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