You broke physics (actually geometry is probably more accurate) when you decided to just make up your own rules for a metric. ...

I'm not making up any rules. In cartesian coords we have 2 distance metrics:

- the pythogorean one (through the earth)

- arclength along a greatcircle.

We will be focusing on the second one as we want to measure distances along the globe.

Next we transform to celestial coords (latitude, longitude).

The distance metric also transforms. It now becomes the haversine formula.

How is this breaking physics or geometry? Isn't this the way any physicist handles coordinate transforms?

I've asked you for several posts now where in my coord transform i'm making any mistakes and i've still not received a reply.

So lets try again. Example will be to calculate the width of Australia.

1. start with a globe in cartesian coords. Distance formula is arclength along a greatcircle

2. convert to celestial coords. Distance formula becomes haversine. Width of Australia is still correct

3. Render latitude on a straight rather than a curvy axis. All coordinates and formulas stay the same. The width of Australia is still correct as mathematically nothing changed.

Please tell me what i'm doing wrong.

In differential geometry, distances are measured by a metric. Euclidean and non-Euclidean spaces use different metrics. They have to because the definition of “the shortest distance between two points is different”. For Euclidean space it is a straight line, for non-Euclidean it is a curve. For Euclidean space the metric is simply the Pythagorean theorem. In non-Euclidean space it is

Obviously, very different formulas

I don't agree pythagorean distances are the only distance metric. In our discussion we're more interested in arclength along a greatcircle distances.

But fine, let's go with straight distances through the earth.

When you do a coordinate transformation between two Euclidean spaces, the metric remains the same so lengths and angles are preserved. When you do a coordinate transformation from a Euclidean space to a non-Euclidean space (Cartesian on a flat disc to Celestial on a sphere, for example), you aren’t just transforming the coordinate system, you are also transforming the metric from Euclidean to non-Euclidean. You seem to have just arbitrarily transformed the metric by “updating the formula”. You can’t do that. The metric transforms when you do the coordinate transformation. You can’t just decide that you don’t like the way it transforms and invent your own metric.

I believe i'm doing the distance transform correct. Here's how i do it, please tell me where i go wrong:

Imagine you have coordinates (x,y,z) and a distance metric d(p1, p2). (say pythagorean)

Take a reversible coordinate transformation f(), and it's inverse f_1().

Then the transformed coordinates (x', y', z') are defined as f(x,y,z). Or inversely: (x,y,z) = f_1(x', y', z').

We define the distance metric d' as

d'(p1', p2') = d(f_1(p1'), f_1(p2'))

So to find the distance between 2 points in the transformed space, we go back to the original space, and calculate the distance there.

This by definition give the same distance between transformed and original space. Where am i going wrong?

These are not arbitrary formulas. This is the way you're supposed to do coord transforms.

But even when done correctly, because the bases of the two different metrics (the underlying geometry) are different, it will never transform exactly. Non-Euclidean metric defines distances in terms of angles, Euclidean doesn’t.

So haversine is wrong, and physics should immediately stop using celestial coordinates?