# 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.