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Homothety

In mathematics, a homothety (or homothecy, or homogeneous dilation) is a transformation of an affine space determined by a point S called its center and a nonzero number called its ratio, which sends point to a point by the rule for a fixed number . Using position vectors: In case of (Origin): which is a uniform scaling and shows the meaning of special choices for : for one gets the identity mapping, for one gets the reflection at the center, For one gets the inverse mapping defined by . 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. These are precisely the affine transformations with the property that the image of every line g is a line parallel to g. In projective geometry, a homothetic transformation is a similarity transformation (i.e., fixes a given elliptic involution) that leaves the line at infinity pointwise invariant. In Euclidean geometry, a homothety of ratio multiplies distances between points by , areas by and volumes by . Here is the ratio of magnification or dilation factor or scale factor or similitude ratio. Such a transformation can be called an enlargement if the scale factor exceeds 1. The above-mentioned fixed point S is called homothetic center or center of similarity or center of similitude. The term, coined by French mathematician Michel Chasles, is derived from two Greek elements: the prefix homo- (όμο), meaning "similar", and thesis (Θέσις), meaning "position". It describes the relationship between two figures of the same shape and orientation. For example, two Russian dolls looking in the same direction can be considered homothetic. Homotheties are used to scale the contents of computer screens; for example, smartphones, notebooks, and laptops. The following properties hold in any dimension. A homothety has the following properties: A line is mapped onto a parallel line.

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Related concepts (12)
Invariant (mathematics)
In mathematics, an invariant is a property of a mathematical object (or a class of mathematical objects) which remains unchanged after operations or transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used. For example, the area of a triangle is an invariant with respect to isometries of the Euclidean plane. The phrases "invariant under" and "invariant to" a transformation are both used.
Scaling (geometry)
In affine geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a scale factor that is the same in all directions. The result of uniform scaling is similar (in the geometric sense) to the original. A scale factor of 1 is normally allowed, so that congruent shapes are also classed as similar. Uniform scaling happens, for example, when enlarging or reducing a photograph, or when creating a scale model of a building, car, airplane, etc.
Cross-ratio
In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points A, B, C, D on a line, their cross ratio is defined as where an orientation of the line determines the sign of each distance and the distance is measured as projected into Euclidean space. (If one of the four points is the line's point at infinity, then the two distances involving that point are dropped from the formula.
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