Find the Orthogonal Trajectories of the Family of Curves

next curve

previous curve

2D curves

3D curves

surfaces

fractals

polyhedra

FIELD LINES, ORTHOGONAL LINES, DOUBLE ORTHOGONAL SYSTEM

2 families of curves are said to be orthogonal when at every point mutual to a bend of each family, the tangents are orthogonal, and i 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 outset family of curves is defined by:	so the orthogonal trajectories are defined past:
Geometrical definition	f( M ) = abiding	g (1000) = abiding with
Cartesian implicit equation	P(x, y) = constant	Q(x, y ) = abiding with
Harmonic Cartesian implicit equation	P(x, y) = constant with P harmonic	Q(ten, y ) = constant with
Circuitous implicit equation	Re (f (z) ) = abiding with f holomorphic (hence conformal)	Im (f (z) ) = abiding
Polar implicit equation	P(r,) = abiding	Q(r,) = constant with
Cartesian differential equation	y' = f(x, y)	y' = -i / f(ten, y)
Polar differential equation	r ' = f(r,)	r ' = - r ² / f(r,)
Field lines of the Cartesian field:	(f(x, y), m(x, y))	(grand(x, y), -f(10, y))
Field lines of the polar field:	(f(r,), g(r,))	(1000(r,), -f(r,))

The orthogonal trajectories of a family of lines are the involutes of the envelope of this family; therefore, they are parallel curves (run into instance xiii beneath).

Examples :

Definitions	Mutual parametric expression (red curves: u = constant blue curves: v = abiding)	Inverse images of the Cartesian coordinate lines by the conformal map f divers by	Physical estimation of the crimson curves	Plot
1		so	Magnetic field lines induced by a compatible linear current orthogonal at O to xOy. Electrostatic equipotential induced past 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 = ivu²(u²-x) r=twou²/ (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		so	Approximate view of the example north° 8 below in a neighbourhood of O. They are the profile lines of the hyperbolic paraboloid
4			limit example of the example due north°seven below when the conductors are infinitely close. See also the Smith chart.	= ii pencils of orthogonal singular circles
5 initial curves (cherry cardioids) orthogonal curves (blueish cardioids)			Remark: figure obtained by inversion of that of the instance n° 2.
vi The four previous cases correspond to n = ane/2, northward = 2, n = -one , due north =-1/two		so	Opposite, view for n = 4 and northward = -4.
7 Remark: the examples 4 and 7 are part of the more than general example of Clairaut's curves: r = acos north and r due north = an sin .	????	????	Field lines of a magnetic dipole Field lines of an electrostatic dipole (limit instance, inverting the red and blue curves, of the example nine below).
viii initial curves (red circles) orthogonal curves (blue circles) geometrical definition MA/MB = constant with A(1,0) and B(-one,0) implicit equation (x - i)² + y² = cte. ((x+ ane)² + y²) geometrical definition (MA, MB) = constant field MA/MA² - MB/MB²		and so	Magnetic field lines induced by a uniform linear current orthogonal at A to xOy and a parallel current, in the opposite management, passing by B. Electrostatic equipotential lines induced by charges uniformly distributed on a line orthogonal at A to xOy and contrary charges uniformly distributed on a line orthogonal at B to xOy.	= two pencils of orthogonal circles
9 initial curves (crimson Cassinian ovals) orthogonal curves (blue rectangular hyperbolas) geometrical definition MA .MB = constant   geometrical definition (Ox, AM) + (Ox, BM) = 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 same management, 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 blueish curves: case where.
10			Electrostatic equipotential lines induced by two opposite charges placed at A and B, in other words, an electrostatic dipole.
11			Electrostatic equipotential lines induced past 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 ten²/(i-cte)-y²/cte=1		f(z) = argcosh(z) f- ane (z) = cosh (z) (image of the first lattice by the Joukovski transformation: j(z) = (z + ane/z)/ii)	Electrostatic equipotential lines induced by charges uniformly distributed on the segment line [AB]??? Interference pattern
13 Involute of circles and their generatrices	????	????
fourteen initial curves (carmine quartics) orthogonal curves (blue quartics) Cartesian implicit equation y² = a²(1 + ane / (a²+x²)) parametrization x = a tan t y² = a² + cos²t Cartesian implicit equation x² = a²(1 - i / (a²+y²)) parametrization x² = a² - cos²t y = a tan t			Streamlines of a compatible flow perturbed by an obstruction (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 bend
fifteen initial curves (cherry cubic hyperbolas) orthogonal curves (blue cubic hyperbolas) Cartesian implicit equation (y - constant) (x²+y²) = y Cartesian implicit equation (ten - constant) (x²+y²) + 10 =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, nosotros get:			Streamlines of a uniform flow in a bent tube.

Other examples:

Lemniscates of Bernoulli : and (instance n = �2 of the example 6 above)	Quatrefoils ; and their orthogonal trajectories.
Logarithmic spirals: and.	Parabolas and ellipses
Cherry parabolas and semicubical parabolas.	cycloids: and symmetric cycloids

Encounter also tractrix, as well as this article.

The project on a horizontal plane of the gradient and contour lines of a surface grade 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 5 when, at each bespeak mutual to the families, the tangents course an angle V, and one of the families is composed of the trajectories at bending V of the other ane.

For case, the trajectories under the bending V of the pencil of lines passing by O are the logarithmic spirals; :

© Robert FERRÉOL 2017

vincentwitim1990.blogspot.com

Source: https://mathcurve.com/courbes2d.gb/orthogonale/orthogonale.shtml

Share this post