[GRASSLIST:3246] Re: help with theory of map projections

[Glynn Clements]

chris2 wrote:

> I have been trying to learn the theory behind map projections from the
> book "Map Projections, A reference Manual" by Bugayevskiy and Snyder. 
> I am stuck on page 2 in Chapter 1, if you can believe it, where they
> take an ellipsoid of revolution, and a point on its surface at
> latitude phi, and mention that the radius of curvature of the meridian
> through the point is:
> 
> M= a*(1-e^2) / [(1-e^2*sin(phi)^2)^(3/2)]
> 
> Where e, a, and b are the eccentricity and semimajor and semiminor
> axes of the ellipse of revolution.
> 
> I set out to verify this formula as a way of reviewing my vector
> caclulus and getting started with the book.
> 
> 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.

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