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Isogonal trajectory

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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

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

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

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

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 \ref{1} passing through it. The orthogonal trajectories for the given system \ref{1} form a one-parameter family of plane curves — the general integral of equation \ref{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

[1] W.W. [V.V. Stepanov] Stepanow, "Lehrbuch der Differentialgleichungen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)


Comments

References

[a1] H.S.M. Coxeter, "Introduction to geometry" , Wiley (1961)
[a2] D. Hilbert, S.E. Cohn-Vossen, "Geometry and the imagination" , Chelsea (1952) (Translated from German)
How to Cite This Entry:
Isogonal trajectory. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Isogonal_trajectory&oldid=32810
This article was adapted from an original article by r equation','../w/w097310.htm','Whittaker equation','../w/w097840.htm','Wronskian','../w/w098180.htm')" style="background-color:yellow;">N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article