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Something is called planar if it is made up of flat planess, or pertaining to planes. A graph is called planar if it can be drawn in the plane without any edge intersections; see planar graph.

In the context of computer graphics, Planar is method of representing pixel information with several bitplanes. Each individual bit in a bitplane is related to a single pixel on the screen. Unlike Chunky, Highcolour or Truecolour graphics, the data for an individual pixel isn't in one specific location in RAM, but spread across the bitplanes that make up the display.

For example, on a Chunky display, each byte will represent one pixel. So, if colour zero is black, colour one is blue, and colour two is green, a byte of chunky pixel data would look like this:

00000000 = Black pixel
00000001 = Blue pixel
00000010 = Green pixel

Whereas planar data would look like this

Plane 0, Byte 0: 00000000 = 8 black pixels
Plane 1, Byte 0: 00000000

Plane 0, Byte 0: 10000000 = 6 black pixels, one blue pixel, one green pixel
Plane 1, Byte 0: 00010000

Planar graphics were used a lot in the 80s and early 90s because displays tended to only be able to show less than 256 colours. For chunky displays, there is always 1 byte per pixel regardless of the colour depth of the display. This uses a lot of precious RAM. If you only need to display 8 colours, you can use 3 bitplanes, and each pixel only has 3 bits assigned to it instead of 8.

Planar graphics are also easier to scroll than chunky graphics in hardware, which is why systems like the Commodore Amiga were able to move large playfields around with little effort.

Note: Perhaps some drawn images would better show how planar graphics are stored in ram