Radius of curvature - there is a problem here.
EF,
I screwed up. I meant big R. :-$
Dave
Dave,
Thanks for the article links. I have read them and they are helpful. The links in the posts by EFBurhkolder are very helpful as well as I try to get my head around the various coordinate systems and projections.
Radius of curvature - there is a problem here.
Several clarifications:
1. I too get things mixed up. The older I get the more time I need to spend checking and rechecking my work. For example, in a previous post I recommeded using items #35 and #54 on a link of references. I got them reversed - item #35 relates to elevation factor computation. Equation (3) in that link shows how to compute the geometrical mean radius of curvature. Question - it that what you mean by "Big R"?
2. The geometry of the GSDM is more straight-forward than working on the ellipsoid and the equations are not as complex as geodesy equations. Yet, using the GSDM one can develop a better solution showing local directions and distances than with using map projections (and state plane or UTM coordinates). For the amount of effort you are devoting to understanding the concepts and geometry, I'll suggest you will get more "bang for your buck" investigating the GSDM - item #54 in the previous post.
3. In an early post I made reference to item #2 on the list of articles. Someone mentioned that the article was not readily available - I have posted a pdf file of that article. See item #2 again. Incidentally, I had access only to a 32 bit main-frame computer when I wrote that article. On that same issue, Dennis Milbert provides many more digits and notes using a 128 bit machine. Oh the impact of progress! Thanks Dennis!
Keep the questions coming - that is how we make progress!
Radius of curvature - there is a problem here.
EF,
Equation (3) looks do-able, but when I saw equation (2) I was like, Shoot. Me. Now.
Dave
Equation (3) Looks Promising
EF,
Reading. I'll plug this into Excel and see what comes up:
Dave
#35
EF,
"Big R" calc'ed from your formula in #35 looks good. For a Latitude of N34-58-50 I get a Big R of 6370769.59362787m, which is 7400 meters shorter than the Semi-Major. I'll keep reading.
Dave
Definitive Findings
There is a "simple" answer to the question, "How do I calculate the Radius of Curvature of this ellipsoid called Earth". The question I should have asked, as [msg=279764]GeeOddMike[/msg] put it is:
"...given that the elevation factor published on NGS datasheets is defined as R/R+h, what is R?"
Well, there's GeoDetic:
GeoCentric:
And Reduced Latitude:
Oy veh! So, pick one. OK, I'll pick the Geodetic Radius of Curvature as my best guess as to the Radius I'll need to design a Low Distortion Projection. Yes, I know that 20,906,000 feet, aka "any old number", is what most people use as a "radius".
Now, on to selecting a formula to calculate the Radius. I read this and thought Oh great!:
So I decided (guessed, threw a dart, picked a formula out of a hat) that this was the number I wanted:
So, out of all the formulas out there to compute Radius,
I finally settled on this formula:
Am I right? I don't know. I think I'm 95% certain, which is good enough for RTK. (Humor, awk, awk!)
Dave
And the Radius is...
Geodetic Latitude: N34-58-50
Semi-Major Axis: 6378137m
Semi-Minor Axis: 6356752.31414035
Prime Vertical Radius of Curvature: 6385165.34501882
Notice that the PVRofC is 7028 meters longer than the Semi-Major Axis. That's because ?(distance below equator) is 24505 meters.
Dave
for Dave
I really like this topic but haven't had the time to chime in in a coherent fashion.
I like your enthusiasm.
Here are my old Geodesy notes on that subject (pardon the coffee stains 🙂 ) maybe you can find them useful.
Cheers
for Dave
Thanks, Ralph. Did you scan those? Those coffee stains are quite realistic.
Good stuff! Time to put on a pot of coffee and settle in to some reading.
Dave
for Dave
> Thanks, Ralph. Did you scan those? Those coffee stains are quite realistic.
>
> Good stuff! Time to put on a pot of coffee and settle in to some reading.
>
> Dave
Yeah, scanned and uploaded to my server
Enjoy:-)
Ralph
Ralph,
I just finished reading it. Excellent, excellent, excellent! The Reader's Digest Version of exactly what I wanted to know. It firmed up my understanding of a couple of key concepts that I was struggling with. Thank you.
Dave
Ralph
Dear Dave,
First, let me say "thanks Ralph" for posting and highlighting
those notes.
Also, for those interested, the Rapp/OSU Geometric Geodesy I
notes can be found at:
http://kb.osu.edu/dspace/handle/1811/24333
-- or --
http://kb.osu.edu/rest/bitstreams/53179/retrieve
Next, this discussion on the elevation factor and radii of
curvature has occurred before. Note the thread at:
https://surveyorconnect.com/index.php?mode=thread&id=234188#p262241
So, let's deconstruct this....
Elevation factor = R/(R+h) (a unitless quantity, by inspection)
R = Gaussian Radius of Curvature (more on this below)
h = ellipsoidal height
Pretty simple. The Elevation Factor is really just an exercise
in finding the Gaussian Radius of Curvature.
Now, go to page 9 of Ralph's notes -- where the purple highlight
is. The key phase is that on an ellipsoid, the radius of
curvature is *not* a constant. In fact it varies with both
geodetic latitude, and with the geodetic azimuth.
Now, it's possible to compute all kinds of "mean" radii as well
as the radii along a specific geodesic and along normal sections
between two points. Section 3.9 of the Rapp/OSU notes shows a few.
Now, most, but not all of them, are built to represent a
spherical model (constant R) for the Earth. And, they typically
return a value close to 6371km. But, suppose we want to do a
little better.
Suppose we want R = R(glat)
where glat = geodetic latitude.
This way, we get the latitude variation.
Well, the R for that is the Gaussian Mean Radius, found at
equation 3.140 of the Rapp notes.
R(glat) = sqrt(M * N)
where
M = radius of curvature on the meridian (varies with lat)
N = radius of curvature on the prime vertical (varies with lat)
(We see it is an interesting way to get a mean -- by means of
multiplication.)
There is also an identical, alternative expression
R(glat) = a * sqrt(1-e2)/(1-e2*sin(glat)^2)
where a is the ellipsoid semimajor axis
and e2 is the square of the first eccentricity of the ellipsoid
Dave, I think you are good about what the "M" is. Imagine standing
on the ellipsoid. Your body axis denotes an ellipsoidal normal.
Pivot until your nose points towards the North (or South) pole.
You are now aligned along a meridian, and the meridian plane
would bisect your body into left and right halves, and bisect the
ellipsoid. That intersection of the meridian plane with the
ellipsoid will describe a Meridian Ellipse (section 3.3 of Rapp
notes). You can see the varying curvature along the ellipsoid.
Now, with your nose still pointed towards the North Pole,
extend your arms to the left and right. The plane slicing through
your head, both arms, and feet will intersect the ellipsoid.
This intersection is a symmetric, closed figure -- but, it is
not (usually) an ellipse. Despite that, it does have a varying
curvature. This is the radius of curvature on the prime vertical.
The equation of M is 3.33 in Ralph's notes, and 3.88 in Rapp.
The equation of N is 3.38 in Ralph's notes, and 3.99 in Rapp.
Now, back to the Gaussian Mean Radius. (Yes, invented by Gauss.)
Radii of curvature vary with azimuth. We can compute the radius
for an arbitrary azimuth by means of Ralph 3.39/3.40, and by
Rapp 3.103/3.104.
What the Gaussian Mean Radius is -- compute the average of *all*
the radii along all possible azimuths at a single geodetic latitude.
No surprise, this leads to an integral expression -- 3.136 of Rapp.
After doing the calculus and algebra....
R(glat) = sqrt(M * N) (Rapp 3.140)
as shown above.
Dave, the Gaussian Mean Radius is the R used in the NGS datasheet
for the expression of the Elevation Factor.
As you noted above, it is not accurate to 20 decimal digits, since:
a) one does not know ellipsoidal height to a commensurate accuracy
b) the Elevation Factor is part of the reduction of distance for
a plane coordinate system, and those distortions will exceed the
desired 20 digit accuracy, and
c) the radii are for normal sections, and not for a geodesic (which
is probably more appropriate to represent a distance)
d) the Gaussian Mean Radius is just that -- a mean, and will not
capture the variation of radius with respect to azimuth to 20 digits.
😉
But, on a formal basis, you can compute R to as many digits as you like.
All the best,......
Ralph
Good Stuff Dennis!
Radially Tortured
Dennis,
"Pretty simple."
No, it's not. Not at all. There are half-a-dozen different formulas to figure Radius, and they return wildly different numbers. So I guess it's time to put Radius Study on a shelf and learn as much as I can about Projections. Maybe there I can discover what R to use.
I greatly appreciate your efforts to enlighten me.
Dave