coordinates - How to convert from one ellipsoid to another?

Introduction

I have the following problem: I want to convert from WGS84 to "Anguilla 1957 / British West Indies Grid" (EPSG 2000). This seems to be straight-forward, as the transformation is a simple Transverse Mercator and the projection uses the Clarke 1880 Ellipsoid without any datum shift. My result differs significantly (some 400m) from the one i.e. epsg.io is returning.

Problem

The reason seems to be the datum transformation chain. Since the code is implemented such that it can convert from any cs to any cs, it also in this case does

1. WGS84 geographic -> WGS84 Geocentric (using the WGS ellipsoid)



2. WGS84 Geocentric -> Clarke 1880 Geocentric (nothing to do here)


3. Clarke 1880 Geocentric -> Clarke 1880 Geographic (now using the Clarke Ellipsoid)

The last step is giving me a headache now, because the output of step 3 is not equal to the input at step 1, while the same conversion returns the exact input on reference pages I've checked.

Question

Should and can a "Geographic Ellipsoid 1->Geocentric->Geographic Ellipsoid 2" Conversion change latitude and/or longitude?

Update

Here's the steps I take: I basically want to convert from WGS84 to Anguilla, British West Indies Grid (EPSG 2000). According to http://epsg.io/2000 this uses Datum 4600. This datum consists of the Clarke 1880 RGS ellipsoid 7012 and no datum transformation.

I'm only considering the conversion from WGS84 to Datum 4600 here, because the rest is not relevant here.

I'm getting -63.000000000000014, 18.001851635270203, -78.860902368091047, which is the exact same result as mkennedy below using a 0 0 0 transformation. However, If I input the same coordinates at http://epsg.io/4600/map, The output is precisely -63, 18. That's where the confusion comes from. May I assume that the web site is wrong and whatever coordinate system library they're using is incorrectly assuming that a conversion whose datum parameters are 0 can be skipped altogether?

Answer

Note: simplified explanation

The relationship between two datums (geographic coordinate reference systems) is not just that they may have different ellipsoids. Differing ellipsoids mean different sizes and shapes. If you treat both datums as being earth-centered (center at 0,0,0 in 3D Cartesian coordinates), and convert between them, you may see latitude and height differences.

However, particularly older GeoCRS aren't earth-centered. It's easiest for me to think of them as attached at the surface and their centers as offset from the earth's center. See this diagram as an example.

When converting between two GeoCRS, you also have to take into account the offsets/rotations/scaling. The EPSG online registry lists one transformation between Anguilla 1957 and WGS 1984. It's applied as arc-second shifts:

latitude: -18"  
longitude: +4.4"

This is from Anguilla 1957 to WGS 1984.

One place to find more information on transformation including the equations and worked examples is EPSG's Guidance Note 7-2.

Edit: Including Esri projection engine test values

From WGS84 to Anguilla 1957 using the EPSG:1447 transformation
Input: 18.0N 63.0W
Output: 18.005N 63.0012222222222W

From Anguilla 1957 to the ProjCRS EPSG:2000:
Output: 294014.7725 1990652.6476

If I use the original coordinates instead (no transformation):
Output: 294141.1785 1990098.7974


If I set up a 0,0,0 (same as +wgs84=0,0,0,0,0,0,0) transformation between Anguilla 1957 and WGS84:
Output: 18.00185163527N 63.0W -78.86 m
and this value to EPSG:2000:
Output: 294142.2835 1990303.6444

The first and last test cases differ because the first just changes the CRS info on the data--literally nothing happens. On the last, because we're converting to/from XYZ space, the differences in flattening and size between the two ellipsoids are handled. If you can try to picture it, one ellipsoid is within the other. Even though their semimajor radii are different, there's no way to model it, so the longitude is unchanged. The semiminor axes (or flattening) are also different but the differences it reflected in a different latitude and new ellipsoid height value.

The Y/Northing difference on the no-transformation example may be to different Clarke 1880 definitions. EPSG has Clarke 1880 (RGS). I don't remember which version PROJ4 uses for clrk80.

Disclosure: I'm on the subcommittee that maintains the EPSG registry.