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This concept is based on a kinematics concept of a run.. Also, a concept of an oriented angle in such a space is considered.. Next, it is shown that the adequacy of these concepts holds

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To Andrzej Zajtz, on the occasion of His 70th birthday

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The concept of a smooth oriented angle in an arbitrary affine space is introduced This concept is based on a kinematics concept of a run Also, a concept of an oriented angle in such a space is considered Next, it is shown that the adequacy of these concepts holds if and only

if the affine space, in question, is of dimension 2 or 1

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Let us consider an arbitrary affine space, i.e a triple

(see [B–B]), where E is a set, V is an arbitrary vector space over reals and →

way that

all vectors of the vector space V will be denoted by V The fact that W is a

AMS (2000) Subject Classification: 51N10, 51N20, 51L10.

A subset H of E is a hyperplane in an affine space (0) iff there exist p∈ E

space

1 in V We say that H is a hyperplane of codimension k in the affine space (0)

is said to be a halfspace of (0) The hyperplane H in (3) uniquely determined

opposite one to P It is easy to verify that (3) yields also

e∈B

where the sign of addition in (6) denotes of course a finite operation This formula will be very useful

For any topology T (see [K]) the set of all points of T will be denoted by T , i.e by definition we have

For any set A ⊂ T the induced to A topology from the topology T will be

For any affine space (0) the smallest topology containing the set of all sets

(0) and denoted by top(E, V, →) It is easy to check that for any hyperplane

H in (0) we have

a set of ordered pairs, and then

and

f tends to p at t in the affine space (0) and we write

Proposition 1

−−−−→pf (x)

For any vector space V we have well defined the affine space aff V as

Setting

where for any set A⊂ R, A0 denotes the set of all cluster points of A, we have

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o

Œ0`Zf0hŽJgHf0‘h0de’0’

V“

’0`Zc a0f

a0‘”gHf0•0b a0h

Before introducing the concept of smooth oriented angle in an arbitrary affine space we introduce a concept of a run and a turn Any function f

or

is said to be an o-turn in (E, V, →) The set of all o-turns in (E, V, →) will

such that

(ii) λ(t) > 0, ϕ0(t) > 0 and−−−−−−→og(ϕ(t)) = λ(t)−−−−→of (t) for t

o∈E

Any element of this set is said to be a smooth oriented angle in the affine space (E, V, →)

Proposition 2

p∈gD g

where

Proof Let f ∈ a We have then f ≡o g Taking any q ∈ fDf we get

og(t)

Proposition 3

it follows that µ is continuous Thus, µ is bounded So, there exists m > 0 such

0 < s < mβ

Proposition 4

2 we get

→ [

p∈U \{o}

Now, we will prove that A⊂ (o q ∞) Assume that there exists a point x ∈

p∈P +

of a, respectively

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a0‘zgHf0•0b a0h

ho f (t) ∞) is defined by (13), and one of the following two conditions

(2 L) for any t∈ DLthere exists δ > 0 for which

o∈E

ho; E, V, →)

angles in the affine space (0) The point o such that the equality in (L) is satisfied depending only on the oriented angle for which L belongs is called the vertex of this oriented angle Any oriented angle for which constant function L belongs is said to be zero angle in the affine space (0)

Proposition 5

For any smooth oriented angle a in the affine space (0) we have the oriented angle <a> well defined by the formula

hE, V, →)/ ≡ for L ∈ hE, V, →) The function

is 1–1 If dim V > 2, then there exists an oriented angle in (0) which is not of the form <a>, where a is a smooth oriented angle in (0)

Lemma

in V and

g1(x) f2(x)B(e1)

g2(x) f2(x)B(e2)

vB(e1) δ21

vB(e2) δ22

and

f1(x)B(e1) g1(x)

f1(x)B(e2) g2(x)

δ11vB(e1)

δ12vB(e2)

= c2,

Proof of Proposition 5 Correctness of the definition of <a> by (14) is

This yields, in turn,

of (t) and

−−−−−−−→

of (t + s) −−−→s−→0 −−−−→of (t) 6= 0

also

1

ϕ(t+s)−ϕ(t)

−−−−−−−−−−−−−−→g(ϕ(t))g(ϕ(t + s))

s

−−−−−−−−−→f(t) f(t + s) ,

1 ϕ(t+s)−ϕ(t)

−−−−−−−−−−−−−−→

and

1 s

−−−−−−−−−→

First, we consider the case when o–turns f and g satisfy conditions (o2f ) and (o2g), respectively Then by Lemma we have

Thus,

1

s

−−−−−−−−−−→

indepen-dent in V Let us set

f0(t1) = α0

1(t1)e1+ α0

2(t1)e2= β0

2(t1)e2+ β0

3(t1)e3

3(t1) So, f0(t1) = α0

−−−−−→of (t

1) = λ(t1

1 )

−−−−−−−→og(ϕ(t

1)) = λ(t1

1 )

λ(t 1 )e2

i.e −−−−→og(u) = λ(ϕ−1(u))−−−−−→of (t

Ÿ ]_™"`Zc a0f

a0‘”gHf0•0b a0hc f”gHf 0Œ0¡Sb c ‘0a0gHf¢0b gHf0a

Let us consider an Euclidean plane, i.e an affine space (0), dim V = 2,

|f| (t) = sup

X

i=0

−−−−−−−−−→

f (ti)f (ti+1)  ; a = t0< < tk= t & k∈ N

)

Proposition 6

ho f (·) ∞) ∈ A,

−−−−→of (s)

the oriented angle defined by (14)

−−−−−−→

oh(a+s)

−−−−−−−→

we see that

and

ho f (·) ∞) = (s 7→ L(a + s)) ∈ A

The condition (0; f ) holds in this case From (0; f ) it follows (21) In the

that τ1< < τl and DL⊂Sl

2 , bj = τj+δτj2

We have then 1–1 functions

−−−−→

oh(t) 

−−−−→

|g| (t) ≤ |g| (b) ≤

l

X

j=1

−−−−→of (s) = cos s

A = <a>

To prove that f is uniquely determined we take a continuous function

1(t)  = 1

1(ϕ(s))  = λ(s)−−−−→of (s)

Corollary

If (0) is an affine plane, i.e dim V = 2, then the function in (15) is 1–1 and

Indeed, taking any positively defined scalar product in V we get an Eu-clidean space and we may apply Proposition 6

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The case when the affine space is 1-dimensional is not of importance however

Remark

Proposition 5, Corollary to Proposition 6 and the above Remark allows us

to conclude our consideration by

Theorem

For any affine space (0) the function (15) is 1–1 This function maps the set soa(E, V, →) of all smooth oriented angles in the affine space (0) onto the set

hE, V, →)/ ≡ of all oriented angles in (0) if and only if dim V = 2 or dim V = 1

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[B–B] A Bia lynicki-Birula, Linear Algebra with Geometry (in Polish), Biblioteka Matematyczna [Mathematics Library] 48, PWN, Warszawa, 1974

[K] J.L Kelley, General Topology, D Van Nostrand Company, Inc., Toronto – New York – London, 1955

Department of Mathematics University of L´od´z

Banacha 22 90–238 L´od´z Poland

1< /small>(t1< /small>)e1< /small>+ α0

2(t1< /small>)e2= β0

2(t1< /small>)e2+ β0...

f1< /small>(x)B(e1< /small>) g1< /small>(x)

f1< /small>(x)B(e2) g2(x)

δ11 vB(e1< /small>)...

vB(e1< /small>) δ 21< /small>

vB(e2) δ22

and

f1< /small>(x)B(e1< /small>)