Centric elongation

Centric stretching with a positive stretching factor
Centric stretching with a negative stretching factor

In geometry, a centric stretching is an image that enlarges or reduces all lines in a certain, given ratio , the image lines being parallel to the original lines . Centric stretches are special similarity maps .

definition

A point on the plane of the drawing or in space and a real number are given . The centric stretching with center and stretching factor (mapping factor) is that mapping of the plane of the drawing or the space itself, in which the image point of a point has the following properties: ${\ displaystyle Z}$ ${\ displaystyle m \ neq 0}$${\ displaystyle Z}$${\ displaystyle m}$${\ displaystyle P '}$${\ displaystyle P}$

• ${\ displaystyle Z}$, and lie on a straight line.${\ displaystyle P}$${\ displaystyle P '}$
• For lying and on the same side of , for on different sides.${\ displaystyle m> 0}$${\ displaystyle P}$${\ displaystyle P '}$${\ displaystyle Z}$${\ displaystyle m <0}$
• The route length is equal to times the route length .${\ displaystyle {\ overline {ZP '}}}$${\ displaystyle | m |}$${\ displaystyle {\ overline {ZP}}}$

The two sketches show the application of two centric stretchings (with and ) to a triangle ABC each . ${\ displaystyle m = 3}$${\ displaystyle m = -0 {,} 5}$

properties

• Centric elongations are straight, circular and angular .
• The length ratios are retained.
• The image path of any path is times the length.${\ displaystyle | m |}$
• Any geometrical figure is mapped onto a figure with times the area .${\ displaystyle m ^ {2}}$
• Any body is mapped onto a body with -fold volume .${\ displaystyle | m | ^ {3}}$
• The centric stretches with a certain center form a group from an algebraic point of view .
• The image of a straight line is parallel to the straight line.
• In vector notation, the centric stretching with the center and the stretching factor is described by${\ displaystyle Z}$${\ displaystyle m}$
${\ displaystyle P '= Z + m (PZ) = mP + (1-m) Z}$.
• A centric stretching is the affinity that is described by the matrix and the displacement vector .${\ displaystyle mE_ {n}}$${\ displaystyle (1-m) Z}$
• The identical mapping is also counted as a stretching with the stretching factor in the stretching. A non-identical extension has exactly one fixed point , that is its extension center, and its fixed straight lines are precisely the straight lines that go through this center.${\ displaystyle m = 1}$

Special cases

For the identical mapping (identity) results for a point reflection . The case is not allowed, since otherwise all points would have the same image point, namely the center. ${\ displaystyle m = 1}$${\ displaystyle m = -1}$${\ displaystyle m = 0}$

Generalizations

• The centric extension is an example of a dilation . In axiomatic affine geometry , this term is defined using parallelism .
• The centric stretching is the special case of a rotational stretching with a rotation angle of 0.
• Instead of the affine 2- or 3- dimensional space over the real numbers, one can also define centric stretching more generally in any finite-dimensional affine space over any body and even over any inclined body . The "vectorial" representation is the same as in the real case, but the parallel displacements that are stretched from a center generally only form a left vector space over the oblique coordinate body.
• In the flat, two-dimensional case, the term centric stretching is still somewhat more general when the parallel displacements (as coordinate “vectors”) of an affine translation plane over a quasi- body are stretched with a “scalar” from the core of the quasi-body.

In the last two cases mentioned, one can generally speak of neither angle nor aspect ratio, since neither an angle nor a length need to exist. Here, too, the centric extensions always belong to the dilatations and the affinities, and the same applies to fixed points and fixed lines as in the real case.

literature

• H. Schupp: Elementary Geometry. UTB Schoeningh 1977