# Definition:Orthogonal Trajectories

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## Contents

## Definition

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.

## Sources

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{VI}$: On the Seashore: Footnote - 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 1.3$: Families of Curves. Orthogonal Trajectories