Orthogonal Trajectories of One-Parameter Family of Curves

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Theorem

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


Proof

Let $f \left({x, y, z}\right)$ define a one-parameter family of curves $\mathcal F$.

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

$F \left({x, y, \dfrac{\mathrm d y}{\mathrm d x}}\right)$

whose solution is $\mathcal F$.


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

So take the equation:

$F \left({x, y, \dfrac{\mathrm d y}{\mathrm d x}}\right)$

and from it create the equation:

$F \left({x, y, -\dfrac{\mathrm d x}{\mathrm d y}}\right)$

i.e. replace $\dfrac{\mathrm d y}{\mathrm d x}$ with $-\dfrac{\mathrm d x}{\mathrm d y}$.

This is also a first order ODE, which corresponds with a one-parameter family of curves $\mathcal G$ defined by the implicit function $f \left({x, y, z}\right)$.


There is clearly one way of doing the above.

Hence the result.

$\blacksquare$


Sources