Teach,
From any angle I attack this, it just doesn't seem possible. Stem is impenetrably dense. I can make no steam on his publication.
I can figure a tangent line on the ellipse from this formula:
But first I have to know where on the ellipse the desired Latitude lands. So I need to calc a "Z":
From those two, in theory, I should be able to calculate my Radius of Curvature. Breakfast, first.
Next up is how the Grid Scale Factor and Elevation Factor work in a Low Distortion Projection. After reading Shawn's article, I thought I had the subject under control. He says, and I'm going from memory here so I might be wrong, that the Grid Scale Factor is Ellipsoid to Projection and Elevation Factor is Ellipsoid to Surface. I'll probably make a new post Monday to solidify that understanding.
Dave
I'm laboring either under the extension of last week's cold or a new cold so I may need some slack!
The plane in question is tangent to the meridian at a point...oh, I just saw it. A plane tangent at the surface *is* perpendicular to the plane of the meridian, but the plane used to define the transverse or prime vertical radius of curvature is perpendicular to both!
Another way to describe it is that you need the normal [vector] to the surface plus a plane perpendicular to the plane that contains the meridian.
If you're devolving to an ellipse, use longitude = 0 instead. That will cause Y to be zero and simplify your calculations (I hope!).
EDIT: Calculating the Radius of Curvature
> I'm laboring either under the extension of last week's cold or a new cold so I may need some slack!
>
> The plane in question is tangent to the meridian at a point...oh, I just saw it. A plane tangent at the surface *is* perpendicular to the plane of the meridian, but the plane used to define the transverse or prime vertical radius of curvature is perpendicular to both!
>
> Another way to describe it is that you need the normal [vector] to the surface plus a plane perpendicular to the plane that contains the meridian.
Let me rephrase that last sentence.
Another way to describe it is that you need the normal [vector] to the surface plus a plane that contains the normal and is perpendicular to the plane that contains the meridian.
Radius of curvature - there is a problem here.
There is a problem here. The geocentric "Z" you've computed is the perpendicular distance from the equator to the point on the ellipsoid. That "Z" bears no relation to the distance above or below the ellipsoid as might be used for radius of curvature, grid scale factor, or elevation reduction factor computations.
Prior to that, you quote an equation for an ellipse in terms of x and y. That is OK for a generic ellipse, but, in the meridian ellipse and using ECEF X/Y/Z values, your ellipse should be in terms of x and z. Again, no relationship to above or below the ellipsoid.
For What Purpose or how close is close enough?
I applaud the fact you are sorting it out because you want to decide for yourself. Several comments, suggestion, and references.
1. The radius of curvature at a point varies according to the azimuth of the line. That gets to be quite onerous so most people use the geometrical mean radius for a given latitude.
2. How much difference does it make if I use the geometrical mean radius approximation instead of the "best" radius of curvature at that point? For the answer, see item #54 at this link.
3. Low Distortion Projections (LDP) can have legitimate applications. But, remember, a map projection (including a LDP) is strictly a two-dimensional model and we work with three-dimensional data. The global spatial data model (GSDM) accommodates all the benefits of a LDP (and more) while preserving the geometrical integrity of the data. See #35 at the same link.
Melita,
I hope you get feeling better. Colds are no fun!
I'll try using zero for Longitude and see if that changes anything. So far I've been unsuccessful using every formula and link in this thread. For a N34 latitude I keep getting a Radius about 7,000 meters longer than the Semi-Major Axis.
Dave
Radius of curvature - there is a problem here.
EF,
"That "Z" bears no relation to the distance above or below the ellipsoid..."
I thought this was the "Z" dimension:
Yes, I soon discovered why my general ellipse equation wouldn't help me. I'm stumped right now. I've been working on this since 1 a.m. last night, and I'm no closer than I was yesterday.
I don't quite understand what you've said about "above or below the ellipsoid". I want to figure smack dab on the ellipsoid, not above or below it. Baby steps first.
Dave
For What Purpose or how close is close enough?
EF,
"The radius of curvature at a point varies according to the azimuth of the line."
I have a hard time with that one. Technically, the radius of curvature can't be perpendicular to a point. What the radius is perpendicular to is a line going through that point. So how is a "geometrical mean radius" defined?
Thanks for the link to your work. Looks like I've got some reading to do. And that's a good thing.
Dave
Yep, that's about what I get (values below; NOT verified). This was using an existing program (just needed recompiled) so I didn't have to think too hard. Just to throw another reference work into the mix, Dr. Jekeli at Ohio State has a book/lecture notes available: Geometric Reference Systems in Geodesy. The section on the radii of curvature starts on page 28.
Don't forget that the radius can drop below the equatorial plane.
GRS_1980, a = 6378137.0, 1/f = 298.257222101
M = meridional radius of curvature
N = prime vertical radius of curvature
Latitude (deg) M (m) N (m)
1 6335458.7042 6378143.5026
2 6335516.8127 6378163.0025
3 6335613.5834 6378195.4764
4 6335748.9015 6378240.8852
5 6335922.6061 6378299.1746
6 6336134.4910 6378370.2744
7 6336384.3042 6378454.0995
8 6336671.7489 6378550.5491
9 6336996.4833 6378659.5074
10 6337358.1214 6378780.8437
11 6337756.2329 6378914.4121
12 6338190.3442 6379060.0523
13 6338659.9387 6379217.5892
14 6339164.4572 6379386.8336
15 6339703.2989 6379567.5820
16 6340275.8215 6379759.6172
17 6340881.3427 6379962.7081
18 6341519.1403 6380176.6104
19 6342188.4532 6380401.0668
20 6342888.4823 6380635.8072
21 6343618.3914 6380880.5487
22 6344377.3082 6381134.9968
23 6345164.3251 6381398.8448
24 6345978.5001 6381671.7746
25 6346818.8586 6381953.4572
26 6347684.3934 6382243.5527
27 6348574.0670 6382541.7112
28 6349486.8116 6382847.5727
29 6350421.5316 6383160.7676
30 6351377.1036 6383480.9177
31 6352352.3786 6383807.6359
32 6353346.1830 6384140.5270
33 6354357.3199 6384479.1883
34 6355384.5706 6384823.2098
35 6356426.6958 6385172.1749
36 6357482.4375 6385525.6607
37 6358550.5202 6385883.2387
38 6359629.6520 6386244.4751
39 6360718.5271 6386608.9316
40 6361815.8264 6386976.1657
41 6362920.2195 6387345.7313
42 6364030.3663 6387717.1792
43 6365144.9186 6388090.0576
44 6366262.5216 6388463.9130
45 6367381.8156 6388838.2902
46 6368501.4376 6389212.7332
47 6369620.0230 6389586.7857
48 6370736.2075 6389959.9917
49 6371848.6281 6390331.8958
50 6372955.9257 6390702.0443
51 6374056.7459 6391069.9849
52 6375149.7413 6391435.2683
53 6376233.5727 6391797.4478
54 6377306.9112 6392156.0805
55 6378368.4396 6392510.7275
56 6379416.8540 6392860.9547
57 6380450.8658 6393206.3330
58 6381469.2029 6393546.4391
59 6382470.6113 6393880.8562
60 6383453.8573 6394209.1739
61 6384417.7282 6394530.9894
62 6385361.0347 6394845.9074
63 6386282.6117 6395153.5413
64 6387181.3202 6395453.5129
65 6388056.0489 6395745.4534
66 6388905.7150 6396029.0038
67 6389729.2664 6396303.8152
68 6390525.6824 6396569.5493
69 6391293.9755 6396825.8789
70 6392033.1924 6397072.4883
71 6392742.4153 6397309.0736
72 6393420.7633 6397535.3431
73 6394067.3933 6397751.0179
74 6394681.5012 6397955.8319
75 6395262.3229 6398149.5324
76 6395809.1356 6398331.8803
77 6396321.2582 6398502.6507
78 6396798.0530 6398661.6325
79 6397238.9256 6398808.6295
80 6397643.3265 6398943.4600
81 6398010.7514 6399065.9575
82 6398340.7418 6399175.9706
83 6398632.8862 6399273.3633
84 6398886.8198 6399358.0152
85 6399102.2256 6399429.8216
86 6399278.8348 6399488.6938
87 6399416.4268 6399534.5590
88 6399514.8297 6399567.3604
89 6399573.9207 6399587.0575
90 6399593.6259 6399593.6259
Radius of curvature - there is a problem here.
Does this help?
lat = 35N
lon = 0
h = 0
X : 5230.427 km
Y : 0 km
Z : 3637.867 km
The 3D Cartesian coordinates should map to X = x, Z = y.
EDIT: Converter was the first one found in Google.
The Elithorp book mentioned in your other thread has a nice discussion of th various radii used in geodetic computations. It includes some good illustrations.
You might also want to review this lecture: http://geodesyattamucc.pbworks.com/f/Ellipse_16Feb2K10.pdf
Another interesting question, given that the elevation factor published on NGS datasheets is defined as R/R+h, what is R?
Melita,
From the NGS Glossary:
"radius of curvature in the meridian - The radius of curvature at a point on an ellipsoid with respect to the meridian through that point.
radius of curvature in the prime vertical - The radius of curvature at a point on an ellipsoid with respect to the prime vertical through that point."
What's the difference between these two?
"Don't forget that the radius can drop below the equatorial plane."
I was beginning to suspect that accounts for the larger-than-expected Radius.
Thanks for the link. Looks like I'll find a few answers in there:
Dave
Radius of curvature - there is a problem here.
Melita,
I got about the same with the Converter I used.
X:5230426.84020036m
Y:0m
Z:3637866.90932638m
I'll fire up Excel and see if I get a different radius.
Dave
Shawn,
Do you have a link to your article? It has been referenced several times in the thread and I am interested in what it says. I am a geodesy neophyte and find the discussion fascinating.
Thanks,
Tim
Odd,
"Another interesting question, given that the elevation factor published on NGS datasheets is defined as R/R+h, what is R?"
Everyone seems to agree that "any old number" is close enough for R, but I'm far too anal to settle for less than 20 decimal places. In the age-old question of accuracy vs precision, I say can't we have BOTH?
Thanks for the link.
Dave
Radius of curvature - there is a problem here.
The following links may help in seeing the difference:
1. Shows coordinate systems
2. Shows same things differently
3. Two views. One view is meridian ellipse (X and Z) while the other view is in plane of equator (X and Y).
We should not "advertise" here but it is all described in the book, "The 3-D Global Spatial Data Model." Find source to purchase by "googling" "global spatial data model."
Radius of curvature - there is a problem here.
EF,
OK, now I'm starting wonder whether little h stops at the plane of the equator or keeps going until it intersects the Polar Axis. :-S
Thanks for the links and I'll add your book to my search list.
Dave
Radius of curvature - there is a problem here.
Little h as shown on the diagram is called "ellipsoid height" and is the distance above or below the ellipsoid along the ellipsoid normal.