The Skew Tangent Space

Sphere

Define a Spherical Coordinate System. For a given point take:

$r =$ radius
$\theta =$ polar angle (angle from the north pole)
$\phi =$ azimuth angle from the prime meridian, which we will take as the great circle through the north pole and the geomagnetic pole at a distance $\gamma$ from the north pole.

In our example we will let $r=1$ .

The lines of longitude ( $\theta$ is constant) are great circles through the poles
The lines of latitude ( $\phi$ is constant) are at different angles to the longitudes.

At each pont you have basis vectors of the tangent space: $\{e^1,e^2\}$ .

For our sphere, the basis vectors are not in general perpendicular, so this is a non-cartesian coordinate system.

Sphere '

$e^1$ is a tangent along a great circle through the magnetic pole, so take that as our unit length a in the cartesian case.

$e^2$ is the same as our cartesian case, so we can take it as a unit vector in the x / horizontal direction of magnitude $\sin\theta$ .

We need the angle between $e^1$ and $e^2$ . This is an exercise in spherical trigonometry:

Given

Let

Let's figure out some of the angles here:

Both arc $PA$ and arc $PMB$ are arcs of great circles through the pole $P$ , so they intersect the longitude arc $AB$ at right angles.
arc $AM$ and $PMB$ are arcs of great circles, so $\angle PMA + \angle AMB = \pi$

And now we can apply the spherical law of sines and some basic trig

\frac{\sin(\pi/2-\alpha)}{\gamma} = \frac{\sin\pi-\phi}{\theta}

\frac{\cos\alpha}{\gamma} = \frac{\sin\phi}{\theta}

\cos\alpha= \frac{\gamma}{\theta}\sin\phi

Which is the angle we seek.

So in summary we have

Or formulating via the dot product

\frac{e^1 \cdot e^2}{|e^1||e^2|} = \cos\alpha= \frac{\gamma}{\theta}\sin\phi

e^1 \cdot e^2=|e^1||e^2|\cos\alpha= \frac{\gamma}{\theta} \sin\phi \sin\theta

Next Up

Next up will be "The Riemannian Metric"