Wikipedia
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 g(x, y) = C, where C is a constant, the orthogonal trajectories may be found as the level curves of a new function f(x, y) by solving the partial differential equation
∇f ⋅ ∇g = 0
for f(x, y). This is literally a statement that the gradients of the functions (which are perpendicular to the curves) are orthogonal.
The partial differential equation may be avoided by instead equating the tangent of a parametric curve $\vec r(t)$ with the gradient of g(x, y):
$$\frac{d}{d t}\left(\vec r(t)\right) = \nabla g$$
which will result in two possibly coupled ordinary differential equations, whose solutions are the orthogonal trajectories. Note that with this formula, if g is a function of three variables its level sets are surfaces, and the family of curves $\vec r(t)$ are orthogonal to the surfaces.