It has been observed for centuries and reported in many books by Muslims around the world that two times a year the sun comes overhead above Ka'bah. This is observational fact for centuries, and is used to set the correct Qibla direction in places far from Makkah by Muslims for last so many centuries. Those two dates and times are:

When you observe the sun at these times (after converting it to your local time), you will be facing the Ka'bah giving you Qibla direction, because if there were a very high minaret over Ka'bah reaching up to the sky, then you will see it just like you are seeing the sun. Now, let us take a few examples. If you are in Islamabad, Pakistan (+5 hours time difference from Greenwich), the local time to observe the sun would be 2:18pm on May 28, and 2:27pm on July 16.

If you are at a location that you cannot see the sun on the above mentioned two dates, then you can locate Qibla from the sun when it comes overhead at a point diametrically opposite of Makkah on the globe and look for the following two dates and times:

November 28 at 21:09UT
January 16 at 21:29UT

Face toward the shadow from the sun at these times (after converting it to local time) and you will be facing Ka'bah. If you can see the sun but cannot see the shadow, put your back towards the sun and your face will be towards Qibla.

Qibla direction from the sun: Every day, at a certain time the Qibla can be determined from the sun, either by facing it or by facing the shadow from it, or else there is a time when Qibla would be 90 degress left or right of the sun. This time can be calculated for everyday for any desired location. This is the most accurate method. Remember, the compass, cannot tell you the True North, because its needle points to Magnetic North, and there is no way to know exactly how many degrees it is away from Ture North. Estimated values of the difference between True North and Magnetic North can be used but they can have errors, because magnetic field is not only changing continuously, but also can be erratic due to other physical changes present inside the earth. Moreover, the presence of other magnetic fields or metallic objects can deflect compass needle pointing in the wrong direction.

 In the last 1000 years, several Muslim mathematicians and In the last 1000 years, several Muslim mathematicians and astronomers, such as Al-Biruni, have discussed the correct way of determining the qibla direction from any point on the Earth's surface. All of them agree that at the two moments in each year when the Sun is directly overhead the Kaaba, the direction of shadows in any sunlit place will point directly away from the qibla. This happens on May 28 at 9:18 GMT and on July 16 at 9:27 GMT.

Of course, at any time, only half of the globe is sunlit. Fortunately, there are two moments in each year when the Sun is directly over the antipodes of the Kaaba. At that time, the direction of shadows in any sunlit place will point directly towards the qibla. This happens on November 28 at 21:09 GMT and January 16 at 21:29 GMT.

Because the Earth is almost a sphere, this is almost the same as saying that the qibla from a place is the direction in which a bird would start flying in order to get to the Kaaba by the shortest possible way.

In recent years, some Muslims from North America have argued that the traditional rule is nonsensical, because it leads to apparently absurd results. A Muslim praying from Alaska, they say, should pray almost due North if determining the qibla according to the traditional rule, because a plane going directly from Alaska to Makkah would take that route. Now, anybody who sees a Mercator projection map of the world can see plainly that Mecca is not North of Alaska, but rather to the southeast (or the southwest, depending on the map used). Therefore the "shortest travel distance" rule traditionally used in the old world makes no sense for North Americans.

These Muslims claim that the only correct way of determining the true qibla from a place is drawing a line from that place to Mecca in a Mercator projection map and that the fact that this line is not the shortest possible line has no relevance whatsoever to the matter.

The Muslim communities in North America are currently divided by the dispute: some of them pray to the Northeast according to the traditional doctrine, and some to the Southeast, according to the Mercator doctrine.

This dispute is far from trivial, because the fact that all Muslims pray towards the Kaaba is traditionally considered to be symbol of the unity of all Muslims worldwide under the law of God.

Most qibla calculating programs just use the traditional method and don't even mention the Mercator map method.