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Some transformations that are non-linear on an n-dimensional Euclidean space R n can be represented as linear transformations on the n+1-dimensional space R n+1. These include both affine transformations (such as translation) and projective transformations. For this reason, 4×4 transformation matrices are widely used in 3D computer graphics.
Rotation matrices provide an algebraic description of such rotations, and are used extensively for computations in geometry, physics, and computer graphics. In some literature, the term rotation is generalized to include improper rotations, characterized by orthogonal matrices with a determinant of −1 (instead of +1).
In plane geometry, a shear mapping is an affine transformation that displaces each point in a fixed direction by an amount proportional to its signed distance from a given line parallel to that direction. [1] This type of mapping is also called shear transformation, transvection, or just shearing.
A specific application of linear maps is for geometric transformations, such as those performed in computer graphics, where the translation, rotation and scaling of 2D or 3D objects is performed by the use of a transformation matrix. Linear mappings also are used as a mechanism for describing change: for example in calculus correspond to ...
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 ...
Using basic linear algebra that intersection point can be determined in a straightforward way. The image to the right shows the real case. The position of the image points y 1 {\displaystyle \mathbf {y} _{1}} and y 2 {\displaystyle \mathbf {y} _{2}} cannot be measured exactly.
Mathematically, planar projections are linear transformations acting on a point in three-dimensional space ,, to give a point , on the projection plane. These transformations consist of various compositions of the five transformations: orthographic projection , rotation , shear , translation and perspective .
3D visualization of a sphere and a rotation about an Euler axis (^) by an angle of In 3-dimensional space, according to Euler's rotation theorem, any rotation or sequence of rotations of a rigid body or coordinate system about a fixed point is equivalent to a single rotation by a given angle about a fixed axis (called the Euler axis) that runs through the fixed point. [6]