Orthogonal Trajectories of One-Parameter Family of Curves

Theorem

Every one-parameter family of curves has a unique family of orthogonal trajectories.

Proof

Let $\map f {x, y, z}$ define a one-parameter family of curves $\FF$.

From One-Parameter Family of Curves for First Order ODE‎, there is a corresponding first order ODE:

$\map F {x, y, \dfrac {\d y} {\d x} }$

whose solution is $\FF$.

From Slope of Orthogonal Curves, the slope of one curve is the negative reciprocal of any curve orthogonal to it.

So take the equation:

$\map F {x, y, \dfrac {\d y} {\d x} }$

and from it create the equation:

$\map F {x, y, -\dfrac {\d x} {\d y} }$

that is, replace $\dfrac {\d y} {\d x}$ with $-\dfrac {\d x} {\d y}$.

This is also a first order ODE, which corresponds with a one-parameter family of curves $\GG$ defined by the implicit function $\map f {x, y, z}$.

There is clearly one way of doing the above.

Hence the result.

$\blacksquare$

Sources

• 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $1$: The Nature of Differential Equations: $\S 3$: Families of Curves. Orthogonal Trajectories