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Draw distance is a computer graphics term, defined as the distance in a 3 dimensional scene that is still drawn by the rendering engine. Polygons that lie behind the draw distance won't be drawn to the screen.
As the draw distance increases more polygons need to be drawn onto the screen which requires more computing power. This means the graphic quality and realism of the scene will increase as draw distance increases, but the overall performance (frames per second) will decrease. Many games and applications will allow users to manually set the draw distance to balance performance and visuals.
Older games had far shorter draw distances, most noticeable in vast, open scenes. Racing arcade games were particularly infamous, as the open highways and roads often led to "pop-up graphics" - an effect where distant objects suddenly appear without warning as the camera gets closer to it. This is a hallmark of poor draw distance, and still plagues large, open-ended games like the Grand Theft Auto series.
A common trick used in games to disguise a short draw distance is to obscure the area with a distance fog. Alternative methods have been developed to sidestep the problem altogether using level of detail manipulation. Black & White was one of the earlier games to use adaptive level of detail to decrease the number of polygons in objects as they moved away from the camera, allowing it to have a massive draw distance while maintaining detail in close-up views.
Grand Theft Auto 3 made particular use of fogging, however, this made the game less playable when driving or flying at high speed, as objects would pop-up out of the fog and cause the player to crash into them.
Halo 3 is claimed by its creators at Bungie Studios to have a draw distance upwards of 14 miles. This is an example of the vastly improved draw distances made capable by more recent game consoles.