Affine Transformation
Any Transformation preserving Collinearity (i.e., all points lying on a Line initially still lie on a Line after Transformation). An affine transformation is also called anAffinity. An affine transformation of 
is a Map 
of theform
 | (1) |
, where 
is a linear transformation of . If 
, thetransformation is Orientation-Preserving; if 
, it is Orientation-Reversing.Dilation (Contraction, Homothecy), Expansion, Reflection,Rotation, and Translation are all affine transformations, as are their combinations. A particular examplecombining Rotation and Expansion is the rotation-enlargement transformation
 |  |  | |
|  | (2) |
Separating the equations,
 | |  | (3) |
 | |  | |
| (4) |
This can be also written as
| |  | (5) |
| |  | (6) |
where
 | |  | (7) |
 | |  | (8) |
The scale factor
is then defined by | (9) |
 | (10) |
References
Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, p. 105, 1993.
- Jacobi Triple Product
- Jacobi Zeta Function
- Jacobsthal-Lucas Number
- Jacobsthal-Lucas Polynomial
- Jacobsthal Number
- Jacobsthal Polynomial
- Jaco-Shalen-Johannson Torus Decomposition
- Janko Groups
- Japanese Triangulation Theorem
- Jarnick's Inequality
- j-Conductor
- Jeep Problem
- Jenkins-Traub Method
- Jensen Polynomial
- Jensen's Formula
- Jensen's Inequality
- Jerabek's Hyperbola
- Jerk
- j-Function
- Jinc Function
- j-Invariant
- Jitter
- Joachimsthal's Equation
- Johnson Circle
- Johnson's Equation