I need to project data which can be nearby the poles, and I want to use a custom transverse mercator projection for a number of reasons.

Using GeographicLib (either TransverseMercator or TransverseMercatorExact), I see I can only set a projection origin for longitude, but not for latitude. However, what kind of projection accuracy can I expect at the poles?

 * Forward projection, from geographic to transverse Mercator.
 * @param[in] lon0 central meridian of the projection (degrees).
 * @param[in] lat latitude of point (degrees).
 * @param[in] lon longitude of point (degrees).
 * @param[out] x easting of point (meters).
 * @param[out] y northing of point (meters).
 * @param[out] gamma meridian convergence at point (degrees).
 * @param[out] k scale of projection at point.
 * No false easting or northing is added. \e lat should be in the range
 * [−90°, 90°].
void Forward(real lon0, real lat, real lon,
             real& x, real& y, real& gamma, real& k) const;

If Earth were a sphere, you would get the same deformation at the poles as at the equator.

When projecting from a flattened at the poles ellipsoid, there is less deformation at the poles than at the equator, because its radius of curvature is greater.

There is no point in defining a latitude origin in a Transverse Mercator projection, because its relationship to the reference object is the same throughout the entire central meridian.

It is not a simple or trivial thing to see, because the form that meridians and parallels take changes with latitude. But it is the meridians that converge at the poles and the parallels that shrink that are different, not the projection. If you forgot the meridians and parallels drawing and only thought about land shapes and distances, there are no latitude modifications (except the radius of curvature difference) in the Transverse Mercator representation.

It deforms to infinite at the equator, 90 degrees to East and West from the central meridian.

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