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Homogeneous coordinates are ubiquitous in computer graphics because they allow common vector operations such as translation, rotation, scaling and perspective projection to be represented as a matrix by which the vector is multiplied. By the chain rule, any sequence of such operations can be multiplied out into a single matrix, allowing simple ...
The camera matrix derived in the previous section has a null space which is spanned by the vector = This is also the homogeneous representation of the 3D point which has coordinates (0,0,0), that is, the "camera center" (aka the entrance pupil; the position of the pinhole of a pinhole camera) is at O.
As with reflections, the orthogonal projection onto a line that does not pass through the origin is an affine, not linear, transformation. Parallel projections are also linear transformations and can be represented simply by a matrix. However, perspective projections are not, and to represent these with a matrix, homogeneous coordinates can be ...
This may be written in terms of homogeneous coordinates in the following way: A homography φ may be defined by a nonsingular (n+1) × (n+1) matrix [a i,j], called the matrix of the homography. This matrix is defined up to the multiplication by a nonzero element of K.
A matrix, has its column space depicted as the green line. The projection of some vector onto the column space of is the vector . From the figure, it is clear that the closest point from the vector onto the column space of , is , and is one where we can draw a line orthogonal to the column space of .
The camera projection matrix is derived from the intrinsic and extrinsic parameters of the camera, and is often represented by the series of transformations; e.g., a matrix of camera intrinsic parameters, a 3 × 3 rotation matrix, and a translation vector. The camera projection matrix can be used to associate points in a camera's image space ...
Switching to homogeneous coordinates using the embedding (a, b) ↦ (a, b, 1), the extension to the real projective plane is obtained by permitting the last coordinate to be 0. Recalling that point coordinates are written as column vectors and line coordinates as row vectors, we may express this polarity by:
The coordinates in the higher-dimensional space are an example of homogeneous coordinates. If the original space is Euclidean , the higher dimensional space is a real projective space . The advantage of using homogeneous coordinates is that one can combine any number of affine transformations into one by multiplying the respective matrices.