The most of the geometrical notions of this chapter will be studied in spaces for which the notions of point and vector are indispensable. The notion of affine space allows using the two notions in a well-defined environment.
§1. Definition and Examples
Let’s consider the non-empty set A = and agree to name each of its elements points, and bipoint of A any element (A, B) I A A . Point A will be named the origin of the bipoint (A, B). Bipoints (A, B) and (B, A) will be named symmetric bipoints.
It is called affine space the triplet A, V, j), where A is a non-empty set of points, V an K-vector space, and the function j AA A, ,which satisfies the following conditions:
A A, B, C I A j (A, B) + j (B, C) = j (A, C)
A A there is a point B I A, unique determined by the relation.
The set A is named support set of the affine space, and its elements will be named points of the affine space. The vector space V is named the directorial vector space of the affine space, and its elements will be named vectors of the affine space. The function j is named the affine structure function.
The elements of an affine space are points and vectors.
The affine space (A, V, j) is named real or complex by the vector space V which can be real or complex.
If we consider A = B = C in the axiom A1) , then j (A, A) =, A I A. Therefore, thru the structure function, to any bipoint (A, A) corresponds the null vector.
The corresponding vectors of a pair of symmetrical bipoints are opposite vectors. Indeed, if we consider C = B, in axiom A1), we have
j (A, B) = - j (B, A).
Function j is surjective and moreover,
for each fixed point O I A jO A V ,
jO (A) = j (O, A), A I A, is bijective.
The proof followes considering axioms A1) and A2).
In an affine space (A, V, j), the function j determines an equivalence relation on the bipoints set of A, which we will name relation of equipollence.
We will say that the bipoint (A, B) is equipollent with the bipoint (C, D) if they have the same image thru j
(A, B) ~ (C, D) j (A, B) = j (C, D) (1.1)
It is easy to verify that the relation “~” is reflexive, symmetric and transitive, meaning it is a relation of equipollence over A A.
The factor space A A is in a bijective correspondence with the vector space V. To each vector V, corresponds a single equivalence class of equipollent bipoints, and that is:
() = (1.2)
When we identify the factor space A A with the vector space V thru this bijection, the class of the bipoint (A, B) denoted by , is named free vector of the affine space.
Considering this, the axioms A1) and A2) can be written as follows:
A, B, C I A (1.3)
A, BI A unique such that
Let O IA be a fixed point and A A = the set of the bipoints with O as origin.
Taking into consideration the Consequence 1.2 and that the relation A I A (O, A)IA° is a bijective correspondence, it results that A° can identify itself as well with A as with the directorial vector space V.
When A° is identified with the vector space V, a vector structure from V is induced on A°. The vectors of this space are called tied vectors of the affine space, or tangent vectors in O to A, and will be denoted by .
When A is identified with the vector space A° thru the bijection A I A (O, A) I Ao, it means that A was considered a vector space having as origin the point O.
The vector will be named position vector.
Actually in any point O I A of an affine space (A V j), a vector space A°, which identifies with A°, can be constructed.
Following these identifications, the notion of dimension of an affine space is justified as being the dimension of the directorial vector space V.
If dimV= n, then the affine space of dimension n will be denoted by (A n, Vn), shortly A n
° If the vector space V is an euclidean vector space, then the affine space (A, V, j) is called euclidean punctual space. If dimV=n, then we will denote by En the corresponding euclidean vector space. The euclidean structure of the directorial vector space V will allow studying the metric properties of certain subsets of the euclidean punctual space En
There are affine spaces that are not vector spaces. But, any vector space is an affine space, because the function j: V V V, verifies the axioms A1) and A2). The affine space (V, V, j) such defined is named canonical affine space associated to the vector space V.
1° The standard affine space
Let’s consider the standard space Kn. This space can be organized as a vector space for which we can associate the canonical affine space (Kn, Kn, j ), where the affine structure function j is defined by the relation j ( A, B) = ( b1 - a1, b2 - a2 , , bn - an ) , for A = ( a1, a2 , , an ) and B = (b1, b2, , bn). This affine space is named standard affine space and will be denoted with Kn as well.
Particularly for K = R, we have the affine standard space (Rn, Rn, j) where the directorial vector space Rn is an euclidean space, therefore the affine space (Rn, Rn, j) becomes an euclidean punctual space.
2° The geometrical affine space of the free vectors
Let’s consider as, support set, the punctual space of the elementary geometry, denoted by E , as directorial vector space , the vector space of the free vectors V and as affine structure function j : E3 E3 V j(A,B) =V
In this way we obtain the geometrical affine space of the free vectors A = (E3, V j). This space constituted the model for the affine spaces. We will study in detail this space in the next chapter.
3° The linear variety of a vector space V are affine spaces.
A linear variety of a vector space V is a subset L, , where V is a vector subspace of V.
If we consider the function:
j : L L V , ,
then the axioms A1) and A2) are satisfied and, therefore, the triplet (L, V j is an affine space.
In particular, any vector subspace is an affine space.
A cartesian frame is a pair R , where O is a fixed point in An, and B is a basis of the directorial vector space.
Let B be a basis of the vector space Vn. Then, for each point P I An, the position vector can be written in an unique way as follows:
The scalars x1, x2,, xn are called cartesian coordinates of the point P with respect to frame R , and the bijection is called function of coordinates with respect to frame R
Let R be a cartesian frame in An . Another frame R of An will be determined in an unique way if we know the position vector of the point O with respect to the initial frame R and the relation between the basis B and the initial basis
B , that is:
If P I An is a point and (xi), (x’j), i,j = are its coordinates in the frame R respectively R , then from the relation we obtain the formulas:
called the coordinate transformation formula.
By denoting X = t[x1, x2, , xn], X = t[x , x , , x n], A0 = (ai0),
A = (aij) we can write the coordinate transformation formula in the following form:
The matrix of n + 1 rank is called the matrix of change of frames from R to R
In particular, if B B then A = I and the equations (2.11) are written as:
The changing of the frame R with R governed by the coordinate transformation formula (11) is called translation.
If O = O, then changing the frame R with the frame R , meaning ai0 = 0, , is called centro-affinity and is characterised by the following equations:
Let (A , V, j) be an affine space, A a non-empty subset of A and j its restriction to A A If V j A A ) is a vector subspace of V then the axioms A1) and A2) are satisfied for the triplet
A , V j
We call affine subspace of the affine space
A , V, j a triplet (A , V j ) , with the properties:
-A A is a non-empty subset
- V j A A ) is a vector subspace of V
j is the restriction of j to A A
An affine subspace of the affine space (A , V, j) is determined either by the subset A A for which j A A ) = V V is a vector subspace, or by a point P0 I A and a vector subspace V V. In this case the support set is given by .
Let us consider A and two affine spaces.
An application t: A with the property
t aP bQ at(P) + bt(Q), P, Q I A and
a b I K, a b=1 is called an affine application.
An affine application t:A uniquely determines the morphism T: V between the associated vector spaces. Knowing that for and A I A, B I A such that j (A, B) = , we can define the associated linear map T : V by , where is affine structure function of the space . The definition does not depend on the choice of the point A.
The set of all bijective affine applications on A represents a group GA(A), called the affine group.
We will name object of the affine space A any subset of points belonging to A
By affine geometry of the affine space A, means the study of all objects, and their invariant properties with respect to the affine group.
The simplest and yet most important affine properties are:
the property collinearity of three points
the property of two affine spaces to be parallel
These mentioned affine properties aid in determining other properties, therefore they will be called fundamental affine properties.
§4. The geometrical affine space of the free vectors
Let us consider E3 the punctual space of the elementary geometry and V the vector space of the free vectors.
If we associate to any bipoint (A, B) I E3 E3 the free vectorI V , then the application j : E3 E3 V j (A, B) = satisfies properties A1) and A2) of the affine space definition:
A1) A, B, C I E3 ,
A2) I V A I E3 exists B I E3 uniquely determined by .
The triplet A (E3, V j) is called the affine geometrical space of the free vectors.
The elements of the affine space A are points and vectors. The points of the affine space A are points of the support set E3 and we will denote them by capital letters A, B, C, , O, P, , and the vectors of the affine space A are vectors of the directorial vector space V , free vectors that we’ll denote by , , , or by . The application j : E3 E3 V which satisfies axioms A1) and A2) represents the affine structure function, and the equivalence relation defined by it on the set E3 is exactly the relation of equipollence “~” of oriented segments, like it was defined in the euclidean geometry.
Consider O I E3 a fixed point. The application j : E3 E3 V defined by j (A) = j(O, A), A I E3 is bijective allowing the identification of the punctual space E3 with the vector space of the free vectors.
The point O I E3, corresponding to the null vector , will be considered as origin of the affine space A . Furthermore, IV3 there exists an unique point A I E3 determined by the relation . The vector is called the position vector of point A.
The set of the position vectors forms an isomorphic vector space with the vector space of the free vectors.
Lets consider now two distinct points A and B of the space E3. The affine subspace generated by A and B, L() = = d is a subspace of dimension one named affine line, shortly - line, having the directorial vector space the vector line
For each point P I d , collinear with A and B, the system of points is affine-dependent, that means the vectors and are linearly dependent.
We remind that two vectors with the same direction are named collinear vectors. The vectors of the subspace V have the same direction, which justifies the definition of collinearity from an affine space.
Two vectors and I V are collinear if and only if they are linearly dependent, meaning l m I R, such that .
Proof. Consider O I E3 a fixed point. Consider A, B I E3 such that and . If , are collinear then the points O, A, B are collinear, meaning that the system of points is affine dependent. This is equivalent with the linear dependence of the vectors and .
The oriented segments and are the representation of vectors and in O, meaning that are linearly dependent for any choice of point O.
Conversely, if and I V are linearly dependent, then O I E3, the vectors and are linearly dependent, meaning that the system of points is affine dependent. The collinearity of the points O, A, and B is equivalent with the fact that the vectors and have the same direction. (q.e.d.)
If is collinear with then we write:
, l I R (the condition of collinearity) (4.1)
of all collinear vectors with the non-null vector is a one-dimensional vector subspace.
Any three non-collinear points are affine dependent, meaning that any two non-collinear vectors are linearly dependent.
Three non-collinear points A, B, C I E3 determine a plane. The plane generated by these three affine independent points is determined by:
affine subspace having as directorial vector space the vector plane:
For any point PI p , the system is affine dependent, which means that the vectors , and are linearly dependent.
Three vectors , and are considered to be coplanar if they are parallel with a plane.
Three vectors , , I V are coplanar if and only if they are linearly dependent, namely,
l m n I R, l m n such that .
If , I V are two vectors, then I V coplanar with and , can be written as follows:
= , l m I R , (the condition of coplanarity) (4.2)
of all vectors coplanar with the non-null vectors and , is a two-dimensional vector space.
Consider now four non-coplanar points A, B, C, D I E3 . The system of points is affine independent, which means that any three non-coplanar vectors are linearly independent. The affine space generated by four non-coplanar points is of dimension three and any five points of this space will be affine dependent.
The vector space V has three dimensions.
Proof. Any four non-coplanar points form an independent affine system, which is equivalent with the existence of three non-coplanar (linearly independent) vectors ,,. Let us demonstrate that these three non-coplanar vectors generate the vector space of the free vectors V . For this, let be a fourth vector, O I E3 some point, and , , , the
representatives of the vectors , , , in the point O (fig. 1).
Using twice the parallelogram rule for summing two free vectors in the parallelogram OA1X1B and respectively OX1XC1, it results:
Because and , and , and are collinear then we have
namely is a basis for the vector space of the free vectors, therefore dimV = 3. q.e.d.
For any point O I E3 and a given basis in V , the ensemble R (O; ) represents a cartesian frame in the affine space A ( E3, V j
For any point P I E3 we have the position vector given by:
; x1, x2, x3 I R
The scalars x1, x2, x3 I R are called cartesian coordinates of point P.
If I V is a free vector, then P I E3 is unique, such that , and in the frame R is written as:
where the scalars x1, x2, x3 I R, the coordinates of point P, will be called the coordinates of vector in the frame R
If R(O; ) is a fixed cartesian frame, A is a geometrical affine space and x1, x2, x3 I R are the coordinates of vector I V , we will write that = ( x1, x2, x3) or briefly ( x1, x2, x3).
If we have ( x1, x2, x3) and ( y1, y2, y3) two free vectors, then:
1° is collinear with (||) if and only if their coordinates are proportional (or equal in the particular case of = ).
2° , , are coplanar if and only if the coodinates of one vector represents a linear combination of the other two.
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