Calculating the Radius of Curvature

For What Purpose?

What are you calculating that you need the radius of curvature?
and to what precision?

Both questions are relevant to whether complicated calculations are required.

Paul in PA

For What Purpose?

Paul,

For designing Low Distortion Projections:
Scale Factor = (R + P) / R

Rather than accept a Radius that someone says is "close enough", I'd like to decide for myself.

Dave

For What Purpose?

I would suggest you stay with the radius used in the SPC Zone in which your LDP exists.

Anything else is complicating the work for others.

Paul in PA

For What Purpose?

Dave - What kind of area are you trying to cover? The software on your data collector should be able to give you an adequate projection; leave the geodesy to the geodesists.

A local Transverse Mercator projection with the origin at the center of the project and an ellipsoid height at the base that is accurate to 20' or better will give you almost zero distortion for about a 2 mile radius.

For What Purpose?

It depends on the type of latitude, there is geodetic, geocentric and I think there might be others. Latitude is an iterative calculation.
Been a while

For What Purpose?

:good:

I agree. For that purpose use the standard 'assumed' radius. Others tying to your work or for later work, everyone will use the same numbers. Still, it's worth going through the academic exercise so you understand it better. Another good exercise is to calculate your ground distances using both to see exactly how much difference it makes.

I did those kinds of calculations in school and would have to reference my material to remember.

For What Purpose?

Actually SPC systems do not specify a radius as they use lines of latitude. Two different radii for projection on transverse Mercator.

I'll have to rethink that.

Paul in PA

For What Purpose?

Paul,

"...stay with the radius used in the SPC Zone in which your LDP exists."

A very good point.

Dave

For What Purpose?

Mark,

"Still, it's worth going through the academic exercise so you understand it better. Another good exercise is to calculate your ground distances using both to see exactly how much difference it makes."

Exactly.

Dave

For What Purpose?

Paul,

It sounded like such a good idea that I was going to investigate whether the Radius was published in a SPCS.

Dave

For What Purpose?

Lee,

"...leave the geodesy to the geodesists."

You're about the last person on this Forum from which I would expect to hear the equivalent of "Don't you worry your silly little head about it".

I mean to master this subject if I have to crawl over corpses to accomplish it.

"The software on your data collector should be able to give you an adequate projection..."

That's next on the list. I can make a LDP on C3D, but I haven't figured out how to transfer that projection to Survey Controller.

Dave

For What Purpose?

Ralph,

Exactly. Latitudes are Geodetic. If I had the Geocentric Angle I could figure it out. I'll work on finding the relationship between the two.

Dave

> The Radius of Curvature of an Ellipsoid changes by Latitude.

The radius of curvature of the ellipsoid also varies with azimuth at the same point.

For What Purpose?

I didn't mean to imply that it's not important to understand the underlying science and mathematics, it is. I just believe that my software provides me with adequate tools; my knowledge enables me to use them correctly and understand the results.

Dave, the local retired expert on such things is Jack Kessler. He lives in Kingman too, and is pretty willing to help any surveyor.

Me, I'd be throwing darts at the wall hoping I win a prize when I hit a balloon. Don't know why you even care, but not my concern.

If you're going to fiddle with all of the "custom" stuff,

Don't forget the oblique stereographic, the oblique conic, and the oblique Mercator. Depending on the shape of your project area, those are candidates for "custom" projections, too.

It certainly will complicate life for whoever attempts to follow in your footsteps, though.

I apologize for not having the source for this David. I had the same thought you did about this when I wrote the article. After solving this for myself and seeing the practical difference it made to the Elevation factor, I was convinced that it was insignificant. You can also test this by comparing an elevation factor using the Earth radius at the equator (6378137m) and an Earth radius at the pole (6356752.3m, as these values represent the extreme spread of radii, and see the difference it would make.

I found the following formula online, but don't recall where.

SQRT[[(cos(LAT)*6378137^2)^2 + (sin(LAT) * 6356752.3141^2)^2]/[(cos(LAT)*6378137)^2 + (sin(LAT) * 6356752.3141)^2]]

for 32° Latitude, I get a radius of 6372168m

If you're going to fiddle with all of the "custom" stuff,

> Don't forget the oblique stereographic, the oblique conic, and the oblique Mercator. Depending on the shape of your project area, those are candidates for "custom" projections, too.
>
> It certainly will complicate life for whoever attempts to follow in your footsteps, though.

I agree to some extent, Doc. I believe it depends on the usage. We're using LDP's for our terrestrial-GNSS connections. Our plats are reported with LDP bearings and distances (which closely resemble geodetic bearings within a few arc minutes and ground measured horizontal distances). We then label several prominent monuments with latitude and longitude. I've found that my meta data statement is reduced tremendously with this, I don't have to gyrate between GNSS derived positions and terrestrial derived positions. Variety is the spice of life I suppose.

Shawn,

"I had the same thought you did about this when I wrote the article."

By coincidence, I was reading your article on LDP's and wanted to check the math on one of your tables when I looked up from the page and wondered about the effect of radius on the Grid Scale Factor.

Thanks for that big, fat, hairy equation. Good suggestion on testing the two extreme radii.

Dave