What is isogonal family of a given family of curves?

Question

I searched in Wikipedia isogonal trajectories about the definition but I do not understand what does it mean by fixed "angle". Angle with the tangents of the curves? Clockwise angle? Orientated Angle? Thanks in advance.

Answer 1

An isogonal trajectory is a plane curve intersecting the curves of a given one-parameter family in the plane at one and the same angle.

If

is the differential equation of the given family of curves, then an isogonal trajectory of this family intersecting it at an angle α, where $0<α<π$, $α≠π/2$, satisfies one of the following two equations:

In particular, the equation

is satisfied by an orthogonal trajectory, that is, a plane curve that forms a right angle at each of its points with any curve of the family (1) passing through it. The orthogonal trajectories for the given system (1) form a one-parameter family of plane curves — the general integral of equation (1). For example, if the family of lines of force of a plane electrostatic field is considered, then the family of orthogonal trajectories are the equipotential lines.

Here is another example of isogonal (and orthogonal) trajectory:

P.s. The word isogonal derives from the ancient Greek, isos = equal, and gon = angle, as seen in the English word knee, which forms an angle in a leg!

References: W.W. [V.V. Stepanov] Stepanow, "Lehrbuch der Differentialgleichungen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)

Answer 2

I think that for two curves $y=f(x)$ and $y=g(x)$ which intersect at $(x_0,y_0)$, they are defining the angle between the curves to be $\ \mathrm{arctan}(f'(x_0)) - \mathrm{arctan}(g'(x_0))$.