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 :
Definitions
Common parametric expression
(red curves: u = constant
blue curves: v = constant)
Inverse images of the Cartesian coordinate lines by the conformal map f defined by
Physical interpretation of the red curves
Plot
1
initial curves (red circles) orthogonal curves (blue lines)
 Cartesian polar implicit equation x² + y² = cte r = a differential equation yy' + x = 0 r' = 0 field (y, -x) (0 , 1)
 Cartesian polar implicit equation y = kx = q0 differential equation xy' - y = 0 d/dr = 0 field (x, y) (1 , 0)

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

2
Family of homofocal parabolas
initial curves (red parabolas) orthogonal curves (blue parabolas)
 Cartesian polar implicit equation y² = 4u²(u²-x) r=2u²/ (1+cosq) differential equation yy'²+2xy'-y=0
 Cartesian polar implicit equation y² = 4v²(v²+x) r=-2v²/ (1+cos) differential equation yy'²-2xy'-y=0

so

3
initial curves (red hyperbolas) orthogonal curves (blue hyperbolas)
 Cartesian polar implicit equation x² - y² = cte r²cos2= cte differential equation yy' - x = 0 r' = r tan2 field (y, x) (sin2, cos2)
 Cartesian polar implicit equation xy = cte r²sin2= cte differential equation xy' + y = 0 r' = -r cot2 field (-x, y) (-cos2 , sin2)

so

Approximate view of the example n° 8 below in a neighbourhood of O.

They are the contour lines of the hyperbolic paraboloid

4
initial curves (red circles) orthogonal curves (blue circles)
 Cartesian polar implicit equation x² + y² = ax r = acos differential equation 2xyy' =   y²-x² r' + r tan = 0 field (2xy,  y² - x²) (sin, -cos)
 Cartesian polar implicit equation (x² + y²) = ay r = asin differential equation ( y² - x²) y' + 2xy = 0 r' = r cot field (x² - y², 2xy) (cos, sin)
limit case of the example n°7 below when the conductors are infinitely close.

= two pencils of orthogonal singular circles

5
initial curves (red cardioids) orthogonal curves (blue cardioids)
 polar implicit equation r=acos²/2 differential equation r' = r tan /2 field (sin/2, cos/2)
 polar implicit equation r=asin²/2 differential equation r' = -r cot/2 field (-cos/2 , sin/2)
Remark: figure obtained by inversion of that of the example n° 2.
6
initial curves (red sinusoidal spirals of index -n) orthogonal curves (blue sinusoidal spirals of index -n)
 polar implicit equation rncosn= cte differential equation r' = r tan n field (sinn, cosn)
 polar implicit equation rnsinn= cte differential equation r' = -r cotn field (-cosn , sinn)
The four previous cases correspond to n = 1/2, n = 2, n = -1 , n =-1/2

so

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

7
initial curves (red double eggs) orthogonal curves (blue curves of the dipole)
 Cartesian polar implicit equation (x² + y²)3 = a²x4 r = acos² differential equation 3xyy' =  2 y²-x² r' + 2r tanq = 0 field (3xy, 2 y² - x²) (2sin, -cos)
 Cartesian polar implicit equation (x² + y²)3 = a4y² r² = a²sin differential equation (2y²-x² ) y' + 3xy = 0 2r' = r cot field (x² - 2 y², 3xy) (cos, 2sin)
Remark: the examples 4 and 7 are part of the more general example of Clairaut's curves: r = acosn and rn= ansin.
???? ???? Field lines of a magnetic dipole

Field lines of an electrostatic dipole
(limit case, inverting the red and blue curves, of the example 9 below).

8
initial curves (red circles) orthogonal curves (blue circles)
 geometrical definition MA/MB = constant with A(1,0) and B(-1,0) implicit equation (x - 1)² + y² = cte. ((x+ 1)² + y²)
 geometrical definition (MA, MB) = constant field MA/MA² - MB/MB²

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
9
initial curves (red Cassinian ovals) orthogonal curves (blue rectangular hyperbolas)
 geometrical definition MA .MB = constant

 polar implicit equation r4 - 2r² cos2= cte differential equation r'( r2 - cos2) +rsin2=0 field (sin2, cos2 - r²)
 geometrical definition (Ox, AM) + (Ox, BM) = constant field MA/MA² - MB/MB²
 polar implicit equation r² = cos2a / cos(2(-a)) differential equation sin2r'= r3 - rcos2q field (r² - cos2, sin2)

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 .

10
initial curves (red Cayley equipotential lines) orthogonal curves (in blue)
 geometrical definition 1/MA-1/MB = cte field
 geometrical definition field MA/MA3 - MB/MB3
Electrostatic equipotential lines induced by two opposite charges placed at A and B, in other words, an electrostatic dipole.
11
initial curves (red Cayley ovals) orthogonal curves (in blue)
 geometrical definition 1/MA + 1/MB = constant field
 geometrical definition field MA/MA3 + MB/MB3
Electrostatic equipotential lines induced by two equal charges placed at A and B.
12
Lattice of homofocal conics
initial curves (red ellipses) orthogonal curves (blue hyperbolas)
 geometrical definition MA + MB = constant field MA/MA - MB/MB Cartesian equation x²/(1+cte)+y²/cte=1
 geometrical definition MA - MB  = constant field MA/MA + MB/MB Cartesian equation x²/(1-cte)-y²/cte=1
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
13
Involute of circles and their generatrices

initial curves (red half-involutes of a fixed circle) orthogonal curves (blue half-tangents to the circle)
 complex parametrization
 complex parametrization

???? ????
14
initial curves (red quartics) orthogonal curves (blue quartics)
 Cartesian implicit equation y² = a²(1  + 1 / (a²+x²)) parametrization x = a tan t y² = a²  + cos²t
 Cartesian implicit equation x² = a²(1 - 1 / (a²+y²)) parametrization x² = a²  - cos²t y = a tan t
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

15
initial curves (red cubic hyperbolas) orthogonal curves (blue cubic hyperbolas)
 Cartesian implicit equation (y - constant) (x²+y²) = y
 Cartesian implicit equation (x - constant) (x²+y²) + x =0

where j is the Joukovski transformation

Streamlines of a uniform flow perturbed by the disk with centre O and radius 1.
16

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