homothetic transformation
Also known as homogeneous dilation, homothety, homothecy
thumb|upright=1|Homothety: Example with . corresponds to (no point is moved); an ; a thumb|upright=1|Example with . corresponds to a point reflection at point thumb|upright=1.2|Homothety of a pyramid In mathematics, a homothety (or homothecy, or homogeneous dilation) is a transformation of an affine space determined by a point called its center and a nonzero number called its ratio, which sends point to a point by the rule, \overrightarrow{SX'}=k\overrightarrow{SX} for a fixed number . Using position vectors: \mathbf x'=\mathbf s + k(\mathbf x -\mathbf s).
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- Zentrische Streckung Konstruktionsprotokoll.pdf
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Article
16 sectionsContents
- Properties
- Mapping lines, line segments and angles
- Derivation of the properties
- Graphical construction (using the intercept theorem)
- Graphical construction (using a pantograph)
- Composition
- Composition of two homotheties with the same center
- Composition of two homotheties with different centers
- Derivation
- Composition of a homothety and a translation
- Derivation
- In homogeneous coordinates
- See also
- Notes
- References
- External links
thumb|upright=1|Homothety: Example with . corresponds to (no point is moved); an ; a thumb|upright=1|Example with . corresponds to a point reflection at point thumb|upright=1.2|Homothety of a pyramid In mathematics, a homothety (or homothecy, or homogeneous dilation) is a transformation of an affine space determined by a point called its center and a nonzero number called its ratio, which sends point to a point by the rule, \overrightarrow{SX'}=k\overrightarrow{SX} for a fixed number . Using position vectors: \mathbf x'=\mathbf s + k(\mathbf x -\mathbf s).
In case of S=O (Origin): \mathbf x'=k\mathbf x, which is a uniform scaling and shows the meaning of special choices for : for k=1 one gets the identity mapping; for k=-1 one gets the reflection at the center; for 1/k one gets the inverse mapping defined by .