Isogonal trajectory

In differential geometry, a plane curve that crosses every member of a family of curves with the same crossing angle is called an isogonal trajectory of the family. If two families of curves all cross at the same angle, then each family is called a family of isogonal trajectories with respect to the other family.^[1] If the crossing angle is a right angle, the curves or families of curves are orthogonal trajectories.^[2]

If one given family is described by the differential equation

$\frac{dy}{dx} = f(x,y),$

then its isogonal family with crossing angle $\alpha$ satisfies the differential equation

$\frac{dy}{dx} = \frac{f(x,y)+\tan\alpha}{1-f(x,y)\tan\alpha}.$^[3]

References

1. Tenenbaum, Morris; Pollard, Harry (2012), Ordinary Differential Equations, Dover Books on Mathematics, Courier Dover, p. 115, ISBN 9780486134642.
2. Tenenbaum & Pollard (2012), p. 112.
3. Tenenbaum & Pollard (2012), Ex. 5, p. 120.