December 17th marked the beginning of the hajj, the annual pilgrimage to Mecca. The hajj is the fifth pillar of Islam, and is expected at least once in the lifetime of every Muslim who can afford to undertake the journey. (This year an estimated 2 million people will take part in the hajj.)
One of the rituals of the hajj involves walking seven times counterclockwise around the Kaaba, a cubical stone building draped in black cloth, regarded by Muslims as the holiest place on Earth.
Islam calls for the faithful to pray at five specified times each day; prayers are said while facing the Kaaba in Mecca. The appropriate direction for prayer is known as the qibla; Islamic interest in spherical trigonometry throughout the medieval era was driven in part by the desire to refine methods of determining the qibla for any given location.
David King’s World-maps for Finding the Direction and Distance to Mecca has an extensive discussion of the history of the problem, including mathematical methods introduced in its approximate and exact solution over the millenia.
According to the wikipedia entry on the qibla, within the North American Islamic communities, there is some controversy over how the qibla should be determined: some hold that the traditional great-circle route is the appropriate direction; others argue for a straight-line path on a Mercator projection as the correct qibla. As an issue of doctrine, apparently this is an issue of ongoing controversy.
The Qibla Locator can determine the distance and direction from any location on the Earth to Mecca. (It gives the traditional great-circle route.)