4×4Homogenitätsmatrix

The 4x4 Homogenitätsmatrix, also known as the 4x4 homogeneous transformation matrix, is a fundamental concept in computer graphics, robotics, and computer vision. It is used to represent the transformation of points and vectors in a three-dimensional space, including translation, rotation, and scaling.

The 4x4 homogeneous transformation matrix is a 4x4 matrix that can be used to represent a combination of translation, rotation, and scaling in a single matrix. It is an extension of the 3x3 rotation matrix and the 3x1 translation vector, which are used to represent rotation and translation separately.

The 4x4 homogeneous transformation matrix is defined as follows:

[ R t ]

[ 0 1 ]

where R is a 3x3 rotation matrix, t is a 3x1 translation vector, and the bottom row is a 1x4 vector with the last element being 1 and the rest being 0.

The 4x4 homogeneous transformation matrix can be used to transform a point or vector in a three-dimensional space by multiplying it with the matrix. The resulting point or vector will be transformed according to the rotation, translation, and scaling represented by the matrix.

The 4x4 homogeneous transformation matrix is widely used in computer graphics, robotics, and computer vision, where it is used to represent the transformation of objects, cameras, and other entities in a three-dimensional space. It is also used in the calculation of the perspective projection matrix, which is used to project a three-dimensional scene onto a two-dimensional screen.