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
for
The partial differential equation may be avoided by instead equating the tangent of a parametric curve
which will result in two possibly coupled ordinary differential equations, whose solutions are the orthogonal trajectories. Note that with this formula, if
Example 1: Circle
Suppose we are given the family of circles centered about the origin:
The orthogonal trajectories of this family are the family of curves such that intersect the circle at right angles. Given a line, the negative reciprocal of its slope is the slope of a perpendicular line. In calculus terms, if y and k are two perpendicular lines:
Orthogonal trajectories of the given family of circles are no different. The tangent line of an orthogonal trajectory is perpendicular to the tangent line of the circle, and vice versa.
By implicit differentiation,
Solving the equation, one acquires the following:
Therefore, an orthogonal trajectory must satisfy the differential equation:
Solving the equation for y, we find the orthogonal trajectories:
The orthogonal trajectories of a family of circles centered at the origin are linear equations containing the origin.
In polar coordinates, the family of circles centered about the origin is the level curves of
where
The lack of complete boundary data prevents determining
The absence of boundary data is a good thing, as it makes solving the PDE simple as one doesn't need to contort the solution to any boundary. In general, though, it must be ensured that all of the trajectories are found.