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isogonal trajectory
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(Derivation)
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Let a one-parametric family of plane curves $\gamma$ have the differential equation
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(1) |
We want to determine the isogonal trajectories of this family, i.e. the curves $\iota$ intersecting all members of the family under a given angle, which is denoted by $\omega$ . For this purpose, we denote the slope angle of any curve $\gamma$ at such an intersection point by $\alpha$ and the slope angle of
$\iota$ at the same point by $\beta$ . Then $$\beta-\alpha \;=\; \omega \quad(\mbox{or alternatively\;\;} -\omega),$$ and accordingly $$\frac{dy}{dx} \;=\; \tan\alpha \;=\; \frac{\tan\beta-\tan\omega}{1+\tan\beta\tan\omega} \;=\; \frac{y'-\tan\omega}{1+y'\tan\omega},$$ where $y'$ means the slope of $\iota$ . Thus the equation
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(2) |
is satisfied by the derivative $y'$ of the ordinate of $\iota$ . In other words, (2) is the differential equation of all isogonal trajectories of the given family of curves.
Note. In the special case $\omega = \frac{\pi}{2}$ , it's a question of orthogonal trajectories.
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"isogonal trajectory" is owned by pahio.
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Cross-references: orthogonal trajectories, ordinate, derivative, equation, slope, point, intersection, slope angle, angle, members, curves, differential equation, plane curves
There are 2 references to this entry.
This is version 4 of isogonal trajectory, born on 2009-08-05, modified 2009-08-05.
Object id is 11855, canonical name is IsogonalTrajectory.
Accessed 547 times total.
Classification:
| AMS MSC: | 34A09 (Ordinary differential equations :: General theory :: Implicit equations, differential-algebraic equations) | | | 34A26 (Ordinary differential equations :: General theory :: Geometric methods in differential equations) | | | 51N20 (Geometry :: Analytic and descriptive geometry :: Euclidean analytic geometry) |
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Pending Errata and Addenda
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