In which direction do Muslims face when they pray?

Muslims direct their bodies and prayers towards the KAABA in Mecca, Saudi Arabia, but unlike the Jews who generally use cardinal directions to pray towards Jerusalem, the Muslims try to position themselves at the exact angle to best look towards Mecca.

This need to face at an exact angle has resulted in some theological disputes as to whether Muslims should look in the direction that forms the straightest line to Mecca on a Mercator projection or if they should look in the direction that would have the shortest distance over the planet's surface. This issue arises prominently in places that are very distant from Mecca in the northern hemisphere (like New York City or Tokyo). The straightest Mercator line would be to look southeast (from New York) or southwest (from Tokyo). However, because the Earth is curved, the shortest distance between the cities would be to look northeast (from New York) and northwest (from Tokyo).