Homogeneous Coordinates and Matrix Representation
Homogeneous Coordinates and Matrix Representation
Homogeneous Coordinates
Homogenous coordinates utilize a mathematical trick to embed three-dimensional coordinates and transformations into a four-dimensional matrix format. As a result, inversions or combinations of linear transformations are simplified to inversion or multiplication of the corresponding matrices. Homogenous coordinates also make it possible to define perspective transformations. Homogenous coordinates allow each point (x, y, z) to be represented by any of an infinite number of four dimensional vectors:
The three-dimensional vector corresponding to any four-dimensional vector can be computed by dividing the first three elements by the fourth, and a four-dimensional vector corresponding to any three-dimensional vector can be created by simply adding a fourth element and setting it equal to one.
* Coordinate System
We need a coordinate system to describe an image, the coordinate system used to place elements in relation to each other is called user space, since this is the coordinates the user uses to define elements and position them in relation to each other.
The coordinate system has the origin in the upper left, with the x axis extending to the right and y axis extending downwards.
Matrix Representation
Matrices allow arbitrary linear transformations to be represented in a consistent format, suitable for computation. This also allows transformations to be concatenated easily (by multiplying their matrices).
Linear transformations are not the only ones that can be represented by matrices. Some transformations that are non-linear on a n-dimensional Euclidean space Rn, can be represented as linear transformations on the n+1-dimensional space Rn+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. These n+1-dimensional transformation
Homogeneous Coordinates
Homogenous coordinates utilize a mathematical trick to embed three-dimensional coordinates and transformations into a four-dimensional matrix format. As a result, inversions or combinations of linear transformations are simplified to inversion or multiplication of the corresponding matrices. Homogenous coordinates also make it possible to define perspective transformations. Homogenous coordinates allow each point (x, y, z) to be represented by any of an infinite number of four dimensional vectors:
The three-dimensional vector corresponding to any four-dimensional vector can be computed by dividing the first three elements by the fourth, and a four-dimensional vector corresponding to any three-dimensional vector can be created by simply adding a fourth element and setting it equal to one.
* Coordinate System
We need a coordinate system to describe an image, the coordinate system used to place elements in relation to each other is called user space, since this is the coordinates the user uses to define elements and position them in relation to each other.
The coordinate system has the origin in the upper left, with the x axis extending to the right and y axis extending downwards.
Matrix Representation
Matrices allow arbitrary linear transformations to be represented in a consistent format, suitable for computation. This also allows transformations to be concatenated easily (by multiplying their matrices).
Linear transformations are not the only ones that can be represented by matrices. Some transformations that are non-linear on a n-dimensional Euclidean space Rn, can be represented as linear transformations on the n+1-dimensional space Rn+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. These n+1-dimensional transformation