Orthogonal Trajectories/Examples

Examples of Families of Orthogonal Trajectories

Concentric Circles

Consider the one-parameter family of curves:

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

Its family of orthogonal trajectories is given by the equation:

$y = c x$

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:

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

Parabolas with Focus at Origin

Consider the one-parameter family of curves of parabolas whose focus is at the origin and whose axis is the $x$-axis:

$(1): \quad y^2 = 4 c \paren {x + c}$

Its family of orthogonal trajectories is given by the equation:

$y^2 = 4 c \paren {x + c}$

Rectangular Hyperbolas

Consider the one-parameter family of curves of rectangular hyperbolas:

$(1): \quad x y = c$

Its family of orthogonal trajectories is given by the equation:

$x^2 - y^2 = c$

Cardioids

Consider the one-parameter family of curves of cardioids given in polar form as:

$(1): \quad r = c \paren {1 + \cos \theta}$

Its family of orthogonal trajectories is given by the equation:

$r = c \paren {1 - \cos \theta}$

Example: $x + C e^{-x}$

Consider the one-parameter family of curves:

$(1): \quad y = x + C e^{-x}$

Its family of orthogonal trajectories is given by the equation:

$x = y - 2 + C e^{-y}$