Definition:Orthogonal Trajectories

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

Let $\map g {x, y, c}$ also define a one-parameter family of curves $G$, with the property that:

Every curve in $F$ is orthogonal to every curve in $G$.

Then $F$ is a family of (reciprocal) orthogonal trajectories of $G$, and contrariwise.

Also see

  • Results about orthogonal trajectories can be found here.

Historical Note

The origin of the problem of orthogonal trajectories is uncertain.

Some sources say that it was posed by Nicolaus II Bernoulli in $1720$ as a challenge to the English Newtonian school of mathematics.

Others suggest that it was the work of Gottfried Wilhelm von Leibniz in $1716$, who was aiming the problem specifically at Isaac Newton himself.

Legend has it that Newton solved the problem in an evening, having returned from his day's work at the Mint.