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A plane curve intersecting the curves of a given one-parameter family in the plane at one and the same angle. If

$$F(x,y,y')=0\label{1}\tag{1}$$

is the differential equation of the given family of curves, then an isogonal trajectory of this family intersecting it at an angle $\alpha$, where $0<\alpha<\pi$, $\alpha\neq\pi/2$, satisfies one of the following two equations:

$$F\left(x,z,\frac{z'-\tan\alpha}{1+z'\tan\alpha}\right)=0,\quad F\left(x,z\frac{z'+\tan\alpha}{1-z'\tan\alpha}\right)=0.$$

In particular, the equation

$$F\left(x,z,-\frac{1}{z'}\right)=0\label{2}\tag{2}$$

is satisfied by an orthogonal trajectory, that is, a plane curve that forms a right angle at each of its points with any curve of the family \eqref{1} passing through it. The orthogonal trajectories for the given system \eqref{1} form a one-parameter family of plane curves — the [[General integral|general integral]] of equation \eqref{1}. For example, if the family of lines of force of a plane electrostatic field is considered, then the family of orthogonal trajectories are the equipotential lines.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W.W. [V.V. Stepanov] Stepanow,   "Lehrbuch der Differentialgleichungen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.S.M. Coxeter,   "Introduction to geometry" , Wiley  (1961)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Hilbert,   S.E. Cohn-Vossen,   "Geometry and the imagination" , Chelsea  (1952)  (Translated from German)</TD></TR></table>