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Lu Abel wrote:

> I'm not sure what is meant by the "geodesic," but as a curiosity question,
> does anyone know if when a Muslim determines the direction to Mecca is it
> the rhumb line or the great circle??  I asked while on a tour of a mosque,
> but the guide didn't understand the difference.

As was pointed out earlier by Tony, the geodesic line is the shortest
line connecting two points on a given surface. It has the property that
at each point along the line, the radius of curvature (i.e. the radius
of a circle that locally best fits an infinitesimal segment of the line)
is perpendicular to the surface.

On a sphere the geodesic line becomes a great circle and Islamic
astronomers and geographers have always calculated the direction to
Mecca (this is known as the qibla) by determining the great circle
connecting a given locality with Mecca.

Unlike the rhumb line therefore, the azimuths or compass bearings at the
begin and end points can be quite different. I.e. a North-American
Muslim generally prays to the North-East while his prayer actually
'arrives' at Mecca in a South-Easterly direction.

On a axially-symmetric spheroid, which closely approximates the actual
Earth,
the calculation of the Qibla direction becomes more complicated but is
still possible with the help of the so-called Bessel-Helmert equations
which I will not try to reproduce here, but which can be found in the
standard literature on geodesy (i.e. G. Bomford, Geodesy, etc.).
According to the TEX article mentioned in Dan Allen's message, the
difference in qibla direction as measured along a great circle or a
geodesic line can amount up to a few tenths of a degree which does not
seem to be of a magnitude that modern Muslims should worry about to
much.

In a recent personal message, George Huxtable sketched to me the
intriguing image of a Muslim society centred on the anti-Meacca point
(which my not too recent and not too detailed World Atlas places just
North of the island of Mururoa in French Society Islands), where its
inhabitants all correctly direct their daily prayers to different
directions of the compass.

Yesterday I found a Windows program called QiblaCalc that calculates the
qibla direction for any place on Earth, with respect to true North or
magnetic North, at:

  http://www.ummah.net/software/qibla10.zip

The author of this program assumes the Ka'ba of Mecca to be located at:

  Lat = 21� 25.3' North, long. = 39� 49.5' East

List members interested in how Islamic astronomers and geographers
solved the problem of determining the qibla direction in the past with
the aid of tables and instruments (all of course assuming a spherical
Earth) can find all that they ever wanted to know on this subject in:

  King, D.A., World-Maps for Finding the Direction and
  Distance to Mecca: Innovation and Tradition in Islamic
  Science (E.J. Brill, Leiden, 1999).

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