## Engineer's Book

Engineers – Telecoms – Physics – Teaching – Surveying

## Survey 06 – How to Define a Geodesic Curve ?

Written By: Jean-Paul Cipria - Oct• 08•17

Geodesics – Osculating Plane – Normales

How to Define a Geodesic Curve ?

Done ! Raving Rabbids Earth

Complicated but Not Difficult – Bachelord Level

Created :2017-10-08 12:16:53. – Modified : 2018-02-24 19:46:44.

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

## What is another definition for a Geodesic Curve ?

Then now they are two cases :

### Geodesic

$n_C = n_S$. 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 ?

# References

## Survey and Topography

• BRABANT, Michel. Maîtriser la Topographie – Des observations aux plans. Eyrolles., 2008.
http://www.editions-eyrolles.
.
• MILLES, Serges. Topographie et Topométrie modernes – T1 – Techniques de mesure et de représentation. Eyrolles. Vol. 1, 1999.
http://www.editions-eyrolles.com/Livre/9782212022896/topographie-et-topometrie-modernes-tome-1.
.
• MILLES, Serges. Topographie Et Topométrie Modernes – T2 – Calculs. Eyrolles. Vol. 2, 1999.
http://www.editions-eyrolles.com/Livre/9782212022896/topographie-et-topometrie-modernes-tome-2.
.
• Topography Web Sites by Interests for a Student :
http://serge.milles.free.fr/btsgt/sites.htm

.

## Mathematics for Physicist

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

## Scientifics

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Jean-Paul Cipria
08/10/2017

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