Search results
Results From The WOW.Com Content Network
In Euclidean space, such a dilation is a similarity of the space. [2] Dilations change the size but not the shape of an object or figure. Every dilation of a Euclidean space that is not a congruence has a unique fixed point [3] that is called the center of dilation. [4] Some congruences have fixed points and others do not. [5]
In Euclidean geometry homotheties are the similarities that fix a point and either preserve (if >) or reverse (if <) the direction of all vectors. Together with the translations, all homotheties of an affine (or Euclidean) space form a group, the group of dilations or homothety-translations.
Dilation (operator theory), a dilation of an operator on a Hilbert space; Dilation (morphology), an operation in mathematical morphology; Scaling (geometry), including: Homogeneous dilation , the scalar multiplication operator on a vector space or affine space; Inhomogeneous dilation, where scale factors may differ in different directions
A shape (in blue) and its morphological dilation (in green) and erosion (in yellow) by a diamond-shaped structuring element. Mathematical morphology (MM) is a theory and technique for the analysis and processing of geometrical structures, based on set theory, lattice theory, topology, and random functions.
A spiral similarity taking triangle ABC to triangle A'B'C'. Spiral similarity is a plane transformation in mathematics composed of a rotation and a dilation. [1] It is used widely in Euclidean geometry to facilitate the proofs of many theorems and other results in geometry, especially in mathematical competitions and olympiads.
Dilation (usually represented by ⊕) is one of the basic operations in mathematical morphology. Originally developed for binary images, it has been expanded first to grayscale images, and then to complete lattices. The dilation operation usually uses a structuring element for probing and expanding the shapes contained in the input image.
Geometry (from Ancient Greek γεωμετρία (geōmetría) 'land measurement'; from γῆ (gê) 'earth, land' and μέτρον (métron) 'a measure') [1] is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. [2]
If , and the Teichmüller mapping between them has dilatation then the Teichmüller distance between them is by definition . This indeed defines a distance on () which induces its topology, and for which it is complete. This is the metric most commonly used for the study of the metric geometry of Teichmüller space.