List of transformations in mathematics

The term transformation is used in many ways in mathematics .

Geometric transformations

(Coordinate) transformation for an object assumed to be stationary (left) or a coordinate system assumed to be stationary (right)

→ Main articles : Movement (mathematics)  and transformation geometry

In geometry , the term transformation is understood to mean the movement of a set of points (object) in a space thought to be at rest (or in relation to a coordinate system assumed to be at rest), also referred to in English as active or alibi transformation. Typical transformations are:

• Displacement (translation)
• Rotation (rotation)
• reflection
• Rigid movement ( execution of rotation and translation one after the other)
• Isometry (execution of mirroring, rotation and translation one after the other)
• Stretching ( centric stretching )
• Similitude illustration

Coordinate transformations

→ Main article : coordinate transformation

In a coordinate transformation, the coordinates of a point or a set of points (e.g. the graph of a function) are transferred from one coordinate system to another. From a formal point of view, this is the transition from one coordinate system with the original coordinates to a second with the new coordinates , also called passive or alias transformation in English . Typical transformation processes are: ${\displaystyle (x_{1},x_{2},\ldots ,x_{N})}$ ${\displaystyle (x'_{1},x'_{2},\ldots ,x'_{N})}$

• Displacement (translation)
• Rotation (rotation)
• reflection
• Scaling (changing the scale)
• Shear .

As the two lists of transformation examples and the illustration on the right show, one and the same transformation, e.g. B. displacement or rotation, depending on the perspective, one time as a geometric, the other time as a coordinate transformation, which is also reflected in the nature of their mathematical formulation, z. B. the use of the transformation matrix for the geometric as well as the inverse transformation matrix for the associated coordinate transformation (or vice versa). ${\displaystyle A}$ ${\displaystyle A^{-1}}$

See also : linear mapping , affine mapping, and basic transformation

Integral transformations

→ Main article : integral transformation

Certain integral operators are traditionally called integral transformations or frequency transformations . These operators are often invertible. Integral transformations are special functional transformations.

• Fourier transform
• Hankel transformation
• Hilbert transformation
• Cosine transformation
• Short-term Fourier transform
• Laplace transform
• Mellin transformation
• Sine transformation
• Radon transformation
• Wavelet transform
• Two-sided Laplace transformation

Discrete Transformations

With the help of the transformations in this list, certain integral transformations from the previous section can be approximately calculated with the computer . For some integral transformations, there are different algorithms to implement them in the computer, which differ, for example, in their speed.

• Bluestein FFT algorithm
• Discrete Fourier Transform (DFT)
  • Discrete Cosine Transformation (DCT)
  • Discrete Sine Transformation (DST)
  • Fourier transform for time-discrete signals
  • Modified Discrete Cosine Transform (MDCT)
  • Fast Fourier Transform (FFT)
• Discrete Hartley Transform (DHT)
• Hadamard transformation
• Fast wavelet transform
  • Wavelet packet transformation
• Z transformation

statistics

• Standardization (statistics) (also z-transformation)
• Centering (statistics)

Other transformations

• Natural Transformations in Category Theory
• Sequence transformation in numerics
• Legendre transformation

Web links

Commons : Transformations (Geometry)  - Collection of images, videos and audio files

Individual evidence

1. ^ Weisstein, Eric W. "Alibi Transformation." From MathWorld - A Wolfram Web Resource.
2. ^ Weisstein, Eric W. "Alias ​​Transformation." From MathWorld - A Wolfram Web Resource.