# Definition:Orthogonal Trajectories

## 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.

## Examples

### Circles Tangent to $y$-axis

Consider the one-parameter family of curves:

- $(1): \quad x^2 + y^2 = 2 c x$

which describes the loci of circles tangent to the $y$-axis at the origin.

Its family of orthogonal trajectories is given by the equation:

- $x^2 + y^2 = 2 c y$

which describes the loci of circles tangent to the $x$-axis at the origin.

### Exponential Functions

Consider the one-parameter family of curves of graphs of the exponential function:

- $(1): \quad y = c e^x$

Its family of orthogonal trajectories is given by the equation:

- $y^2 = -2 x + c$

### Parabolas Tangent to $x$-axis

Consider the one-parameter family of curves of parabolas which are tangent to the $x$-axis at the origin:

- $(1): \quad y = c x^2$

Its family of orthogonal trajectories is given by the equation:

- $x^2 + 2 y^2 = c$

## 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): $1$: The Nature of Differential Equations: $\S 3$: Families of Curves. Orthogonal Trajectories