Transformation (function)
Wherein a coordinate transformation coordinates from one coordinate system to be transferred to another. Formally speaking, this is the transition from the original coordinates to the new coordinates. The most common applications are in the geometry, geodesy, photogrammetry and technical tasks.
Typical coordinate transformations caused by rotation (rotation), scaling ( changing the scale ), and shear displacement (translation ) of the coordinate system, which can also be combined.
The new coordinates can be arbitrary functions of the old coordinates. In general, using special transformations, in which these functions to certain restrictions - eg Differentiability, linearity or form loyalty - are subject. Coordinate transformations can be applied, can be solved more easily if a different coordinate system, a problem, such as in the transformation of cartesian coordinates into spherical coordinates and vice versa.
A special case of the co-ordinate transformation is the base change in a vector space.
- 2.1 displacement (translation ) 2.1.1 example
- 3.1 Cartesian coordinates and polar coordinates
- 3.2 Other Applications
Linear transformations
For linear transformations are linear functions of the new coordinates of the original, so
Or
The origin of the new coordinate system will coincide this match that of the original coordinate system.
Rotation ( rotation)
Upon rotation of the coordinate system is rotated. In two dimensions, there is only a rotation angle as a parameter. In 3D space, you can rotate around all three coordinate axes. Rotation is represented by a rotation matrix.
Example
We consider two ( here: three-dimensional ) Cartesian coordinate systems S and S ' with a common z- axis and a common origin. S ' is compared to S is rotated by the angle about the Z -axis. A point P in the coordinate system S, the coordinates, has the coordinate system S ', the coordinates:
In matrix notation results with the inverse rotation matrix for this rotation of the coordinate system:
Scaling
When scaling the " units " of the axes are changed. That is, the numerical values of the coordinates are multiplied by constant factors ( " scales " )
The parameters of this transformation are the numbers. A special case is the " scaling " in which all the elements have the same value
The scaling is a special case of the linear transformation in which all of coordinate values are multiplied by the same factor. The matrix is in this case once the identity matrix.
Shear
Wherein the shear angle between the coordinate axes changed. In 2D space, there is therefore a parameter and in 3D space three parameters.
Affine transformations
Affine transformations consist of one or more simple transformations.
Are both involved in coordinate systems linearly (ie, in principle, given by a coordinate origin, and equally divided coordinate axes ), as is available an affine transformation. Here, the new coordinate affine functions of the original, so
This can be compactly represented as a matrix multiplication of the vector with the old coordinates of the matrix containing the coefficients, and adding a vector which contains,
Translation is a special case of an affine transformation, in which A is the unit matrix.
Displacement (translation )
A translation can ( easier to imagine ) be interpreted as a shift in the depicted objects as either a shift of the coordinate origin or. In 2D space, a translation requires two parameters: displacement in the x- direction ( tx) and in the y- direction (ty ), by analogy, it is in 3D as the third parameter, the displacement in the z - direction ( tz).
Example
We consider two coordinate systems S and S '. S ' is shifted with respect to S by the vector. A point P in the coordinate system S, the coordinates, has the coordinate system S ', the coordinates.
Examples
Cartesian coordinates and polar coordinates
A point in the plane is determined in a Cartesian coordinate system by the coordinates (x, y) and in the polar coordinate system by the distance from the origin and the (positive) angle to the x-axis.
The following applies for the conversion of polar coordinates into Cartesian coordinates:
For the conversion from Cartesian coordinates to polar coordinates holds:
Other applications
In physics, the invariance of certain natural laws under coordinate transformations plays a special role, please refer to symmetry transformation. Of particular importance are the fundamental Galilean transformation, Lorentz transformation and the gauge transformation. Often also used transformations of operators and vectors:
- The transformation of differential operators
- The transformation of vector fields
In the earth sciences - in particular geodesy and cartography, there are other transformations that represent the formal coordinate transformations.
- Transformation of latitude and longitude in Gauss-Krueger - coordinates
- Conversions between astronomical coordinates
- 7- parameter transformation (translation, rotation, scaling between two coordinate systems on the same or other reference ellipsoid (s), also Helmert transformation ( " rotation - extension " ) ).
In the field of robotics, the Denavit -Hartenberg transformation is considered the standard method.