Translasjonsmatriser

Translasjonsmatriser, or translation matrices, are mathematical tools used in linear algebra to represent translation operations in space. A translation is a geometric transformation that moves every point of a figure or space by the same distance in the same direction. In a 2D plane, a translation can be represented by adding a vector (tx, ty) to each point (x, y), resulting in a new point (x+tx, y+ty).

To represent this translation using matrices, we often employ homogeneous coordinates. In homogeneous coordinates, a 2D point (x, y) is represented as a 3D vector [x, y, 1]T. A translation by (tx, ty) can then be achieved by multiplying this vector by a 3x3 translation matrix. This matrix has the form:

[[1, 0, tx],

[0, 1, ty],

[0, 0, 1]]

When this matrix is multiplied by the homogeneous coordinate vector [x, y, 1]T, the result is [x + tx, y + ty, 1]T, which corresponds to the translated point (x+tx, y+ty) in homogeneous coordinates.

Similarly, in 3D space, a translation by (tx, ty, tz) can be represented by a 4x4 translation matrix operating on 4D homogeneous coordinates [x, y, z, 1]T:

[[1, 0, 0, tx],

[0, 1, 0, ty],

[0, 0, 1, tz],

[0, 0, 0, 1]]

Translasjonsmatriser are fundamental in computer graphics, robotics, and various engineering fields for manipulating objects in virtual or physical spaces. They are often combined with other transformation matrices, such as rotation and scaling matrices, to perform complex geometric operations.