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Direct3D and OpenGL

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Direct3D and OpenGL are two technologies of rendering 3D graphics on modern GPUs. Often incorrectly compared as DirectX and OpenGL; DirectX includes may functions for reading input, for sound, etc, which isn't in OpenGL. The direct competitor for OpenGL is Direct3D.

Direct3D is limited to Microsoft Windows platform, while OpenGL can be run on many.

Direct3D uses row-major notation, while OpenGL uses column major.

Direct3D left-multiplies matrices, OpenGL right-multiplies.

As for the speed and extension support - there aren't that man differences. Mostly the choice comes from whether the user is planning to use platforms other than Windows.

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