isogonal trajectory
Let a one-parametric family of plane curves have the differential equation
(1) |
We want to determine the isogonal trajectories of this family, i.e. the curves intersecting all members of the family under a given angle, which is denoted by . For this purpose, we denote the slope angle of any curve at such an intersection point by and the slope angle of at the same point by . Then
and accordingly
where means the slope of . Thus the equation
(2) |
is satisfied by the derivative of the ordinate of . In other , (2) is the differential equation of all isogonal trajectories of the given family of curves.
Note. In the special case , 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 |