[GRASSLIST:3251] Re: help with theory of map projections

[Glynn Clements]

Eric G. Miller wrote:

> > > I used the following parameterization of the ellipse:
> > > 
> > > x = a*cos(t)
> > > y = b*sin(t)
> > > 
> > > My t, by the way, is equivalent to their phi, I am pretty sure. That
> > > is, t is the angle between a normal line to the ellipse through the
> > > given point, and the x-axis.
> > 
> > I don't think so. Differentiating gives:
> > 
> > dx/dt	= -a*sin(t)
> > dy/dt	=  b*cos(t)
> > 
> > => dy/dx	= -(b/a).cot(t)		[cot(x) = 1/tan(x) = cos(x)/sin(x)]
> > 
> > => normal grad.	= (a/b).tan(t)		[normal grad. * tangent grad. = -1]
> > 
> > Clearly, the normal gradient is tan(phi), by the definition of phi.
> > 
> > If it's any consolation, I also got stuck right at the beginning of
> > that book. My conclusion is that geodetic longitude sucks from a
> > coordinate geometry perspective.

Sorry, I meant "geodetic latitude".

> AFAIK, it's based on gravity potential.  You might be interested in
> looking at the bibliography in NIMA TR8350.2 "World Geodetic System of
> 1984".  It gives a reference to:

I probably should have said "geographic"; I was using the term in the
context of Chapter 1 of Bugayevskiy and Snyder, which approximates the
geoid to an ellipsoid, so geodetic and geographic latitudes are
equivalent.

The point is that the definition of latitude as the angle between the
surface normal (for some surface, be it the geoid or an ellipsoid) and
the equatorial plane isn't particularly convenient from the
perspective of coordinate geometry, where it would be simpler to deal
with the angle between a radial and the equatorial plane (but this
doesn't correspond to Chris' "t" parameter either).

BTW, I can understand why latitude is defined the way it is, I'm just
saying that it makes the equations more complex.

-- 
Glynn Clements <glynn.clements at virgin.net>