Isogonal trajectory

In mathematics an orthogonal trajectory is

For example, the orthogonal trajectories of a pencil of concentric circles are the lines through their common center (see diagram).

Suitable methods for the determination of orthogonal trajectories are provided by solving differential equations. The standard method establishes a first order ordinary differential equation and solves it by separation of variables. Both steps may be difficult or even impossible. In such cases one has to apply numerical methods.

Orthogonal trajectories are used in mathematics for example as curved coordinate systems (i.e. elliptic coordinates) or appear in physics as electric fields and their equipotential curves.

If the trajectory intersects the given curves by an arbitrary (but fixed) angle, one gets an isogonal trajectory.

Generally one assumes, that the pencil of curves is implicitly given by an equation

where ${\displaystyle c}$ is the parameter of the pencil. If the pencil is given explicitly by an equation ${\displaystyle y=f(x,c)}$, one can change the representation into an implicit one: ${\displaystyle y-f(x,c)=0}$. For the consideration below it is supposed that all necessary derivatives do exist.

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