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A reflection about a line or plane that does not go through the origin is not a linear transformation — it is an affine transformation — as a 4×4 affine transformation matrix, it can be expressed as follows (assuming the normal is a unit vector): [′ ′ ′] = [] [] where = for some point on the plane, or equivalently, + + + =.
Let X be an affine space over a field k, and V be its associated vector space. An affine transformation is a bijection f from X onto itself that is an affine map; this means that a linear map g from V to V is well defined by the equation () = (); here, as usual, the subtraction of two points denotes the free vector from the second point to the first one, and "well-defined" means that ...
In this case in all formulas below all arguments in θ should have sine and cosine exchanged, and as derivative also a plus and minus exchanged. All divisions by zero result in special cases of being directions along one of the main axes and are in practice most easily solved by observation.
Noting that any identity matrix is a rotation matrix, and that matrix multiplication is associative, we may summarize all these properties by saying that the n × n rotation matrices form a group, which for n > 2 is non-abelian, called a special orthogonal group, and denoted by SO(n), SO(n,R), SO n, or SO n (R), the group of n × n rotation ...
Formulas involving homogeneous coordinates are often simpler and more symmetric than their Cartesian counterparts. Homogeneous coordinates have a range of applications, including computer graphics and 3D computer vision , where they allow affine transformations and, in general, projective transformations to be easily represented by a matrix .
Affine involutions can be categorized by the dimension of the affine space of fixed points; this corresponds to the number of values 1 on the diagonal of the similar matrix D (see above), i.e., the dimension of the eigenspace for eigenvalue 1. The affine involutions in 3D are: the identity; the oblique reflection in respect to a plane
An indirect isometry is an affine transformation with an orthogonal matrix that has a determinant of −1. A proper rotation is an ordinary rotation. In the wider sense, a proper rotation is defined as a direct isometry; i.e., an element of E + (3): it can also be the identity, a rotation with a translation along the axis, or a pure translation ...
Point set registration is the process of aligning two point sets. Here, the blue fish is being registered to the red fish. In computer vision, pattern recognition, and robotics, point-set registration, also known as point-cloud registration or scan matching, is the process of finding a spatial transformation (e.g., scaling, rotation and translation) that aligns two point clouds.