For What Purpose?
I haven't used all software. I use TBC. In a points report you can set it to show the scale factors. So for a small scale LDP (say less than 10 miles wide) you can get a scale factor simply from the inverse of the elevation factor. Simply enter a point into the software at the elevation you want grid to equal ground in your LDP. Let the software do all the math. Print out the points report with the elevation factor turned on. Inverse the elevation factor and then enter that into your projection as the scale factor. Reprint your points report and the combined scale factor should now be 1.0000000 or very near.
Any geodetic software worth using should have this capability. Setting up an small area LDP doesn't need to be all that complicated. I suppose you should “roll your own” a few times to verify your confidence in the software you are using.
Kent,
"The radius of curvature of the ellipsoid also varies with azimuth at the same point."
I see your point, but I'm still chewing on that one.
Dave
Wayne,
There are no prizes for almost hitting the balloon. I'd like to know how to hit that balloon to 15 decimal places. 
Dave
For What Purpose?
Leon,
"I suppose you should “roll your own” a few times to verify your confidence in the software you are using."
That's a good way to put it. I use AutoCAD Civil 3D and TGO 1.6, but I have to understand the theory before I move on to automating the results.
Dave
If you're going to fiddle with all of the "custom" stuff,
Cliff,
My mind starts to smoke when I ponder how to oblique a stereographic or a conic projection. I think I'll stick to the "standard" projections until I get a really good handle on this subject.
Dave
This thread reminds me of that great old movie, "Journey to the Center (RP) of the Earth."
Brad,
Was that "Journey to the Geodetic Center of the Earth" or "Journey to the GeoCentric Center of the Earth"? 
Dave
I am glad you saw the humor. Whew, relief.
For What Purpose?
From my view TGO was better than TBC in some ways. In TBC they took out some really good stuff like formatting custom reports. If you dig deep into the report options in TGO you can get at all the good stuff and format a custom report to print all the factors (maybe even the radius you seek). In TBC they limited it more but you can turn on or off some options in the points report. You can turn on the scale factors and get them for any point you decide to list.
The combined factor is the scale factor times the elevation factor. At you origin for a small scale LDP I like the combined factor to be 1.000000000 (grid distance equals ground distance). So punch in a point for your LDP origin point at the elevation you want. List out the points report to get the elevation factor. Inverse it. Then reenter your projection with this scale factor. Now the scale factor times the elevation factor equals 1.000000000 at your LDP chosen elevation. You can list the points report to verify.
As long as you are using the the same ellipsoid (say GRS80) in the various projections you use (including SPC and UTM) the elevation factor will be the same at a lat, long and elevation in any projection. Geodetic software can crank these out by the millions per second. Inside of that I'm sure the radius is calculated.
Response to question "... how do you calculate that Radius?" Specifically, if understood correctly, "that Radius" is the radius of curvature referred in the discussion below the diagram.
Given a latitude, an ellipsoid and its semi-major and semi-minor diameters as "A" and "B" respectively,
Rn = A/sqrt(1 - e^2 * sin^2(latitude)) .... [Rn = R-subscript-N in diagram]
where e^2 = (A^2 - B^2)/A^2
The primary basis for derivation of the Rn equation is that the slope of the tangent to an ellipse at a point is the negative reciprocal of the slope of the normal and that the slope of the normal is the trigonometric tangent of the latitude.
A couple comments:
1. I think the definition stated below the diagram "The physical radius of the earth, R," is not quite accurate. From a personal perspective, I think it would be more accurate to state that R is the distance from a point on the ellipsoid surface to the ellipsoid center and possibly add that the ellipsoid center is an approximation for the location of the earth's center of mass.
2. Autocad, unless it has changed recently, does not draw an ellipse, but instead, draws a polyline that approximates an ellipse. In its database, Autocad stores a series of coordinates for points on the ellipse but, does not store the ellipse's equation. That is probably the reason the "perpendicular command" does not work.
Dave,
A lot of these equations are in John P. Snyder's Map Projections: A Working Manual. It's online at http://pubs.er.usgs.gov/djvu/PP/PP_1395.pdf (22 MB). See pp 24 and 25 for the radii of curvature.
The radius of curvature of a normal section at azimuth alpha is
1/R_alpha = (cos(alpha)*cos(alpha)) / M + (sin(alpha)*sin(alpha)) / N
where
M = meridian radius of curvature
N = prime vertical radius of curvature
Torge, Wolfgang. Geodesy. 2nd Edition. de Gruyter. pp 46-47
Back to Snyder, he has the conversion between geodetic and geocentric latitudes on page 17 (bottom).
I'm definitely in agreement with the others--don't mess with the ellipsoid, instead adjust the projection parameters to lessen distortion. Changing/using a different ellipsoid than GRS80 (or Clarke 1866) means to many software packages that there's a datum transformation involved. The workflows can get...messy. :excruciating:
Melita
Melita,
"...don't mess with the ellipsoid..."
OK, I swear I won't. Well, maybe just a little.
Thanks for the link. That's good stuff.
Dave
ML,
Thank you for the equations, and the explanation of why AutoCAD's ellipses are so lame.
Dave
Kent,
Melita's link backs you up:
Dave
Are those figures from one ofJames Clynch's papers? Pretty good place to start.
For mean radius of curvature, go to page 30 of James Stem's Manual NGS 5. This is in the Lambert section of the manual under arc to chord correction.
His formula for little r sub zero is the mean radius of curvature at the latitude of the natural origin multiplied by the scale factor at the natural origin. Simply (sorry) drop the k sub zero from the formula and you will have the formula you want.
One nerdy-neat use is determine how far below the ellipsoid the state plane is at its natural origin in a Lambert state. For North Carolina, my home and the home of the ECU Pirates, little r sub zero is 6,370,148 meters. Dividing that by k sub zero, the scale factor at the natural origin, gives mean radius of curvature of 6,370,959 meters. Subtracting the two puts the state plane 811 meters below the ellipsoid at the natural origin. See page 100 of Stem for the numbers.
Studying Lambert projections in Stem will give you insight into the geometry of plane projections, either SPC or LDP. Mercator SPCs are done by series expansions, so there's not much of instructional value in Stem for Mercator.
Keep studying! There does not have to be an immediate need for knowledge in order for you to seek it.
>
> Keep studying! There does not have to be an immediate need for knowledge in order for you to seek it.
:good: :good: :good:
Teach,
Yes, the figures are from James R. Clynch's white paper. I love his illustrations. Sometimes a good illustration is worth 1,000 words.
Stem's white paper is too dense for me. I've tried tackling it twice, and both times ended up unconsious and drooling on the desk. But I'll go back and look at page 30. 
Dave
Melita,
From pages 24 and 25 of Map Projections - A Working Manual, there are two formulas for figuring the Radius of Curvature. I don't understand the distinction between R' and N:
R' is "in the plane of the meridian":
N is "in a plane perpendicular to the meridian and also perpendicular to a plane tangent to the surface":
Wouldn't a plane tangent to the surface also be perpendicular to the meridian?
Dave
For a plane perpendicular to the meridian, think about a guillotine blade poised over the meridian with the business end of the blade running in an east-west direction. As it slices through the ellipsoid, it will cut the major axis at an angle equal to the geodetic latitude and then slice through the minor axis.
Do not try this at home.
The mean radius of curvature, using Snyder's notation is sqrt(N*R-prime). If you do the algebra, you will get the formula in Stem. without the k sub zero.
For SPCs and LDPs, the mean radius of curvature is used to calculate elevation factors. Scale factors come from the geometry of the ellipsoid and utilize triangle similarity in their derivation.
> Do not try this at home.
haha:-D
