According to this source, an optimal projection for small, roughly square areas is the stereographic projection. The wikipedia page of the stereographic projection shows how to convert from spherical coordinates to polar or xy coordinates. However, I would like to convert WGS84 data (latitude and longitude) to xy coordinates and WGS84 uses an ellipsoidal model of Earth instead of a sphere.

My questions is:

What are the formulae for a stereographic projection on an ellipsoid centred at lat0,lon0 (point tangent to the plane) projected from the antipode of lat0,lon0?

Bonus questions:

1) What is the inverse transformation?

2) Is this projection still conformal?

I have read section of the Geomatics Guidance Note 7, part 2 "Coordinate Conversions & Transformations including Formulas" (March 2019) and while I believe it may contain the answers to questions (1) and (3), I'm not sure what does it mean by "For the ellipsoid the parameters defining the conformal sphere at the tangent point as origin are first derived". Does it first compute the parameters for a sphere tangent to lat0,lon0? Then uses the antipode of lat0,lon0 in that sphere instead of the ellipsoid? If so, why does it do it that way? To preserve conformality? If this is true then would there be any advantage in projecting stereographically from the antipode on the ellipsoid instead of projecting from the antipode in the conformal sphere?

closed as too broad by nmtoken, Erik, J.R, PolyGeo Apr 24 at 8:53

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    Please refrain from asking multiple questions at once. It's both harder to answer, and later on to find the correct answer for one of your questions - especially for others seeking help on similar problems. – Erik Apr 24 at 8:41
  • There are two (at least) ellipsoidal stereographic implementations. One, used in the US, can be found in Snyder. It converts directly from ellipsoid to plane. The other, used more in Europe (and Canada), converts geodetic lat-lon to conformal versions on a sphere, then to a plane. Both versions are conformal but give different values. EPSG documents the second one. – mkennedy Apr 24 at 15:37
  • I edited the question @Erik. I hope it's ok now. I believe the question could be rephrased into just one question like "What are the formulae and properties for a stereographic projection on an ellipsoid centred at lat0,lon0 (point tangent to the plane) projected from the antipode of lat0,lon0?" and then go on to explain to what I refer by properties, but I prefer to leave it like this as I find it clearer. – Ricardo Apr 27 at 13:54
  • I don't understand why this question has been closed. The question is one and very specific. – Ricardo Apr 29 at 15:13