isogonal trajectory

Let a one-parametric family of plane curvesMathworldPlanetmathγ  have the differential equationMathworldPlanetmath

F(x,y,dydx)= 0. (1)

We want to determine the isogonal trajectories of this family, i.e. the curves  ι  intersecting all members of the family under a given angle, which is denoted by ω. For this purpose, we denote the slope angle of any curve γ at such an intersection point by α and the slope angle of ι at the same point by β.  Then

β-α=ω(or alternatively -ω),

and accordingly


where y means the slope of ι.  Thus the equation

F(x,y,y-tanω1+ytanω)= 0 (2)

is satisfied by the derivative y of the ordinate of ι.  In other , (2) is the differential equation of all isogonal trajectories of the given family of curves.

Note.  In the special case  ω=π2,  it’s a question of orthogonal trajectories.

Title isogonal trajectory
Canonical name IsogonalTrajectory
Date of creation 2013-03-22 18:59:20
Last modified on 2013-03-22 18:59:20
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Derivation
Classification msc 51N20
Classification msc 34A26
Classification msc 34A09
Related topic AngleBetweenTwoCurves
Related topic OrthogonalCurves
Related topic ExampleOfIsogonalTrajectory
Related topic AngleBetweenTwoLines
Defines isogonal trajectory