example of isogonal trajectory
Determine the curves which intersect the origin-centered circles at an angle of .
The differential equation of the circles is , i.e.
Thus, by the model (2) of the parent entry (http://planetmath.org/IsogonalTrajectory), the differential equation of the isogonal trajectory reads
which can be rewritten as
Here, one may take as a new variable (see ODE types reductible to the variables separable case), when
and in the resulting equation
one can separate the variables (http://planetmath.org/SeparationOfVariables):
Multiplying here by 2 and integrating then give
Consequently, the family of the isogonal trajectories consists of logarithmic spirals.
|Title||example of isogonal trajectory|
|Date of creation||2013-03-22 18:59:23|
|Last modified on||2013-03-22 18:59:23|
|Last modified by||pahio (2872)|
|Synonym||isogonal trajectories of concentric circles|