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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, q) = constant Q(r, q) = constant with 
Cartesian differential equation y' = f(x, y) y' = -1 / f(x, y)
Polar differential equation r' = f(r, q) r' = - r² / f(r, q)
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, q), g(r, q)) (g(r, q), -f(r, q))
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 q = q0
differential equation xy' - y = 0 dq/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+cosq)
differential
equation
yy'²-2xy'-y=0

so

 
3
initial curves (red hyperbolas) orthogonal curves (blue hyperbolas)
Cartesian polar
implicit equation x² - y² = cte r²cos2q= cte
differential equation yy' - x = 0 r' = r tan2q
field (y, x) (sin2q, cos2q)
Cartesian polar
implicit equation xy = cte r²sin2q = cte 
differential equation xy' + y = 0 r' = -r cot2q
field (-x, y) (-cos2q , sin2q)

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 = acosq
differential equation 2xyy' =   y²-x² r' + r tanq = 0
field (2xyy² - x²) (sinq, -cosq)
Cartesian polar
implicit equation (x² + y²) = ay r = asinq
differential equation  ( y² - x²) y' + 2xy = 0 r' = r cot
field (x² - y², 2xy) (cosq, sinq)
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

5
initial curves (red cardioids) orthogonal curves (blue cardioids)
polar
implicit equation r=acos² q/2
differential equation r' = r tan q/2
field (sinq/2, cosq/2)
polar
implicit equation r=asin² q/2
differential equation r' = -r cot q/2
field (-cosq/2 , sinq/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 rncosnq= cte
differential equation r' = r tan nq
field (sinnq, cosnq)
polar
implicit equation rnsinnq = cte 
differential equation r' = -r cotnq
field (-cosnq , sinnq)
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 = acos²q
differential equation 3xyy' =  2 y²-x² r' + 2r tanq = 0
field (3xy, 2 y² - x²) (2sinq, -cosq)
Cartesian polar
implicit equation (x² + y²)3 = a4y² r² = a²sinq
differential equation  (2y²-x² ) y' + 3xy = 0 2r' = r cot
field (x² - 2 y², 3xy) (cosq, 2sinq)
Remark: the examples 4 and 7 are part of the more general example of Clairaut's curves: r = acosnq and rn= ansinq.
 ???? ???? 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
cos2q = cte
differential equation sin2qr'= r3 - rcos2q
field (sin2q, cos2q - r²)
geometrical definition (Ox, AM) + (Ox, BM) = constant
field MA/MA² - MB/MB²
polar
implicit equation r² = cos2a /cos(2q+2a)
differential equation sin2qr'= r3 - rcos2q
field (r² - cos2q, sin2q)

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

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.
 
 
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© Robert FERRÉOL 2017