Orthogonal Trajectories

The equation

Φ (x, y, c) = 0,

in which $c$ is a parameter, represents a family of plane curves. To this family of curves there is related a second family, namely, the family of orthogonal trajectories or curves which cut every curve of the given family at right angles. To return to the instance given in § 1.4, the first family of curves may be considered as the lines of force due to a given plane magnetic or electrostatic distribution. The family of orthogonal trajectories will then represent the equipotential lines in the given plane.

Let

F (x, y, p) = 0

be the differential equation of the given family of curves; it determines the gradient $p$ of any curve of the family which passes through the point $(x, y)$ . The gradient $ϖ$ of the orthogonal curve through $(x, y)$ is connected with $p$ by the relation

p ϖ = - 1,

and consequently the differential equation of the family of orthogonal trajectories is

F (x, y, - \frac{1}{p}) = 0.

Since the differential equation of the given family is obtained by eliminating $c$ between the two equations

Φ = 0, \frac{\partial Φ}{\partial x} + p \frac{\partial Φ}{\partial y} = 0,

the differential equation of the orthogonal trajectories arises through the elimination of $c$ between the equations

Φ = 0, p \frac{\partial Φ}{\partial x} - \frac{\partial Φ}{\partial y} = 0.

Examples.
(i) The family of parabolas,

y^{2} = 4 c x,

where $c$ is a parameter, are integral-curves of the differential equation

2 x p = y .

The differential equation of the orthogonal trajectories is therefore

2 x + p y = 0,

and the trajectories themselves are the curves

2 x^{2} + y^{2} = c^{2};

they compose a family of similar ellipses whose axes lie along the co-ordinate axes.

(ii) The family of confocal conics,

\frac{x^{2}}{a^{2} - λ} + \frac{y^{2}}{b^{2} - λ} = 1,

where $λ$ is the parameter, are integral-curves of the differential equation

(x + p y) (y - p x) + (a^{2} - b^{2}) p = 0.

This equation is unaltered by the substitution of $- p^{- 1}$ for $p$ . The family is therefore self-orthogonal.

2.3.1 Oblique Trajectories

An oblique trajectory is a curve which cuts the curves of a family at a given angle. Let the given angle be $\arctan m$ . Then if $p$ and $ϖ$ are respectively the gradients of a curve of the given family and the trajectory at a point where they intersect,

ϖ = \frac{p - m}{1 + m p} .

If the differential equation of the given family is

F (x, y, p) = 0,

that of the family of oblique trajectories will be

F (x, y, \frac{p - m}{1 + m p}) = 0.

Example. Consider the family of concentric circles,

x^{2} + y^{2} = c^{2};

their differential equation is

x + y p = 0.

The family of curves which cut the circles at the angle $\arctan m$ is therefore

x + \frac{p - m}{1 + m p} y = 0

(m x + y) p + x - m y = 0.

This equation is homogeneous: its solution is

\log \sqrt{x^{2} + y^{2}} + m \arctan \frac{y}{x} = const.

In polar co-ordinates, the equation of the trajectories is

r = C e^{- m θ},

the curves are therefore equiangular spirals.

2.3.2 Conformal Representation of a Surface on a Plane

Another important application of differential equations of the first order is to the conformal representation of an algebraic surface upon a plane. The real quadratic form

d S^{2} = E d u^{2} + 2 F d u d v + G d v^{2} (E G - F^{2} \neq 0)

represents an element of surface. Since it is essentially positive, its linear factors,

a\,du+b\,dv, \quad a'\,du+b'\,dv

are such that $a$ and $b$ are, in general, complex functions of $u$ and $v$ , and a' and b' are respectively the conjugate complex functions.

Let $μ (u, v)$ be an integrating factor for $a d u + b d v$ , then the conjugate \mu' will be an integrating factor for a'\,du+b'\,dv. If

\mu(a\,du+b\,dv) = dV, \quad \mu'(a'\,du+b'\,dv) = dV'

then $V$ and V' will be conjugate complexes, and

\mu\mu'dS^2 = dVdV'.

Define $x$ and $y$ as new variables by the equations

V=x+iy, \quad V'=x-iy

and let

\lambda^2 = \mu\mu',

then

d S^{2} = λ^{2} (d x^{2} + d y^{2}) = λ^{2} d s^{2} .

Thus the surface $(u, v)$ is conformally represented on the plane $(x, y)$ .¹

Example. Consider the representation of the sphere

d S^{2} = a^{2} d u^{2} + a^{2} \sin^{2} u d v^{2}

on the plane.

\begin{align*} dS^2 &= a^2(du+i\sin u\,dv)(du-i\sin u\,dv) \\ &= a^2\sin^2u(\csc u\,du+i\,dv)(\csc u\,du-i\,dv) \end{align*}

Let

\csc u d u = d y, d v = d x,

that is

y = \log \tan \frac{1}{2} u, x = v .

Then

d S^{2} = 4 a^{2} {sech}^{2} y (d x^{2} + d y^{2}) .

This correspondence between the sphere and the plane is Mercator's projection.[^fn18] Meridians on the sphere are represented by lines parallel to the $y$ -axis in the plane, and parallels of latitude by lines parallel to the $x$ -axis. The whole sphere is represented by that strip of the plane which lies between $x = - π$ and $x = + π$ . Any straight line in the plane represents a loxodrome on the sphere, that is a curve which cuts all the meridians at a constant angle.

2.3.1 Oblique Trajectories

2.3.2 Conformal Representation of a Surface on a Plane

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