Dilatation

Also known as: dilation.

A mapping T on a euclidian plane is a dilatation iff it is a similarity transformation that maps every line to another line parallel to the original. A similarity transformation maps any shape to another that is similar to the original. The set of all dilatations form a group with composition of dilatations as the group operator. The composition of dilatations is associative but not commutative, hence not abelian. If a dilatation has a fixed point C, the dilatation is known as a central dilatation and the point is known as the dilatation center. The scale factor r of a dilatation measures the change in perimeter length, and is known as the dilatation ratio. If a dilatation has a dilatation ratio of 1, then it is known as a translation. While the set of all translations form a group, the set of all central dilatations are not a group.

Equation for translation: τ_α(x) = x + α
Equation for central dilatation: δ_C,r(x) = r(x - C) + C = rx + (1 - r)C

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Source: "Vectors and Transformations in Plane Geometry" by Philippe Tondeur, Publish or Perish, Inc. 1993

Dil`a*ta"tion (?), n. [OE. dilatacioun, F. dilatation, L. dilatatio, fr. dilatare. See Dilate, and cf. 2d Dilation.]

1.

Prolixity; diffuse discourse.

[Obs.] "What needeth greater dilatation?"

Chaucer.

2.

The act of dilating; expansion; an enlarging on al sides; the state of being dilated; dilation.

3. Anat.

A dilation or enlargement of a canal or other organ.

 

© Webster 1913.