RHUMB LINE

 Notion studied by Wallis in 1741. Other name: loxodrome, from the Greek loksos "oblique" and dromos "race".

 Differential equation for a surface z = f(x, y): (Monge notations) . For a surface of revolution: . For a surface of revolution: , the differential equation can therefore be written: .

Given a vertical direction, the rhumb lines of a surface are the topographic curves traced on the surface that form a constant angle a with the contour lines (and therefore also a constant angle with the slope lines).
Note that the angle between the rhumb line and the contour lines is constant on the surface, but not necessarily when projected on a horizontal plane.
Another mistake: the rhumb lines do not necessarily form a constant angle with a horizontal plane (confusion with the helices).
The contour lines ( ) are of course a limit case of rhumb lines and so are the slope lines ( ).
In the case of a surface of revolution with vertical axis, they are the curves forming a constant angle with the meridians (or the parallels).

Examples:
- rhumb lines of the sphere.
- rhumb lines of the vertical cone or cylinder of revolution: they are the helices traced on this cone or cylinder.
- rhumb lines of a vertical or horizontal cylinder: they are the geodesics of this cylinder (curves that develop onto straight lines); the rhumb lines of a vertical or horizontal cylinder of revolution are therefore the cylindrical helices.
- rhumb lines of the torus, including the Villarceau circles.
- rhumb lines of the catenoid.
- rhumb lines of the axial revolution of the catenary.

© Robert FERRÉOL  2018