[Proj] Re: libproj4 stmerc = French Gauss-Laborde projection
Strebe at aol.com
Strebe at aol.com
Thu Jun 15 05:05:13 EDT 2006
I'm not lost. I've analyzed the method and programmed it. It works fine.
Yes, p is the parametric co-latitude in the first formula. It's wrong to use
the same variable in the later formulae because p refers to something else
there -- in fact, it's a complex variable in the later instances.
The "trick" is this: Wallis uses the polar stereographic because it's the
simplest way to get a conformal mapping to the plane. Once the ellipsoid is
mapped, he treats the plane as the complex plane and looks for a complex
"co-latitude" which can be used with the polar stereographic, but this time treating the
polar stereographic as function of a complex variable. The reason he does
this is (a) to preserve conformality; and (b) so that the central meridian (in
which the imaginary axis is 0) maps back to the parametric colatitude. At this
point the ellipsoid is mapped conformally in such a way that leaves the central
meridian effectively unmapped.
Leaving the complex plane aside, using the colatitude as the parameter to the
elliptic integral of the second kind gives the distance from the pole to the
colatitude. Since this odd mapping Wallis contrived effectively leaves the
central meridian unmapped, and since any analytic function applied to a conformal
mapping results in a conformal mapping, and since the elliptic integral has
an analytic form, all that is left is to push the mapping through the complex
form of the elliptic integral of the second kind. This "straightens out" the
central meridian to its true differential lengths whilst dragging the whole
complex plane with it in a conformal fashion. The result must be the transverse
Mercator, since the central meridian is projected with correct scale and since a
conformal projection is unique except with respect to scale and rotation.
-- daan Strebe
In a message dated 6/15/06 01:18:47, martin.vermeer at hut.fi writes:
> On Wed, 2006-06-14 at 13:50 -0400, strebe at aol.com wrote:
> > Hm. I didn't know about that web page. Obviously it's wrong -- for some
> > reason "p" appears in several different roles. I tend to think that's
> > an error in conversion to a web page. (I see that the entire blurb is a
> > single graphic, not HTML mark-up.) Certainly he's been pedantic and
> > precise in all his communications with me.
> > The p/2 exponent should read (e/2), where e is the eccentricity.
> Yes, I agree.
> > Use some other variable (perhaps p') in place of p in "Then, the
> > complex variable tan (p/2) can be obtained..." and "...yields the
> > argument p..."
> Actually the argument p is simply the (ellipsoidal) co-latitude
> 90d - phi.
> The common expression in u and v corresponds to exp(psi), where psi is
> the _isometric latitude_, i.e., essentially the "northing" in a
> traditional (non-transverse) Mercator map plane.
> Isometric latitude and longitude (psi, lambda) together as (x,y)
> co-ordinates in a plane define a conformal mapping from the curved
> Earth's surface. Using (psi, lambda) directly as rectangular
> co-ordinates produces classical Mercator. Using
> u + iv = exp(psi + i * lambda)
> i.e., polar co-ordinates, produces the stereographic projection. This is
> very much what Dr Wallis's formula looks like. Apparently for him it is
> only a trick leading somewhere... but then I also get lost.
> Regards Martin V
> PS you may want to look at
> pp 99-100 and around p. 90. Sorry it's in Fenno-ugrian formulese...
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