Orthogonal Trajectory

Orthogonal Trajectory

In mathematics, orthogonal trajectories are a family of curves in the plane that intersect a given family of curves at right angles. The problem is classical, but is now understood by means of complex analysis; see for example harmonic conjugate.

For a family of level curves described by, where is a constant, the orthogonal trajectories may be found as the level curves of a new function by solving the partial differential equation

for . This is literally a statement that the gradients of the functions (which are perpendicular to the curves) are orthogonal. Note that if and are functions of three variables instead of two, the equation above will be nonlinear and will specify orthogonal surfaces.

The partial differential equation may be avoided by instead equating the tangent of a parametric curve with the gradient of :

which will result in two possibly coupled ordinary differential equations, whose solutions are the orthogonal trajectories. Note that with this formula, if is a function of three variables its level sets are surfaces, and the family of curves are orthogonal to the surfaces.