform linear current orthogonal
at O to xOy.
Electrostatic equipotential induced by a charge placed at O or charges uniformly distributed on a line orthogonal at O to xOy 


Family of homofocal parabolas


so 



so 
Approximate view of the example n° 8 below in a neighbourhood of
O.
They are the contour lines of the hyperbolic paraboloid 




limit case of the example n°7 below when the conductors
are infinitely close.
See also the Smith chart. 
= two pencils of orthogonal singular circles 



Remark: figure obtained by inversion of that of the example n° 2. 



so 
Opposite, view for n = 4 and n = 4. 



????  ????  Field
lines of a magnetic dipole
Field lines of an electrostatic
dipole





so 
Magnetic field lines induced by a uniform linear current orthogonal
at A to xOy and a parallel current, in the opposite direction,
passing by B.
Electrostatic equipotential lines induced by charges uniformly distributed on a line orthogonal at A to xOy and opposite charges uniformly distributed on a line orthogonal at B to xOy. 
= two pencils of orthogonal circles 


so 
Magnetic field lines induced by a uniform linear current orthogonal
at A to xOy and a parallel current, in the same direction,
passing by B.
Electrostatic equipotential lines induced by charges uniformly distributed on a line orthogonal at A to xOy and equal charges uniformly distributed on a line orthogonal at B to xOy. 
See a generalisation at Cassinian curve for the red curves, and at stelloid for the blue curves: case where . 


Electrostatic equipotential lines induced by two opposite charges placed at A and B, in other words, an electrostatic dipole. 



Electrostatic equipotential lines induced by two equal charges placed at A and B. 


Lattice of homofocal conics

f(z) = argcosh(z)
f^{}^{1}(z) = cosh (z) (image of the first lattice by the Joukovski transformation: j(z) = (z + 1/z)/2) 
Electrostatic equipotential lines induced by charges
uniformly distributed on the segment line [AB]???
Interference pattern 


Involute of circles and their generatrices

????  ???? 



Streamlines of a uniform flow perturbed by an obstacle (the segment line [AB] with A(0, 1) and B(0, 1)) 
The blue curve passing by O (obtained for a = 1) is a bullet nose curve 


where j is the Joukovski transformation 
Streamlines of a uniform flow perturbed by the disk with centre O and radius 1. 

where . For example, when v = pi/2, we get: 
Streamlines of a uniform flow in a bent tube. 

Other examples:
Lemniscates of Bernoulli : and (case n = –2 of the example 6 above)  Quatrefoils ; and their orthogonal trajectories . 


Logarithmic spirals: and.  Parabolas and ellipses 


Red parabolas and semicubical parabolas .  cycloids: and symmetric cycloids 


See also tractrix, as well as this article.
The projection on a horizontal plane of the slope and contour lines of a surface form two orthogonal lattices; see for example the egg box.
This notion of orthogonal curves can be generalised to any angle; two families of curves intersect under the angle V when, at each point common to the families, the tangents form an angle V, and one of the families is composed of the trajectories at angle V of the other one.
For example, the trajectories under the angle V of the pencil of lines passing by O are the logarithmic spirals;:
The 3D generalisation of the double orthogonal systems
is the notion of triple
orthogonal system of surfaces.
Map obtained by stereographic projection with the pole
a point on the equator; the parallels and the meridians form two bundles
of orthogonal circles. Bourse du commerce, Paris. 

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© Robert FERRÉOL 2022