Orthogonal Trajectories of One-Parameter Family of Curves

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:
 * $$F \left({x, y, \frac{\mathrm{d}{y}}{\mathrm{d}{x}}}\right)$$

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:
 * $$F \left({x, y, \frac{\mathrm{d}{y}}{\mathrm{d}{x}}}\right)$$

and from it create the equation:
 * $$F \left({x, y, -\frac{\mathrm{d}{x}}{\mathrm{d}{y}}}\right)$$

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

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.