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Survey 06 – How to Define a Geodesic Curve ?

Written By: Jean-Paul Cipria - Oct• 08•17
Geodesics - Osculating Plane - Normales

Geodesics – Osculating Plane – Normales

How to Define a Geodesic Curve ?

Raving Rabbids Earth

Done ! Raving Rabbids Earth

Complicated but Not Difficult - Bachelord Level

Complicated but Not Difficult – Bachelord Level

Created :2017-10-08 12:16:53. – Modified : 2017-10-14 13:01:29.

We are triing to navigate on earth or to « project » distances form a theorical ellipsoïd to a plane. Then how to be sure the distance between two points on ellipsoid is the minimum as possible BEFORE even think to project this distance on a certain map ?

Let’s go ? Do we try to understand some definitions in mathematics books or internet web sites ? Yes we do. And we loss of valuable time. Example ?

??? An osculating plane is a plane in a Euclidean space or affine space which meets a submanifold at a point in such a way as to have a second order of contact at the point. (Internaute site !) ???

Do we understand anything even where you have got a PhD in Physics ? No, we did’nt !

Then ?

Let’s try to find some « equivalent » mathematics definitions with differential methods ?

What is a Geodesic … Curve ?

Geodesic is a curve included on a particular non plane surface where it exists only one minimum path to join two points on the surface whatever the first direction you use on differential dM point.

How do we get a Curve on a Surface with the Osculating Plane ?

First we plot a tangente in M Point in one direction. This is T tangent.

Then we varie dM with another direction from a small differential distance dM and we plot a second tangente. This is T+dT second tangent.

After that we construct a plane who is defined by this two tangents (the grey plane on drawing). We call this plane osculating plane. This is a definition !

This osculating plane crosses the surface S and we can draw a C curve in this intersection. The M point is on this curve.

Then after that we plot a perpendicular vector n_C from M point and this intersection curve C and also a perpendicular vector n_S   from infinitesimal surface dS .

Ouch ! It is not so easy ? Let’s see a hands drawing ?

Geodesics - Osculating Plane - Normales

Geodesics – Osculating Plane – Normales

What is another definition for a Geodesic Curve ?

Then now they are two cases :


n_C = n_S . Those two vectors are the same, then infinitesimal surface dS and dM curve directions are parallel then the curve C defined by the dM displacement is a geodesic. In particular there is the minimum distance between two points on the surface S following the curve C . This is a definition for a geodesic curve C !

Non Geodesic(s)

Then in all the other cases, i.e 99.99% of cases, the two vectors n_C and n_S are not egal then we can find two or more trajectories from point A to point B as minimal distances ! Those different curves on surface S can’t be name geodesic. There are some possibilities to navigate from  A point to B point by different minimal trajectories on different C_1 , C_2 , C_n curves.


What about Geodesic Curves on Ellipsoid ?

We CAN find several minimum distanceS between two points. Then we CAN’T get geodesic curves at the order 0 of physics approximation except with complex formulas on ellipse. Then for simplification we can do first order and second order approximations to get circular small curves on a surface different from a spherical form and we can do some differential infinitesimal calculations. We can then project some small circular arc from ellipsoïd to geoid or to plane map. It is a first method.

That’s all folks ? Sure not !

Engineers solving problems you didn’t know you had in ways you don’t understand ?


Survey and topography

  • BRABANT, Michel. Maîtriser la Topographie – Des observations aux plans. Eyrolles., 2008.
  • MILLES, Serges. Topographie et Topométrie modernes – T1 – Techniques de mesure et de représentation. Eyrolles. Vol. 1, 1999.
  • MILLES, Serges. Topographie Et Topométrie Modernes – T2 – Calculs. Eyrolles. Vol. 2, 1999.
  • Topography Web Sites by Interests for a Student :


Mathematics for Physicist

  • BONVALET, M. (1993). T2 – Les principes variationnels – Principes mathématiques de la physique – Édition Masson (Vol. 2). Masson.



Jean-Paul Cipria

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