This website (Hawkeye's official website) explains it all:
I'll try and summarise:
Many cameras, positioned around the court, continually take 2D pictures of the court at set intervals. By knowing the position of all the cameras, this actually allows a computer to create a 3D model of the court, so you can know exactly where the ball is at any given time.
As the pictures are taken fast and regularly, this allows you to create a series of models of how the ball is travelling over time - so you're making a 4d model of the ball's motion.
Using this technique of monitoring tragectory, you can then analyse accurately where the ball will hit the group - cross reference that with the position of the lines on the court and you have an accurate way of establishing whether a ball was in or out.
According to the manufacturers, the system is successful 100% of the time:
During ITF testing in 2006 Hawk-Eye made the correct call in 100% of all tests, showing an average error of only 3.6mm. The system recorded 100% of all rallies.
Players can actually challenge as many times they like - if the umpire deems a challenge correct, this does not come off their 'Three challenges' tally, but if they are wrong, it does. And since the system is deemed to be so accurate, usually the umpire will go with the Hawkeye decision, which they will have 2-3 seconds after the ball has been played. I suspect this rule is to stop timewasting - even if the spectators are agreeing with the player that it was a bad call, it stops frivolous McEnroe-style tirades time and time again.
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