orthogonal trajectories

Given one family of curves filling the plane, the orthogonal trajectories are a second family that crosses the first at right angles everywhere — every curve of one set meets every curve of the other at ninety degrees. The everyday picture is a contour map: the lines of steepest descent that a raindrop follows run exactly perpendicular to the level contours of constant height. The two families are orthogonal trajectories of each other.

Finding them is a tidy application of first-order differential equations. Start with the family written as F(x, y) = C, differentiate to get its differential equation dy/dx = m(x, y) — the slope of the original curve through each point, with the constant C eliminated. The orthogonal family must have slope equal to the negative reciprocal at every point (perpendicular lines have slopes that multiply to -1), so it satisfies dy/dx = -1/m(x, y). Solve that new differential equation and you have the orthogonal trajectories. The whole method is: get the family's ODE, flip the slope to its negative reciprocal, solve again.

Orthogonal families are everywhere in physics, and the reason is deep: in many fields the lines of force run perpendicular to the surfaces of constant potential. Electric field lines cross equipotential surfaces at right angles; streamlines of an ideal fluid cross the equipotential lines orthogonally; isotherms (curves of equal temperature) and heat-flow lines are orthogonal trajectories. So computing an orthogonal family is, very often, computing the flow that a potential drives — a clean first-order-ODE technique with a direct line into the physics of fields.