Projective base

In the projective plane, four projectively independent points form a projective basis

In mathematics, a projective basis is a set of points in a -dimensional projective space , each of which is projectively independent . Projective bases are used in projective geometry to characterize projectivities and to define projective coordinates . ${\displaystyle n+2}$ ${\displaystyle n}$ ${\displaystyle n+1}$

definition

A - tuple of points of a projective space over a - vector space is called projectively independent if one of the following equivalent conditions is met: ${\displaystyle (n+1)}$ ${\displaystyle (P_{0},\ldots P_{n})}$ ${\displaystyle P(V)}$ ${\displaystyle K}$ ${\displaystyle V}$

• There are linearly independent vectors with for . ${\displaystyle v_{0},\ldots ,v_{n}\in V}$ ${\displaystyle P_{i}=K\,v_{i}}$ ${\displaystyle i=0,\ldots ,n}$
• Each - tuple of vectors from with for is linearly independent. ${\displaystyle (n+1)}$ ${\displaystyle (v_{0},\ldots ,v_{n})}$ ${\displaystyle V}$ ${\displaystyle P_{i}=K\,v_{i}}$ ${\displaystyle i=0,\ldots ,n}$
• The following applies to the dimension of the connecting space of the points . ${\displaystyle \dim(P_{0}\vee P_{1}\vee \ldots \vee P_{n})=n}$

A - tuples of points of a projective space is projective base of the space, if ever points are projectively independent. It then applies . ${\displaystyle (n+2)}$ ${\displaystyle (P_{0},\ldots P_{n+1})}$ ${\displaystyle n+1}$ ${\displaystyle \dim P(V)=n}$

Special cases

• ${\displaystyle n=1}$: three points on a projective line form a projective base if and only if they are different in pairs.
• ${\displaystyle n=2}$: four points on a projective plane form a projective base if and only if no three of them lie on a straight line. The four points define a complete square .
• ${\displaystyle n=3}$: five points in a three-dimensional projective space form a projective base if and only if no four of them lie in one plane.

Standard projective base

The standard projective basis in standard projective space consists of the points generated by the standard basis vectors of the coordinate space ${\displaystyle (E_{0},\ldots ,E_{n+1})}$ ${\displaystyle KP^{n}}$ ${\displaystyle e_{0},\ldots ,e_{n}}$ ${\displaystyle K^{n+1}}$

${\displaystyle E_{i}=K\,e_{i},~i=0,\ldots ,n}$,

along with the unit point

${\displaystyle E_{n+1}=K\,\left(e_{0}+\ldots +e_{n}\right)}$.

In homogeneous coordinates arise, for example the following standard projective bases:

• In the projective straight line over a body , the points and form the projective standard basis. ${\displaystyle KP^{1}}$ ${\displaystyle K}$ ${\displaystyle 3}$ ${\displaystyle \left[1:0\right],\left[0:1\right]}$ ${\displaystyle \left[1:1\right]}$
• In the projective plane above a body , the points and form the projective standard base. ${\displaystyle KP^{2}}$ ${\displaystyle K}$ ${\displaystyle 4}$ ${\displaystyle \left[1:0:0\right],\left[0:1:0\right],\left[0:0:1\right]}$ ${\displaystyle \left[1:1:1\right]}$

${\displaystyle \ldots }$

• In the -dimensional projective space above a body , the points and form the projective standard basis. ${\displaystyle n}$ ${\displaystyle KP^{n}}$ ${\displaystyle K}$ ${\displaystyle n+2}$ ${\displaystyle \left[1:0:\ldots :0\right],\left[0:1:\ldots :0\right],\ldots ,\left[0:0:\ldots :1\right]}$ ${\displaystyle \left[1:1:\ldots :1\right]}$

use

If there is any projective basis of projective space , then there is a basis of such that ${\displaystyle (P_{0},\ldots ,P_{n+1})}$ ${\displaystyle P(V)}$ ${\displaystyle (v_{0},\ldots ,v_{n})}$ ${\displaystyle V}$

${\displaystyle P_{0}=K\,v_{0},\ldots ,P_{n}=K\,v_{n}~{\text{und}}~P_{n+1}=K\,\left(v_{0}+\ldots +v_{n}\right)}$

applies. Now and are two projective spaces of the same dimension with projective bases and , then there is exactly one projective mapping such that ${\displaystyle P(V)}$ ${\displaystyle P(W)}$ ${\displaystyle (P_{0},\ldots ,P_{n+1})}$ ${\displaystyle (Q_{0},\ldots ,Q_{n+1})}$ ${\displaystyle f\colon P(V)\to P(W)}$

${\displaystyle f(P_{i})=Q_{i}}$

for applies. Accordingly, a projective mapping between projective spaces of the same dimension is clearly characterized by specifying the images of the projective base points. Such images can therefore be described by size matrices . Furthermore, in a projective space with the projective base, with the aid of the projective mapping ${\displaystyle i=0,\ldots ,n+1}$ ${\displaystyle (n+1)\times (n+1)}$ ${\displaystyle P(V)}$ ${\displaystyle (P_{0},\ldots ,P_{n+1})}$

${\displaystyle KP^{n}\to P(V),E_{i}\mapsto P_{i}~{\text{für}}~i=0,\ldots ,n+1}$

define homogeneous projective coordinates .

literature

• Gerd Fischer : Analytical Geometry . 3. Edition. Springer, 2013, ISBN 978-3-322-96417-5 , pp. 142 .

Individual evidence

1. ↑ Gerd Fischer: Analytical Geometry . 3. Edition. Springer, 2013, p. 142 .
2. ↑ ^{a b c} Gerd Fischer: Analytical Geometry . 3. Edition. Springer, 2013, p. 143 .
3. ↑ Gerd Fischer: Analytical Geometry . 3. Edition. Springer, 2013, p. 144 .