Affinity (math)

In geometry , affinity is a structure- preserving bijective mapping of an affine space (often the plane of the drawing or the three-dimensional visual space ) on itself. The term encompasses and generalizes the term similarity , in which the relationships of any length of line and the dimensions of angles ( → see angular accuracy ) are retained.

• An affinity is thus an affine mapping of an affine space in itself, which is at the same time a bijection . It always has the property that

1. the points and straight lines of space are mapped onto points or straight lines while maintaining collinearity : points on a straight line are mapped onto points of the associated image line,
2. the partial ratio of any three points on any straight line is maintained (partial ratio accuracy) and
3. every pair of parallel straight lines is mapped onto a pair of parallel straight lines (parallels).

• Every affinity is a collineation , so it has the first-mentioned property of being straight line .
• In Euclidean space , an affinity generally changes the lengths of lines and the dimensions of angles and thus also areas and spaces. Affinities of Euclidean space, which also leave these quantities unchanged, i.e. isometries , are called movements .
• Likewise, an affinity of a Euclidean space generally changes the proportions of distances (length proportions). If they and thus also the angles between straight lines are not changed, then such an affinity is called similarity .

In synthetic geometry , the term affinity is generalized for two- dimensional affine spaces, i.e. planes : a collineation on an affine plane is an affinity if and only if each of its restrictions on a straight line can be represented by a composition of parallel projections. For Desargue planes, this definition is equivalent to the definition “An affinity is a partial ratio-faithful collineation”, which is used in analytic geometry. A generalization is unnecessary for at least three-dimensional affine spaces, since these are always desargue, one-dimensional spaces are not considered in themselves in synthetic geometry.

Coordinate representation

You can use the mapping rule after choosing an affine point base for the position vectors in the form ${\displaystyle {\vec {x}};\;{\vec {x}}'}$

${\displaystyle \alpha \colon {\vec {x}}'={A_{n}}\cdot {\vec {x}}+{\vec {t}}}$

specify. The vector is called the displacement vector and is a square matrix , the so-called mapping matrix . For its determinant is always , i. H. the mapping is bijective . ${\displaystyle {\vec {t}}}$ ${\displaystyle A_{n}}$ ${\displaystyle n\times n}$ ${\displaystyle \det(A_{n})\neq 0}$

Here the affine space is understood as a vector space over a body (mostly in geometry ). The points of the affine space are the vectors from (position vectors ), and affine subspaces are the additive secondary classes of the linear subspaces of this vector space . It is always assumed from the vector space in geometry and also predominantly in linear algebra that its dimension is finite. ${\displaystyle V}$ ${\displaystyle K}$ ${\displaystyle K=\mathbb {R} }$ ${\displaystyle V}$ ${\displaystyle V}$ ${\displaystyle V}$

Classification of affinities

Radial affinities

An affinity is called radial / centric affinity if it has exactly one fixed point , this is equivalent to . ${\displaystyle \mathrm {rang} (A_{n}-E_{n})=n}$

(Rank is explained in Rank .) ${\displaystyle \mathrm {rang} (f)}$

Perspective affinities

An affinity is called perspective affinity if it has exactly one fixed point hyperplane (that is, a hyperplane consisting exclusively of fixed points), which is equivalent to . ${\displaystyle \mathrm {rang} (A_{n}-E_{n})=\mathrm {rang} (A_{n}-E_{n}\mid {\vec {t}})=1}$

A perspective affinity is called parallel stretching if it has an eigenvalue in addition to the eigenvalue (i.e. an eigenvalue of ) . ${\displaystyle \lambda _{1}=1}$ ${\displaystyle A_{n}}$ ${\displaystyle \lambda _{2}\in K\setminus \lbrace 0,1\rbrace }$

A parallel stretch with is called affine mirroring. It is called shear if it only has the eigenvalue . ${\displaystyle \lambda _{2}=-1}$ ${\displaystyle \lambda _{1}=1}$

An invariant pair of right angles has a perspective affinity .

Homotheties

An affinity with

${\displaystyle A_{n}=k\cdot E_{n}}$with means homothetia or dilation . ${\displaystyle k\in K\setminus \lbrace 0\rbrace }$

If also

• ${\displaystyle k\in K\setminus \lbrace 0,1\rbrace }$, is called central stretching . ${\displaystyle \alpha }$
• ${\displaystyle k=1}$, is called displacement or translation ${\displaystyle \alpha }$
• ${\displaystyle k=-1}$is called point reflection . ${\displaystyle \alpha }$

Unimodularity

An affinity is called unimodular if . ${\displaystyle \det(A_{n})=\pm 1}$

It's actually unimodular, though . ${\displaystyle \det(A_{n})=1}$

Content fidelity

If the underlying body is arranged, then an affinity is content-true if . ${\displaystyle \det(A_{n})=\pm 1}$

It is in the same direction if . ${\displaystyle \det(A_{n})>0}$

Properties of general affinities

Affinities have a number of properties that can be used in construction.

Bijectivity

An affinity is both injective and surjective, i.e. bijective.

Straight line

The image of a straight line under an affinity is again a straight line.

Parallels

The images of parallel straight lines under an affinity are again parallel.

Partial relationship loyalty

If there is a point on the line and the images of and are under an affinity, then the division ratio of is equal to the division ratio of . The following applies in particular: If the center of , then the image point of M under an affinity is the center of the segment . ${\displaystyle T}$ ${\displaystyle [AB]}$ ${\displaystyle A',B',T'}$ ${\displaystyle A,B}$ ${\displaystyle T}$ ${\displaystyle (A';T';B')}$ ${\displaystyle (A;T;B)}$ ${\displaystyle M}$ ${\displaystyle [AB]}$ ${\displaystyle [A'B']}$

Properties of plane perspective affinities

In the case of a perspective affinity in a two-dimensional affine space, the plane, the fixed point hyperplane is a straight line, which is also referred to as the axis of affinity. One speaks here of axis affinities .

Straight lines through point and image point are fixed lines

A straight line through a point and its image point is a fixed line . This can be shown with the help of the fixed point line of perspective affinity: ${\displaystyle {\overline {PP'}}}$ ${\displaystyle P}$ ${\displaystyle P'}$ ${\displaystyle a}$

• If the fixed point line intersects at a point , the image from is the straight line due to the fidelity of the line . But this coincides with . ${\displaystyle {\overline {PP'}}}$ ${\displaystyle a}$ ${\displaystyle P_{a}}$ ${\displaystyle {\overline {PP_{a}}}}$ ${\displaystyle {\overline {P_{a}P'}}}$ ${\displaystyle {\overline {PP'}}}$
• If is parallel to , then the image of is parallel to through because of the fidelity to parallel , since the image is equal to itself. However, this parallel coincides with . ${\displaystyle {\overline {PP'}}}$ ${\displaystyle a}$ ${\displaystyle {\overline {PP'}}}$ ${\displaystyle a}$ ${\displaystyle P'}$ ${\displaystyle a}$ ${\displaystyle a}$ ${\displaystyle {\overline {PP'}}}$

Parallels of fixed lines are again fixed lines

Parallels of fixed lines are again fixed lines

The image of a parallel to a fixed line is itself a fixed line again. The statement follows from the parallels and partial relationships: ${\displaystyle p'}$ ${\displaystyle p}$ ${\displaystyle f}$

• Since and are parallel, must also and be parallel. From the transitivity of parallelism it follows that then also and must be parallel. ${\displaystyle f}$ ${\displaystyle p}$ ${\displaystyle f=f'}$ ${\displaystyle p'}$ ${\displaystyle p}$ ${\displaystyle p'}$
• Pick a point on the affinity axis and a point on . ${\displaystyle A}$ ${\displaystyle a}$ ${\displaystyle X}$ ${\displaystyle f}$
• Since and are parallel, the connecting line also intersects at one point . ${\displaystyle f}$ ${\displaystyle p}$ ${\displaystyle {\overline {AX}}}$ ${\displaystyle p}$ ${\displaystyle P}$
• Since there is a fixed line, the image of lies on and the image of is the same . ${\displaystyle f}$ ${\displaystyle X'}$ ${\displaystyle X}$ ${\displaystyle f}$ ${\displaystyle {\overline {AX}}}$ ${\displaystyle {\overline {AX'}}}$
• About relationship fidelity it follows that to how to . ${\displaystyle \left|AX\right|}$ ${\displaystyle \left|AP\right|}$ ${\displaystyle \left|AX'\right|}$ ${\displaystyle \left|AP'\right|}$
• With the inversion of the first ray theorem, it results that then must lie on a parallel to through (i.e. on ). Since and are parallel and have the point in common, they must be identical. ${\displaystyle P'}$ ${\displaystyle f}$ ${\displaystyle P}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle p'}$ ${\displaystyle P'}$

Constructions

Image point under a perspective affinity

Construction of the image point Q 'of Q under a perspective affinity.

Given a perspective affinity through their checkpoint straight and the point / pixel pair , . The image of any point can be constructed as follows: ${\displaystyle a}$ ${\displaystyle P}$ ${\displaystyle P'}$ ${\displaystyle Q}$

• Choose any point on the fixed point line . ${\displaystyle A}$ ${\displaystyle a}$
• Draw the straight line connecting it . ${\displaystyle {\overline {PA}}}$
• The image of is again a straight line due to the fidelity of the figure. The image of is itself because it lies on the fixed line . This is the picture of the straight line . ${\displaystyle {\overline {PA}}}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle a}$ ${\displaystyle {\overline {PA}}}$ ${\displaystyle {\overline {AP'}}}$
• Draw parallels through . This intersects at one point . Because of the parallels of the figure, the image is parallel to through the point . The point sought lies on this parallel straight line. ${\displaystyle {\overline {PA}}}$ ${\displaystyle Q}$ ${\displaystyle a}$ ${\displaystyle A'}$ ${\displaystyle {\overline {QA'}}}$ ${\displaystyle {\overline {AP'}}}$ ${\displaystyle A'}$ ${\displaystyle Q'}$
• Draw the straight line . It cuts at one point (if this is not the case, special treatment is necessary). The picture of this straight line is . The point you are looking for also lies on this straight line and is therefore the intersection of and the parallel of through . ${\displaystyle {\overline {PQ}}}$ ${\displaystyle a}$ ${\displaystyle S}$ ${\displaystyle {\overline {SP'}}}$ ${\displaystyle Q'}$ ${\displaystyle {\overline {SP'}}}$ ${\displaystyle {\overline {AP'}}}$ ${\displaystyle {\overline {A'}}}$

Another possibility of construction saves the auxiliary point and uses the property that straight lines through point and image point are fixed lines : ${\displaystyle A}$

• Draw the straight line . Since it is a straight line through point and image point, the image of this straight line is the straight line itself. ${\displaystyle {\overline {PP'}}}$
• Draw a parallel to through . It intersects the fixed line in . ${\displaystyle {\overline {QQ_{a}}}}$ ${\displaystyle {\overline {PP'}}}$ ${\displaystyle Q}$ ${\displaystyle a}$ ${\displaystyle Q_{a}}$
• The image of is yourself: ${\displaystyle {\overline {QQ_{a}}}}$ ${\displaystyle {\overline {QQ_{a}}}}$
  • Straight line: Since parallel to , the image from runs parallel to the image from . ${\displaystyle {\overline {QQ_{a}}}}$ ${\displaystyle {\overline {PP'}}}$ ${\displaystyle {\overline {QQ_{a}}}}$ ${\displaystyle {\overline {PP'}}}$
  • ${\displaystyle {\overline {PP'}}}$is a fixed line: the image of is itself. It follows that the image of is parallel to itself. ${\displaystyle {\overline {PP'}}}$ ${\displaystyle {\overline {PP'}}}$ ${\displaystyle {\overline {QQ_{a}}}}$
  • Since the point is part of the fixed point line , the image is self-evident. ${\displaystyle Q_{a}}$ ${\displaystyle a}$ ${\displaystyle Q_{a}}$ ${\displaystyle Q_{a}}$
  • Since the image of by extending and parallel to itself, it can only be themselves. ${\displaystyle {\overline {QQ_{a}}}}$ ${\displaystyle Q_{a}}$ ${\displaystyle {\overline {QQ_{a}}}}$
• This is part of . ${\displaystyle Q'}$ ${\displaystyle {\overline {QQ_{a}}}}$
• With the consideration of the first construction lies on the intersection of and (with the intersection of and ). ${\displaystyle Q'}$ ${\displaystyle {\overline {QQ_{a}}}}$ ${\displaystyle {\overline {SP'}}}$ ${\displaystyle S}$ ${\displaystyle {\overline {PQ}}}$ ${\displaystyle a}$

Group structure

The set of affinities over an affine space form a group with regard to the sequential execution . Is the affine space of -dimensional vector space assigned, then this group can be (here abbreviated as written) in the generally linear groups classified as a subgroup. ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle n}$ ${\displaystyle V=K^{n}}$ ${\displaystyle {\mathcal {A}}(K^{n})}$

The group of affinities is also a subgroup of the group of (in-plane) collineations.

Group operations

The properties required of an affinity naturally result in various group operations :

• ${\displaystyle {\mathcal {A}}(K^{n})}$ operates as a mapping group

1. on the point set , ${\displaystyle A}$
2. on the set of affine subspaces of a fixed dimension with , ${\displaystyle A}$ ${\displaystyle m}$ ${\displaystyle 0\leq m\leq n}$
3. on sets of directions in affine space, for example the set of all families of parallel lines.

• The group operates sharply simply transitive on the set of affine point bases of the affine space . This means here: If you specify points in a general position (in such a way that the connection vectors of the first point are linearly independent with the other points ), then there is exactly one affinity with which the standard basis is mapped to these points (in the specified order) becomes. This gives a simple way to compute the number of elements of when is a finite field. ${\displaystyle {\mathcal {A}}(K^{n})}$ ${\displaystyle A}$ ${\displaystyle n+1}$ ${\displaystyle n}$ ${\displaystyle {\mathcal {A}}(K^{n})}$ ${\displaystyle K}$

Group structure

The group ${\displaystyle {\mathcal {A}}(K^{n})}$

1. is always non-commutative , ${\displaystyle n>1}$
2. contains the general linear group as a subgroup - the affinities in which the fixed origin is a fixed point , whose translation component or displacement vector is the zero vector , ${\displaystyle \mathrm {GL} (n,K)}$ ${\displaystyle O}$
3. can be understood as a subgroup of the general linear group , ${\displaystyle \mathrm {GL} (n+1,K)}$
4. can be used as sub-group of projective linear array to be construed - here include those projectivities to which a fixed hyperplane of the projective space, the remote hyperplane mapped as Fixhyperebene on itself, ${\displaystyle \mathrm {PGL} (n,K)}$ ${\displaystyle {\mathcal {A}}(K^{n})}$
5. contains the commutative subgroup of translations (pure displacements, the mapping matrix of which is the identity matrix ) as normal divisors , ${\displaystyle A_{n}}$ ${\displaystyle {\mathcal {T}}(K^{n})}$
6. is an inner semi-direct product of and . ${\displaystyle {\mathcal {T}}(K^{n})}$ ${\displaystyle \mathrm {GL} (n,K)}$
7. The normal divisor of the translations is isomorphic to the additive group of the underlying vector space. ${\displaystyle {\mathcal {T}}(K^{n})}$ ${\displaystyle (K^{n},+)}$
8. ${\displaystyle {\mathcal {T}}(K^{n})}$operates by conjugation sharp simply transitive on the set of subgroups . This is the subgroup of that maps a certain point of the affine space to itself. Each of these subgroups is too isomorphic. ${\displaystyle \lbrace A_{Z}<{\mathcal {A}}(K^{n}):Z\in A\left(\forall \alpha \in A_{Z}:\;\alpha (Z)=Z\right)\rbrace }$ ${\displaystyle A_{Z}}$ ${\displaystyle {\mathcal {A}}(K^{n})}$ ${\displaystyle Z}$ ${\displaystyle \mathrm {GL} (n,K)}$

Group order

If the body is a finite body with elements, then the group of affinities is finite and their order is ${\displaystyle K}$ ${\displaystyle q}$ ${\displaystyle {\mathcal {A}}(K^{n})}$

${\displaystyle \left|{\mathcal {A}}(K^{n})\right|=q^{n}\cdot \prod _{i=0}^{n-1}\left(q^{n}-q^{i}\right)=q^{n}\cdot \left(q^{n}-1\right)\cdot \left(q^{n}-q\right)\cdot \left(q^{n}-q^{2}\right)\cdots \left(q^{n}-q^{n-1}\right)}$.

The factor is the order of the translation group ; it is also the index of the subgroup that maps the origin to itself. The order of this subgroup provides the remaining factors (→ see General linear group # About finite fields ). ${\displaystyle q^{n}}$ ${\displaystyle {\mathcal {T}}(K^{n})\cong (K^{n},+)}$ ${\displaystyle \left[{\mathcal {A}}(K^{n}):A_{O}\right]=q^{n}}$ ${\displaystyle A_{O}\cong \mathrm {GL} (n,k)}$

literature

• Rolf Brandl: Lectures on analytical geometry . Publisher Rolf Brandl, Hof 1996.
• Gerd Fischer : Analytical Geometry . 6th, revised edition. Vieweg Verlag , Braunschweig [u. a.] 1992, ISBN 3-528-57235-3 .
• Thomas W. Hungerford: Algebra . 5. print. Springer-Verlag, 1989, ISBN 0-387-90518-9 .
• Uwe Storch, Hartmut Wiebe: Textbook of Mathematics, Volume II: Linear Algebra . BI-Wissenschafts-Verlag, 1990, ISBN 3-411-14101-8 .
• Günter Scheja, Uwe Storch: Textbook of Algebra: including linear algebra . 2., revised. and exp. Edition. Teubner, Stuttgart 1994, ISBN 3-519-12203-0 ( table of contents [accessed on January 14, 2012]).

Web links

• Hans Walser: Affine images (PDF)

References and comments

1. ↑ Scheja and stork §43
2. ↑ ^{a b} The parallelism follows from the first requirement for an affinity, the preservation of collinearity, except in the case of affine spaces with exactly two points on each straight line that are at least three-dimensional. In all cases it is, if the straight line is assumed, equivalent to the level loyalty (Scheja and Storch §43 and VI). Compare also the detailed explanations in collineation .