Projective illustration

Projective images are images which convert straight lines into straight lines. In projective geometry, they are the analogue of the linear mappings in linear algebra .

definition

The projective space to a - vector space is the set of all straight lines through the zero point in , that is, the quotient space with respect to the equivalence relation . Now let and be vector spaces and and the associated projective spaces, then a mapping is called ${\ displaystyle P (V)}$ ${\ displaystyle K}$ ${\ displaystyle V}$ ${\ displaystyle V}$ ${\ displaystyle V \ setminus \ {0 \} / \ sim}$ ${\ displaystyle x \ sim y \ Leftrightarrow \ exists \ lambda \ in K \ setminus \ {0 \} \ colon x = \ lambda y}$
${\ displaystyle V_ {1}}$ ${\ displaystyle V_ {2}}$ ${\ displaystyle P (V_ {1})}$ ${\ displaystyle P (V_ {2})}$

{\ displaystyle \ pi \ colon P (V_ {1}) \ longrightarrow P (V_ {2})}

projective or projective-linear if there is an injective linear mapping

{\ displaystyle {\ tilde {\ pi}} \ colon V_ {1} \ longrightarrow V_ {2}}

With

{\ displaystyle \ pi (\ langle x \ rangle) = \ langle {\ tilde {\ pi}} (x) \ rangle}

for all

{\ displaystyle \ langle x \ rangle \ in P (V_ {1})}

gives.

The following (non-equivalent) definition can also be found with individual authors:
Let and be projective spaces and a projective subspace of , then a mapping is called ${\ displaystyle P (V_ {1})}$ ${\ displaystyle P (V_ {2})}$ ${\ displaystyle Z}$ ${\ displaystyle P (V_ {1})}$

{\ displaystyle \ pi \ colon P (V_ {1}) \ setminus Z \ longrightarrow P (V_ {2})}

projective if there is a linear map

{\ displaystyle {\ tilde {\ pi}} \ colon V_ {1} \ longrightarrow V_ {2}}

With

{\ displaystyle \ pi (\ langle x \ rangle) = \ langle {\ tilde {\ pi}} (x) \ rangle}

for all

{\ displaystyle \ langle x \ rangle \ in P (V_ {1}) \ setminus Z}

and gives. The subspace is referred to as the exception space. ${\ displaystyle P (\ operatorname {core} ({\ tilde {\ pi}})) = Z}$ ${\ displaystyle Z}$

This article refers to the first definition below.

example

An example of a projective mapping (between projective spaces of different dimensions) is the Veronese embedding . ${\ displaystyle i: P (K ^ {2}) \ rightarrow P (K ^ {3})}$

{\ displaystyle i ([x: y]) = [x ^ {2}: xy: y ^ {2}]}

.

Projective linear group

The invertible projective images of a projective space on themselves form a group which is referred to as a projective linear group . The elements of this group are especially true to the line, i.e. collineations . ${\ displaystyle P (V)}$ ${\ displaystyle \ mathrm {PGL} (V)}$

The projective linear group over a vector space over a field is the factor group , with the normal (even central ) subgroup of the scalar multiples of identity being with off . The names etc. correspond to those of the general linear group. If a finite body are and equally powerful but generally not isomorphic. ${\ displaystyle \ mathrm {PGL} (V)}$ ${\ displaystyle V}$ ${\ displaystyle K}$ ${\ displaystyle \ mathrm {GL} (V) / K ^ {\ times}}$ ${\ displaystyle K ^ {\ times}}$ ${\ displaystyle k \ cdot \ mathrm {id} _ {V}}$ ${\ displaystyle \ mathrm {id}: V \ rightarrow V}$ ${\ displaystyle k}$ ${\ displaystyle K \ setminus \ {0 \}}$ ${\ displaystyle \ mathrm {PGL} (n, K)}$ ${\ displaystyle K}$ ${\ displaystyle \ mathrm {PGL} (n, K)}$ ${\ displaystyle \ mathrm {SL} (n, K)}$

Projective images preserve the incidence structure.

The name comes from projective geometry , where the analogue to the general linear group is the projective linear group , while the group belongs to the -dimensional projective space , it is the group of all projectivities of space. ${\ displaystyle n}$ ${\ displaystyle K}$ ${\ displaystyle \ mathrm {PGL} (n + 1, K)}$

Fractional linear transformations

In the case of the projective straight line , the projective images are precisely the fractional-linear transformations. ${\ displaystyle KP ^ {1}: = P (K ^ {2})}$

After identifying with (through ) works on through . ${\ displaystyle P (K ^ {2})}$ ${\ displaystyle K \ cup \ left \ {\ infty \ right \}}$ ${\ displaystyle [x_ {0}: x_ {1}] \ leftrightarrow x_ {0} x_ {1} ^ {- 1}}$ ${\ displaystyle PGL (2, K)}$ ${\ displaystyle P (K ^ {2})}$ ${\ displaystyle \ left ({\ begin {matrix} a & b \\ c & d \ end {matrix}} \ right) z = {\ frac {az + b} {cz + d}}}$

Furniture transformations

A special case is the group of Möbius transformations , the . These are the projective images of the . Discrete groups of Möbius transformations are called Kleinian groups . Fuchsian groups are Kleinian groups which map the projective subspace onto themselves. ${\ displaystyle \ mathrm {PGL} (2, \ mathbb {C})}$ ${\ displaystyle P (\ mathbb {C} ^ {2})}$ ${\ displaystyle P (\ mathbb {R} ^ {2}) \ subset P (\ mathbb {C} ^ {2})}$

properties

Projective images map projective subspaces onto projective subspaces.

Projective images get the double ratio of 4-tuples of collinear points. This property can be seen as a distinctive feature of projective geometry . See also: Erlanger program . These connections were already known in antiquity and can be found e.g. B. at Pappos . They are the main reason why the term double ratio was even developed.

literature

Heiner Zieschang: Linear Algebra and Geometry . Springer, 1997, ISBN 978-3-519-02230-5 .