# [Proj] Jacobi + Guyou projections; conformal map of ellipsoid to sphere

Charles Karney charles.karney at sri.com
Thu Jan 1 18:12:33 EST 2015

```Other than Nyrstov https://dx.doi.org/10.1007/978-3-642-32618-9_17 ,
there's

http://geocnt.geonet.ru/en/3_axial
J. P. Snyder, Survey Review 38(217), 130-148 (1985)
and references therein

but Snyder's method seems to be much more complicated that Jacobi's
projection (converted into elliptic integrals).  I'm not aware of
any work talking about mapping a trixial ellipsoid to a sphere.
However, once you've mapped the ellipsoid to a rectangle, the
mapping to a sphere is straightforward.

BTW, what does "equivalence" mean in this context?

--Charles

On 01/01/2015 04:38 PM, strebe at aol.com wrote:
> Hello Charles. Good stuff; thanks.
>
> Do you know of any literature around triaxial-to-spherical mappings in
> general (including other techniques to preserve conformality, as well as
> ones to preserve other traits such as equivalence)?
>
> — daan
>
> -----Original Message-----
> From: Charles Karney <charles.karney at sri.com>
> To: proj <proj at lists.maptools.org>
> Sent: Thu, Jan 1, 2015 1:20 pm
> Subject: [Proj] Jacobi + Guyou projections; conformal map of ellipsoid
> to sphere
>
> I've updated my notes on Jacobi's conformal projection of a triaxial
> ellipsoid; see
>
>     http://geographiclib.sourceforge.net/1.41/jacobi.html
>
> New stuff:
>
> * the limits, ellipsoid of revolution and sphere, are easily obtained;
>
> * the Guyou projection (and hence the Peirce quincuncial projection) are
>     special cases of Jacobi's projection (which predates both of these);
>
> * Jacobi + Guyou can be used to map a triaxial ellipsoid conformally
>     onto a sphere.
>
>     --Charles
>
\
```