In mathematics an orthogonal trajectory is a curve, which intersects any curve of a given pencil of (planar) curves orthogonally. 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. Determination of the orthogonal trajectory In cartesian coordinates Generally one assumes, that the pencil of curves is implicitly given by an equation (0) 1. example 2. example 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 (1) in 1. example 2. example Step 2. Now it is assumed, that equation (0) can be solved for parameter , which can thus be eliminated from equation (1). One gets the differential equation of first order (2) in 1. example 2. example 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 (3) in 1. example 2. example Step 4. This differential equation can (hopefully) 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 In polar coordinates If the pencil of curves is represented implicitly in polar coordinates by (0p) one determines, alike the cartesian case, the parameter free differential equation (1p) (2p) of the pencil. The differential equation of the orthogonal trajectories is then (see Redheffer & Port p. 65, Heuser, p. 120) (3p) Example: Cardioids: (0p) (in diagram: blue) (1p) Elimination of yields the differential equation of