For example, the orthogonal trajectories of a pencil of concentric circles are the lines through their common center. 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 or appear in physics as electric fields and their equipotential curves. If the trajectory intersects the given curves by an arbitrary angle, one gets an isogonal trajectory.
Generally one assumes, that the pencil of curves is implicitly given by an equation where is the parameter of the pencil. If the pencil is given explicitly by an equation, one can change the representation into an implicit one:. For the consideration below it is supposed that all necessary derivatives do exist. ;Step 1. Differentiating implicitly for yields ;Step 2. Now it is assumed, that equation can be solved for parameter, which can thus be eliminated from equation. One gets the differential equation of first order which is fulfilled by the given pencil of curves. ;Step 3. Because the slope of the orthogonal trajectory at a point is the negative multiplicative inverse of the slope of the given curve at this point, the orthogonal trajectory satisfies the differential equation of first order ;Step 4. This differential equation can be solved by a suitable method. For both examples separation of variables is suitable. The solutions are: in example 1, the lines and in example 2, the ellipses
If the pencil of curves is represented implicitly in polar coordinates by one determines, alike the cartesian case, the parameter free differential equation of the pencil. The differential equation of the orthogonal trajectories is than Example: Cardioids: Elimination of yields the differential equation of the given pencil: Hence the differential equation of the orthogonal trajectories is: After solving this differentialequation by separation of variables one gets which describes the pencil of cardioids, symmetric to the given pencil.
Isogonal trajectory
A curve, which intersects any curve of a given pencil of curves by a fixed angle is called isogonal trajectory.
Between the slope of an isogonal trajectory and the slope of the curve of the pencil at a point the following relation holds: This relation is due to the formula for List of trigonometric identities#Angle sum and difference identities|. For one gets the condition for the orthogonal trajectory. For the determination of the isogonal trajectory one has to adjust the 3. step of the instruction above: ;3. step The differential equation of the isogonal trajectory is:
For the 1. example and the angle one gets This is a special kind of differential equation, which can be transformed by the substitution into a differential equation, that can be solved by separation of variables. After reversing the substitution one gets the equation of the solution: Introducing polar coordinates leads to the simple equation which describes logarithmic spirals.
Numerical methods
In case that the differential equation of the trajectories can not be solved by theoretical methods, one has to solve it numerically, for example by Runge–Kutta methods.
