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In mathematics, a homothety (or homothecy) is a transformation of space which dilates distances with respect to a fixed point A called the origin. The number c by which distances are multiplied is called the dilation factor or similitude ratio. Such a transformation is also called an enlargement. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
In mathematics, a transformation in elementary terms is any of a variety of different functions from geometry, such as rotations, reflections and translations. ...
Distance is a numerical description of how far apart objects are at any given moment in time. ...
More generally c can be negative; in that case it not only multiplies all distances by | c | , but also inverts all points with respect to the fixed point.
Choose an origin or center A and a real number c (possibly negative). The homothety hA,c maps any point M to a point M' such that In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
- A − M' = c(A − M)
(as vectors).
A homothety is an affine transformation (if the fixed point is the origin: a linear transformation) and also a similarity transformation. It multiplies all distances by | c | , all surface areas by c2, etc. In geometry, an affine transformation or affine map (from the Latin, affinis, connected with) between two vector spaces (strictly speaking, two affine spaces) consists of a linear transformation followed by a translation: In the finite-dimensional case each affine transformation is given by a matrix A and a vector b...
In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...
// Two geometrical objects are called similar if one is congruent to the result of a uniform scaling (enlarging or shrinking) of the other. ...
An open surface with X-, Y-, and Z-contours shown. ...
Homothetic relation
One application is a homothetic relation R. R, then, is homothetic if - for .
An economic application of this is that a utility function which is homogeneous of degree one corresponds to a homothetic preference relation. In economics, utility is a measure of the relative happiness or satisfaction (gratification) gained. ...
Look up Homogeneous in Wiktionary, the free dictionary. ...
Preference (or taste) is a concept, used in the social sciences, particularly economics. ...
See also In mathematics, a dilation is a function f from a metric space into itself that satisfies the identity where d(x,y) is the distance from x to y and r is some positive real number. ...
In abstract algebra, a branch of pure mathematics, the algebraic structure group with operators or Ω-group is a group with a set of group endomorphisms. ...
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