radii.c

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/*!
   \file lib/gis/radii.c
 
   \brief GIS Library - Calculating the Meridional Radius of Curvature
 
   \todo Suggestion: all "lon"s in the file "radii.c" should read as "lat"
 
   Comments:
   on page http://www.mentorsoftwareinc.com/cc/gistips/TIPS0899.HTM
   down where it says "Meridional Radius of Curvature" is the exact formula
   out of "radii.c".
   Quote: "essentially, the radius of curvature, at a specific latitude ...".
 
   See also http://williams.best.vwh.net/ellipsoid/node1.html which has a nice
   picture showning the parametric latitude and phi, the geodetic latitude.
   On the next page,
   http://williams.best.vwh.net/ellipsoid/node2.html, in equation 3, the
   Meridional Radius of Curvature shows up.
 
   So, it looks like you are calculating the Meridional Radius of Curvature
   as a function of GEODETIC LATITUDE.
 
   Various formulas for the ellipsoid.
   Reference: Map Projections by Peter Richardus and Ron K. Alder
   University of Illinois Library Call Number: 526.8 R39m
   Parameters are:
   - lon = longitude of the meridian
   - a   = ellipsoid semi-major axis
   - e2  = ellipsoid eccentricity squared
 
 
   meridional radius of curvature (p. 16)
   \verbatim
   2
   a ( 1 - e )
   M = ------------------
   2   2    3/2
   (1 - e sin lon)
   \endverbatim
   transverse radius of curvature (p. 16)
   \verbatim
   a
   N = ------------------
   2   2    1/2
   (1 - e sin lon)
   \endverbatim
   radius of the tangent sphere onto which angles are mapped
   conformally (p. 24)
   \verbatim
   R = sqrt ( N * M )
   \endverbatim
 
   (C) 2001-2009 by the GRASS Development Team
 
   This program is free software under the GNU General Public License
   (>=v2).  Read the file COPYING that comes with GRASS for details.
 
   \author CERL
 */
 
#include <math.h>
#include <grass/gis.h>
#include "pi.h"
 
/*!
 * \brief Meridional radius of curvature
 *
 * Returns the meridional radius of curvature at a given longitude:
 *
 \f$
 \rho = \frac{a (1-e^2)}{(1-e^2\sin^2 lon)^{3/2}}
 \f$
 *
 *  \param lon longitude
 *  \param a ellipsoid semi-major axis
 *  \param e2 ellipsoid eccentricity squared
 *
 *  \return radius value
 */
double G_meridional_radius_of_curvature(double lon, double a, double e2)
{
    double x;
    double s;
 
    s = sin(Radians(lon));
    x = 1 - e2 * s * s;
 
    return a * (1 - e2) / (x * sqrt(x));
}
 
/*!
 * \brief Transverse radius of curvature
 *
 * Returns the transverse radius of curvature at a given longitude:
 *
 \f$
 \nu = \frac{a}{(1-e^2\sin^2 lon)^{1/2}}
 \f$
 *
 *  \param lon longitude
 *  \param a ellipsoid semi-major axis
 *  \param e2 ellipsoid eccentricity squared
 *
 *  \return radius value
 */
double G_transverse_radius_of_curvature(double lon, double a, double e2)
{
    double x;
    double s;
 
    s = sin(Radians(lon));
    x = 1 - e2 * s * s;
 
    return a / sqrt(x);
}
 
/*!
 * \brief Radius of conformal tangent sphere
 *
 * Returns the radius of the conformal sphere tangent to ellipsoid at
 * a given longitude:
 *
 \f$
 r = \frac{a (1-e^2)^{1/2}}{(1-e^2\sin^2 lon)}
 \f$
 *
 *  \param lon longitude
 *  \param a ellipsoid semi-major axis
 *  \param e2 ellipsoid eccentricity squared
 *
 *  \return radius value
 */
double G_radius_of_conformal_tangent_sphere(double lon, double a, double e2)
{
    double x;
    double s;
 
    s = sin(Radians(lon));
    x = 1 - e2 * s * s;
 
    return a * sqrt(1 - e2) / x;
}