Projective coordinates

A one-to-one correspondence between the elements of a projective space $ \Pi _ {n} ( K) $ (projective subspaces $ S _ {q} $) and the equivalence classes of ordered finite subsets of elements of a skew-field $ K $. Projective coordinates of subspaces $ S _ {q} $ for $ q > 0 $ (also called Grassmann coordinates) are defined in terms of coordinates of the points ( $ 0 $-dimensional subspaces) lying in $ S _ {q} $. Therefore it suffices to define the projective coordinates of the points of a projective space.

Suppose that in the collection of rows $ ( x ^ {0}, \dots, x ^ {n} ) = x $ of elements of a skew-field $ K $ that are not simultaneously zero (they are also called homogeneous point coordinates) a left (right) equivalence relation is introduced: $ x \sim y $ if there is a $ \lambda \in K $ such that $ x ^ {i} = \lambda y ^ {i} $ ($ x ^ {i} = y ^ {i} \lambda $), $ i = 0, \dots, n $. Then the collection of equivalence classes is in one-to-one correspondence with the collection $ {\mathcal P} $ of points of the projective space $ P _ {n} ^ {l} ( K) $ (respectively, $ P _ {n} ^ {r} ( K) $). If $ {\mathcal P} $ is interpreted as the set of straight lines of the left (right) vector space $ A _ {n+1} ^ {l} ( K) $ (respectively, $ A _ {n+1} ^ {r} ( K) $), then the homogeneous coordinates of a point $ M $ have the meaning of the coordinates of the vectors belonging to the straight line that represents this point, and the projective coordinates have the meaning of the collection of all such coordinates.

In the general case, projective coordinates of points of a projective space $ \Pi _ {n} $ relative to some basis are introduced by purely projective means (under the necessary condition that the Desargues assumption holds in $ \Pi _ {n} $) as follows:

A set $ \sigma _ {n} $ of $ n+ 1 $ independent points $ A _ {0}, \dots, A _ {n} $ of the space $ \Pi _ {n} $ is called a simplex. In this case the points $ A _ {0}, \dots, A _ {i-1} , A _ {i+1}, \dots, A _ {n} $ are also independent and determine a subspace $ S _ {n-1} = \Sigma ^ {i} $, which is called a face of this simplex. There exists a point $ E $ that lies on none of the faces $ \Sigma ^ {i} $. Let $ i _ {0}, \dots, i _ {n} $ be any permutation of the numbers $ 0, \dots, n $. The points $ A _ {i _ {k+1} }, \dots, A _ {i _ {n} } $, $ k \geq 0 $, and $ E $ turn out to be independent and determine some $ S _ {n-k} $. Next, the points $ A _ {i _ {0} }, \dots, A _ {i _ {k} } $ also determine some $ S _ {k} $, and since the sum of $ S _ {k} $ and $ S _ {n-k} $ is the entire space $ \Pi _ {n} $, $ S _ {k} $ and $ S _ {n-k} $ have exactly one common point $ E _ {i _ {0} \dots i _ {k} } $ that lies in none of the $ ( k- 1) $-dimensional subspaces determined by the points $ A _ {i _ {0} }, \dots, A _ {i _ {j-1} } $, $ A _ {i _ {j+1} } , \dots, A _ {i _ {k} } $; in this case the points $ E _ {i _ {0} \dots i _ {k} } , A _ {i _ {k+1} }, \dots, A _ {i _ {n} } $ are also independent. Thus one obtains $ 2 ^ {n+1} - 1 $ points $ E _ {i _ {0} \dots i _ {k} } $, including the points $ E _ {i} = A _ {i} $, $ E _ {0 \dots n } = E $, which constitute a frame of the space $ S _ {n} = \Pi _ {n} $; the simplex $ \sigma _ {n} $ is its skeleton.

On each straight line $ A _ {i} A _ {j} $ there are three points $ A _ {i} , A _ {j} , E _ {ij} $; suppose they play the role of the points $ O , U , E $ in the definition of the skew-field $ K $ of the projective geometry under consideration (see Projective algebra). The skew-fields $ K ( A _ {i} , E _ {ij} , A _ {j} ) $ and $ K ( A _ {k} , E _ {kl} , A _ {l} ) $ are isomorphic to one another, and the isomorphism is established by a projective correspondence $ T _ {ij} ^ {kl} $ between the points of the two lines $ A _ {i} A _ {j} $ and $ A _ {k} A _ {l} $ such that the points $ A _ {k} , E _ {kl} , A _ {l} $ correspond to the points $ A _ {i} , E _ {ij} , A _ {j} $. The element of the skew-field $ K $ corresponding to a point $ P $ of the straight line $ A _ {i} A _ {j} $ is called the projective coordinate $ p $ of the point $ P $ in the scale $ ( A _ {i} , E _ {ij} , A _ {j} ) $. In particular, the projective coordinate of $ E _ {ij} $ is always 1, while the projective coordinate of $ P $ in the scale $ ( A _ {j} , E _ {ij} , A _ {i} ) $ is $ p ^ {*} = p ^ {-1} $.

Let $ P $ be a point of the space that lies on none of the faces of the simplex $ \sigma _ {n} $: $ A _ {0}, \dots, A _ {n} $ that together with some point $ E $ forms a frame $ R $. If one uses the point $ P $ instead of $ E $ in the above construction of the frame, then one obtains a sequence of points $ P _ {i} , P _ {ij} , P _ {ijk}, \dots $ where $ P _ {i _ {0} \dots i _ {k} } $ lies in the subspace determined by $ A _ {i _ {0} }, \dots, A _ {i _ {k} } $ (but lies in none of the faces of the simplex $ \sigma _ {k} $ formed by these points). Let $ p _ {ij} $ be the coordinate of a point $ P _ {ij} $ (lying on $ A _ {i} A _ {j} $) in the scale $ ( A _ {i} , E _ {ij} , A _ {j} ) $. If the $ i , j , k $ are distinct, then

1) $ p _ {ij} p _ {ji} = 1 $;

2) $ p _ {ik} p _ {kj} p _ {ji} = 1 $.

Let $ x _ {0} $ be an arbitrary element of $ K $ different from zero, and let $ x _ {i} = x _ {0} p _ {i0} $, $ x \neq 0 $, $ i = 1, \dots, n $ (in this case it turns out that $ p _ {ij} = x _ {j} ^ {-1} x _ {i} $). Then the collection of equivalent rows determined by various elements $ x _ {0} $ gives the projective coordinates of the point $ P $ with respect to the frame $ R $.

Suppose that $ P $ lies in the subspace $ S _ {k} $ determined by the points $ A _ {i _ {0} }, \dots, A _ {i _ {k} } $ but in none of the faces of the simplex determined by these points. Let the collection of equivalent rows $ ( x _ {i _ {0} }, \dots, x _ {i _ {k} } ) $ be the projective coordinates of a point $ P $ with respect to a frame $ R $ of the subspace $ S _ {k} $ determined by a simplex $ \sigma _ {k} $ and a point $ E _ {i _ {0} \dots i _ {k} } $. Then the projective coordinates of the point $ P $ with respect to the frame $ R $ are given as follows: $ y _ {i} = x _ {i} $, $ i = i _ {0}, \dots, i _ {k} $; $ y _ {i} = 0 $, $ i \neq i _ {0}, \dots, i _ {k} $.

Any collection of $ n+ 1 $ left (right) equivalent rows constructed by the above method corresponds to one and only one point $ P $ of the space $ \Pi _ {n} $ and therefore defines projective coordinates in it.

References

[1]	W.V.D. Hodge, D. Pedoe, "Methods of algebraic geometry" , 1 , Cambridge Univ. Press (1947) MR0028055 Zbl 0796.14002 Zbl 0796.14003 Zbl 0796.14001 Zbl 0157.27502 Zbl 0157.27501 Zbl 0055.38705 Zbl 0048.14502

Comments

For transformations of projective coordinates see e.g. Collineation; Projective transformation.