In computer graphics we manipulate objects which may include light sources, cameras, and models in our scenes. Each of these is likely to be defined in its own coordinate system and then placed within the scene we are modeling. We must be able to relate these coordinate systems, both to a global coordinate system and to each other. We place coordinate systems into an affine space through the use of frames.
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Definition of a Frame
Let be an affine space of dimension . Let be a point in this space and let be any basis for . We call the collection a frame for . Frames form coordinate systems in our affine space: If we are given a point , then it can be written as , where is the origin of the frame and is a vector. Since forms a basis for , the vector can be written uniquely as
Examples of Frames
Two-dimensional examples of frames are fairly straightforward to produce. These frames contain two vectors and a point representing the origin of the frame. The vectors form a basis for the vector space of all two-dimensional vectors.
Matrix representation of Points and Vectors
Points and vectors can be uniquely identified by the coordinates relative to a specific frame. Given a frame in an affine space , we can write a point uniquely as
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