Rhumb line/Great Circle vs projection type

[Smurfjet]

Something I missed in ground school last week.
The instructor mentioned that depending on how the chart was "drawn" either GC or Rhumb line is good for plotting the route.

So what goes with what?
Lambert Conformal Conic prjection Transverse Mercator projection

Any other projections you can add to this list...

Next time I will make sure I'll have coffee, good thing the instructor has all those stories to tell so we can stay awake :)

EDIT: I managed to find the answer (RTFM :)), but I'd still appreciate the input

[This message has been edited by Smurfjet (edited 04 February 2001).]

[bookworm]

On a Mercator projection, a straight line on the chart is a rhumb line on the globe (i.e. constant bearing).

On a gnomonic projection, a straight line on the chart is a great circle on the globe.

On a Lambert Conformal Conic, a straight line on the chart is approximately a great circle on the globe, and the difference from a true great circle is usually so small as to be negligible.

[The Nr Fairy]

And the Great Circle line is ALWAYS, irrespective of projection, on the POLAR side of the rhumb line.

[Scallywag]

Bookworm, hope you haven't mislead Smurfjet. He asked about a Tranverse Mercator, not Mercator. On a Tranverse Mercator, a straight line is a Great Circle the same as an Oblique Mercator which also give constant scale within + or - 8 deg of the false equator with no scale reduction factor.

[411A]

Anyone here remember pressure-pattern navigation? Now there was an interesting exercise. Any old Navigators available to explain in concise terms? Cannot remember which chart was used, however.

[bookworm]

Scallywag

I stand corrected. Yes Smurfjet did ask about Transverse Mercator, which is not the same as a Normal Mercator.

Unlike a great circle, what constitutes a rhumb lines is an artefact of the coordinate system that you use on the globe.

A Normal Mercator depicts rhumb lines as straight lines. If you change the aspect and move the reference point by 90 degrees to make it a Transverse Mercator, rhumb lines are no longer straight.

I think, however, it's a little misleading to say "On a Tranverse Mercator, a straight line is a Great Circle the same as an Oblique Mercator which also give constant scale within + or - 8 deg of the false equator with no scale reduction factor. "

On any projection, straight lines approximate to both rhumb lines and great circles over short distances and close to a single point or line, simply because a sphere is locally flat. Even on a simple cylindrical projection, a straight line is very close to a great circle (and a rhumb line) close to the equator.

Over larger distances and further away from an undistorted line, which is the real test, I can't see how a straight line on a Transverse Mercator can possibly be a great circle.

http://www.geometrie.tuwien.ac.at/karto/index.html

is a super resource for getting an idea of what a projection looks like. If you look at the Transverse Mercator (http://www.geometrie.tuwien.ac.at/karto/trans03.html), the meridians are clearly not straight, except at the false equator.

[Alex Whittingham]

Great circles are never straight lines on any two dimensional chart unless they happen to follow the tangent of the chart to the reduced earth in projection or lie at right angles to it. Apart from these cases the best you get is that great circles are nearly straight lines, always concave to the tangent to the earth.

Transverse Mercators and regular Mercators are exactly the same in this respect, near the great circle of tangency great circles are nearly straight but actually slightly concave to the tangent (which is the equator on a regular Mercator).

Lamberts don't have a tangent as such as they cut into the earth in projection but the idea holds, great circles are nearly straight but actually slightly concave to the parallel of origin which is itself parallel to the tangent in projection.

[bookworm]

<font face="Verdana, Arial, Helvetica" size="2">Great circles are never straight lines on any two dimensional chart unless they happen to follow the tangent of the chart to the reduced earth in projection or lie at right angles to it. </font>

Not quite true, I believe (though pretty close). The gnomonic projection really does have great circles as straight lines.

I'm not going to attempt a mathematical proof :) But if, for example, you look at the Oblique Gnomonic (http://www.geometrie.tuwien.ac.at/karto/schief14.html) you'll see that the meridians are straight lines, even though the north pole does not have a "special" role in the projection. Other great circles are similarly straight lines.

The problem with the gnomonic is that it's pretty much useless for navigation as the angles are not preserved (it's not conformal). You can't easily measure a bearing unless the meridians cross the lines of latitude at 90 degrees.

[This message has been edited by bookworm (edited 09 February 2001).]

[This message has been edited by bookworm (edited 09 February 2001).]

[Smurfjet]

Hey guys!

Thanks for all the info! I became interested in maps now, and I bought a few :)

Now since 411A mentioned it, anyone for some pressure-pattern info? :)

Regards

[Alex Whittingham]

Bookworm, You are quite right, my apologies. A Gnomic projection projects on a flat plate from the centre of the earth. As the planes of all great circles pass through this point they all show up straight on the chart. I discounted Gnomics as very rarely used in aviation.

[xyz_pilot]

Bookworm

Please check the geometrie url it sounds interesting.

thanks

xyz_pilot