Geodesic distances away from the ellipsoid

[Jay Hollingsworth]

At the risk of asking a dumb question, do any of the geodesic
algorithms allow calculation of the geodesic distance if the path is
not on the ellipsoid? Like in an airplane or satellite whose path
could be assumed to be a constant height above the ellipsoid?

Or do you have to define an ellipsoid matching the satellite orbit
and use the normal equations on that ellipsoid? If so, how would you
define an ellipsoid for a long airplane flight?

I've poked around and nothing has jumped out at me. I assume
aero-engineers guys figured this out long ago....


Jay Hollingsworth

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Principal Data Architect
Schlumberger Information Solutions
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[Gerald I. Evenden]

On Friday 20 March 2009 12:56:53 pm Jay Hollingsworth wrote:

> At the risk of asking a dumb question, do any of the geodesic
> algorithms allow calculation of the geodesic distance if the path is
> not on the ellipsoid? Like in an airplane or satellite whose path
> could be assumed to be a constant height above the ellipsoid?
>
> Or do you have to define an ellipsoid matching the satellite orbit
> and use the normal equations on that ellipsoid? If so, how would you
> define an ellipsoid for a long airplane flight?
>
> I've poked around and nothing has jumped out at me. I assume
> aero-engineers guys figured this out long ago....

I think the solution is fairly simple: just scale you path by the ratio:

R = approximate radius of the Earth over flights path
h = flight height

geodesic length * (R+h)/R

There is also the 3D geodesic at the NGS site which might apply and give a
better answer.  But I think the above would be good enough for most purposes.  
Try both and see what happens.

PS: I did not convert the NGS 3D because it was hard-wired for WGS84 and the
description was confusing enough that I decided against bothering for the
time being.  Also, I could not get a mental image of what was happening when
there was a difference of altitude at the two end points.

> Jay Hollingsworth
>
> Seabed Portfolio Manager and
> Principal Data Architect
> Schlumberger Information Solutions
> phone: [phone] fax: [phone]
>
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--
The whole religious complexion of the modern world is due
to the absence from Jerusalem of a lunatic asylum.
-- Havelock Ellis (1859-1939) British psychologist
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[Karl Swartz]

> At the risk of asking a dumb question, do any of the geodesic
> algorithms allow calculation of the geodesic distance if the path is
> not on the ellipsoid? Like in an airplane or satellite whose path
> could be assumed to be a constant height above the ellipsoid?

It doesn't answer your real question but for airlines the altitude is
not factored in, at least not anywhere that I've seen.  The geodesic
distance is just a theoretical value since the actual path flown is
often quite different due to winds and other factors.  Time aloft is
what drives fuel burn, pilot legalities, etc., regardless of the
distance flown.  In addition, for topics like frequent flier miles,
it wouldn't go over well if my SFO-LHR flight accrued more miles than
yours because we flew at a higher altitude.

Satellites aren't my area but for most missions I'd guess they just use
the orbital ellipse, defined by apogee and perigee, without regard to
the terrain underneath.  Some missions might notice the gravitational
variation but I doubt most would.  The question goes away for satellites
in geosynchronous orbit, of course, since they stay over the equator --
one expect spot on the equator.

 -- Karl
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[Clifford J Mugnier]

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Jay,

 

There is an old ESSA or NOAA publication that directly addressed "The Great Ellipse" and also included all the math for hyperbolic lattice computations as well as for Rho-Rho systems.  I don't remember the exact title, but it was mainly addressed to the Hydrographic Surveyor.  My guess is that's exactly what you are looking for.  It's somewhere in my office library on campus, but I won't get there until Monday.  Call me and remind me after 10:00 AM and I'll look it up, I think I know the bookcase it's in.  I'll bet John Stigant and Noel have a copy, too.

 

Cliff

 

Clifford J. Mugnier, C.P., C.M.S.
Past National Director (2006-2008),
Photogrammetric Applications Division
American Society for Photogrammetry and Remote Sensing
and
Chief of Geodesy,
CENTER FOR GEOINFORMATICS
Department of Civil Engineering
Patrick F. Taylor Hall 3223A
LOUISIANA STATE UNIVERSITY
Baton Rouge, LA  70803
Voice and Facsimile:  [phone] [Academic]
Honorary Life Member of the
Louisiana Society of Professional Surveyors
Member Emeritus of the ASPRS
Member of the Americas Petroleum Survey Group
======================================================
http://www.asprs.org/resources/GRIDS/
======================================================

 

---

From: [hidden email] on behalf of Jay Hollingsworth
Sent: Fri 20-Mar-09 11:56
To: [hidden email]
Subject: [Proj] Geodesic distances away from the ellipsoid

At the risk of asking a dumb question, do any of the geodesic
algorithms allow calculation of the geodesic distance if the path is
not on the ellipsoid? Like in an airplane or satellite whose path
could be assumed to be a constant height above the ellipsoid?

Or do you have to define an ellipsoid matching the satellite orbit
and use the normal equations on that ellipsoid? If so, how would you
define an ellipsoid for a long airplane flight?

I've poked around and nothing has jumped out at me. I assume
aero-engineers guys figured this out long ago....


Jay Hollingsworth

Seabed Portfolio Manager and
Principal Data Architect
Schlumberger Information Solutions
phone: [phone] fax: [phone]

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[OvV_HN]

----- Original Message -----
From: "Gerald I. Evenden" <[hidden email]>
To: <[hidden email]>
Sent: Friday, March 20, 2009 6:24 PM
Subject: Re: [Proj] Geodesic distances away from the ellipsoid

> There is also the 3D geodesic at the NGS site which might apply and give a
> better answer.  But I think the above would be good enough for most
> purposes.
> Try both and see what happens.
>
> PS: I did not convert the NGS 3D because it was hard-wired for WGS84 and
> the
> description was confusing enough that I decided against bothering for the
> time being.  Also, I could not get a mental image of what was happening
> when
> there was a difference of altitude at the two end points.

The last time I looked at the code the geodesic was calculated for an
altitude of 0 meters above the ellipsoid, all the way. I forgot where the 3D
information was used for, probably the chord distance or the north-east-up
coordinate difference.

Oscar van Vlijmen





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[support.mn]

Hello,

I think one method is to convert all coordinates to Cartesian
using Helmert's equations and forget the earth shape, since you
are more or less in the free space now... where the satellites rule.
The earth reference ellipsoid just adds complexity there.

Now you have vectors (x, y and z) and can calculate anything
using simple vector arithmetic.

A "laser line" distance is now very easy to calculate:

dist=sqrt((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2)

if your path is a set of 3D-curves, just calculate the perimeters
and add them together. This is more efficient especially for
space travellers (?)

regards: Janne.

----------------------------------------------------------

Jay Hollingsworth [[hidden email]] kirjoitti:

> At the risk of asking a dumb question, do any of the geodesic
> algorithms allow calculation of the geodesic distance if the path is
> not on the ellipsoid? Like in an airplane or satellite whose path
> could be assumed to be a constant height above the ellipsoid?
>
> Or do you have to define an ellipsoid matching the satellite orbit
> and use the normal equations on that ellipsoid? If so, how would you
> define an ellipsoid for a long airplane flight?
>
> I've poked around and nothing has jumped out at me. I assume
> aero-engineers guys figured this out long ago....
>
>
> Jay Hollingsworth
>
> Seabed Portfolio Manager and
> Principal Data Architect
> Schlumberger Information Solutions
> phone: [phone] fax: [phone]
>
> _______________________________________________
> Proj mailing list
> [hidden email]
> http://lists.maptools.org/mailman/listinfo/proj
>

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[Karney, Charles]

> From: Jay Hollingsworth <[hidden email]>
> Sent: Fri 20-Mar-09 11:56
> Subject: [Proj] Geodesic distances away from the ellipsoid

> At the risk of asking a dumb question, do any of the geodesic
> algorithms allow calculation of the geodesic distance if the path is
> not on the ellipsoid? Like in an airplane or satellite whose path
> could be assumed to be a constant height above the ellipsoid?

The paper

  Richard Mathar
  Geodetic Line at Constant Altitude above the Ellipsoid
  http://arxiv.org/abs/0711.0642

addresses this problem.  However, I believe there is a simple solution.

Let E be the ellipsoid and S a surface a constant height h above it.
A normal section through a point on E is a normal section through the
corresponding point on S.  Thus mapping a geodesic on E to S by
elevating it by h results in a geodesic on S.  This directly gives you
the course of the geodesic.  A little extra work gives you the azimuth
and length.

A couple of questions suggest themselves:

(1) is this observation true?
(2) is it new?

--
Charles Karney <[hidden email]>
Sarnoff Corporation, Princeton, NJ 08543-5300

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[strebe]

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Charles Karney asks:

>(1) is this observation true?

No.

In order for it to be true, the surface of constant height above an ellipsoid must also be an ellipsoid, and must be the same ellipsoid but scaled by a constant. That this is not true can be seen by the two-dimensional case: If you scale an ellipse, the new ellipse's major axis and minor axis are scaled by the same amount, but since the major and minor axes are different lengths, scaling by the same amount cannot result in the same differential added to both axes. If the differentials are not the same, the height of the new ellipse over the old one cannot be constant. This generalizes to three dimensions.

The exception is the sphere.

Regards,

— daan Strebe

On Jul 11, 2009, at 9:03:17 AM, "Karney, Charles" <[hidden email]> wrote:

Let E be the ellipsoid and S a surface a constant height h above it.
A normal section through a point on E is a normal section through the
corresponding point on S.  Thus mapping a geodesic on E to S by
elevating it by h results in a geodesic on S.  This directly gives you
the course of the geodesic.  A little extra work gives you the azimuth
and length.

A couple of questions suggest themselves:

(1) is this observation true?
(2) is it new?

--
Charles Karney <[hidden email]>
Sarnoff Corporation, Princeton, NJ 08543-5300

Tel: [phone]
Fax: [phone]

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[Karney, Charles]

> > From: Charles Karney
> >
> > Let E be the ellipsoid and S a surface a constant height h above it.
> > A normal section through a point on E is a normal section through the
> > corresponding point on S.  Thus mapping a geodesic on E to S by
> > elevating it by h results in a geodesic on S.  This directly gives you
> > the course of the geodesic.  A little extra work gives you the azimuth
> > and length.
> >
> > A couple of questions suggest themselves:
> >
> > (1) is this observation true?

> From: daan Stebe
>
> No.
>
> In order for it to be true, the surface of constant height above an
> ellipsoid must also be an ellipsoid, and must be the same ellipsoid
> but scaled by a constant.

Indeed my statement is false.

But your way of showing this doesn't work because while an ellipsoid
raised by h stops being an ellipsoid, an ellipsoidal geodesic raised
by h also ceases to have the properties of an ellipsoidal geodesic.

A simple counter-example is a cylinder with a cross section which is a
stadium (two semi circles joined by straight segments).  When the
cylinder is unfolded, the geodesic spiralling up such a surface is a
straight line.  However if the surface is mapped into another cylinder
a distance h away, this straight line maps into a connected sequence
of straight lines which have different slopes corresponding to the
flat and round portions of the stadium.  This obviously is not the
geodesic for the expanded cylinder.

--
Charles Karney <[hidden email]>
Sarnoff Corporation, Princeton, NJ 08543-5300

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[strebe]

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I agree that a geodesic on a projected surface need not be a simple scaling of a geodesic on the original surface even if the paths of the geodesics coincide. While I wrote "the surface of constant height above an ellipsoid must also be an ellipsoid" as if "must be an ellipsoid" were a constraint for any solution, in fact I merely thought it was a constraint of the posed problem. I did not realize (or recall) that other surfaces were up for consideration. Presumably constant height is of interest for... flying? Or...? (I cannot imagine any practical use for a precise solution, but there's no reason that should shut off consideration.)

You originally stated that the geodesics should coincide because a normal section through a point on E is a normal section through the corresponding point on S. Surely that's insufficient to claim a geodesic would coincide, since all it gives is a local direction for the geodesic, not the derivative of the direction that informs the direction of the next infinitesimal section in the geodesic's path. What might be two "adjacent" normal sections on the ellipsoid along its geodesic would belong to two different geodesics on the raised surface. Not a proof by any means, but I think it's analogous to saying the slope along a two-dimensional path tells you nothing about the behavior of the path away from that point. Presumably a rigorous form of this observation is what convinced you of the error.

Regards,

— daan Strebe

On Jul 12, 2009, at 4:10:00 AM, "Karney, Charles" <[hidden email]> wrote:

> From: daan Stebe
>
> No.
>
> In order for it to be true, the surface of constant height above an
> ellipsoid must also be an ellipsoid, and must be the same ellipsoid
> but scaled by a constant.

Indeed my statement is false.

But your way of showing this doesn't work because while an ellipsoid
raised by h stops being an ellipsoid, an ellipsoidal geodesic raised
by h also ceases to have the properties of an ellipsoidal geodesic.

A simple counter-example is a cylinder with a cross section which is a
stadium (two semi circles joined by straight segments). When the
cylinder is unfolded, the geodesic spiralling up such a surface is a
straight line. However if the surface is mapped into another cylinder
a distance h away, this straight line maps into a connected sequence
of straight lines which have different slopes corresponding to the
flat and round portions of the stadium. This obviously is not the
geodesic for the expanded cylinder.

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[support.mn]

strebe [[hidden email]] kirjoitti:
> Presumably constant height is of interest for... flying? Or...? (I cannot imagine any practical use for a precise solution, but there's no reason that should shut off consideration.)
>

Not exactly,
they are talking about flight levels and that is defined as:

"A Flight Level (FL) is a standard nominal altitude of an aircraft, in hundreds of feet. This altitude is calculated from a world-wide fixed pressure datum of 1013.25 hPa (29.921 inHg), the average sea-level pressure, and therefore is not necessarily the same as the aircraft's true altitude either above mean sea level or above ground level."

http://en.wikipedia.org/wiki/Flight_level

The WGS84 ellipsoid and the average sea level can differ several hundreds meters. So to be exact one should first calculate the geoid height and then transform that to an ellipsoid height. And even adjust for the fact that the actual sea level pressure at that point is far from the average.

Janne. / MNS Support

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