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GEODESIC of a surface

From the Greek "Earth" and daiein "share, divide".
Other name (from Navigation vocabulary): orthodromy, from the Greek orthos "right" and dromos "race".

Differential equation:  where  is the normal vector of the surface at M,
i.e. for a surface with parametrized by (u, v) and a curve parametrized by t.

This can be written, for a surface z = f(x, y), with u = x = t and v = y i.e.  (see the notations).
The partial condition  does not limit the generality (it leads to a normal parametrization of the geodesic); we get the differential system:

that can be written,
for a surface z = f(x, y), with u = x and v = y:

The geodesic lines of a surface, which are, in a way, the "straight lines" of this surface, have several equivalent definitions:

DEF #1 (mechanical): they are the trajectories of a massive point moving on the surface and only subject to the normal reaction; we can physically create them by making small balls roll on the surface (on the concave side), in a state of weightlessness (with a weight, these lines become flow lines)

DEF #2 (mechanical): they are the equilibrium figures of an inextensible homogeneous massive wire placed on the surface, in a state of weightlessness (with a weight, these figures become the catenaries)

DEF #3: they are the curves traced on the surface such that, at every point, the principal normal of the curve (if it exists) coincides with the normal of the surface (in other words, such that the osculating plane of the curve contains the normal of the surface, or even that the rectifying plane of the curve is the tangent plane of the surface).

DEF #4: they are the curves traced on the surface such that, at every point, the normal curvature of the curve (i.e. the curvature of the normal section of the surface in the direction of the tangent to the curve) is equal in absolute value to the curvature of the curve.

DEF #5: they are the curves traced on the surface with zero geodesic curvature: figuratively, they are the trajectories of observers moving on the surface by walking straight ahead of them, or of small cars the direction of which is blocked in a linear position.

The geodesics of a surface are the curves the geodesic torsion of which is equal to the torsion: the general case gives the pseudogeodesics.

In general, by every point of the surface passes, in a given direction, one and only one geodesic, and by two points pass at least one geodesic (this property is a generalization of Euclid's axioms, but for a surface that is not isometric to the plane, everything lies in the "in general"!)

It can be proved that any arc joining two points on the surface, of minimal length, is a geodesic, but there can be geodesics joining two points, and not having the minimal length (for example two points of a generatrix of a cylinder of revolution are also joined by a cylindrical helix, which is longer, but is a geodesic nonetheless).

However, for any point B sufficiently close to a point A of the surface, there exists a unique geodesic joining A to B, necessarily realizing the minimum of the geodesic distance from A to B.

     - the geodesics of the plane are the straight lines
     - the geodesics of the sphere are the great circles and are also called orthodromies.
see also
     - the geodesics of a cylinder or a cone, and more generally of developable surfaces are the curves that transform into lines when the surface is applied onto a plane; for the cylinder of revolution, they are the cylindrical helices (or circles or lines), and see the geodesics of the cone of revolution on its own page.
     - the geodesics of the paraboloid of revolution are on its page, same for the ellipsoid of revolution.
     - the geodesics of the torus lead to calculations with elliptic integrals.
     - the meridians of a surface of revolution are geodesics (but not the parallels, except those with extreme radius).
     - the straight lines of a surface are geodesics (and they are the only one to be geodesics and asymptotic lines).
    - a geodesic of a surface is planar if and only if it is a curvature line.
    - the evolutes of a skew curve are geodesics of the polar developable of the curve.
For a surface of revolution:, the cylindrical equation of the geodesics is:([gray] p. 577), but the following Clairaut theorem allows for a better characterization:
Apart from the cases of the meridians, the geodesics of a surface of revolution are the curves such that the product between the distance to the axis of a point M on the curve and the cosine of the angle between the curve and the parallel passing by M is constant, i.e. 
In other words, if we measure a length equal to  on the tangent, the horizontal projection of the segment has a constant length (so the more we move away from the axis, the greater the slope).

See for example the remarkable geodesics of Tannery's pear.

See also the brachistochrones, the timewise shortest curves, and the pseudogeodesics, curves the osculating plane of which forms a constant angle with the tangent plane (right angle in the case of geodesics).
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© Robert FERRÉOL  2018