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Συστήματα αναφοράς και χρόνου E ισαγωγικές έννοιες PowerPoint PPT Presentation


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2. Συστήματα αναφοράς και χρόνου E ισαγωγικές έννοιες. Γεωμετρικές και αναλυτικές μέθοδοι στα μαθηματικά. Descartes (1596-1650): συντεταγμένες για την περιγραφή σημείων του χώρου με αριθμούς Σύνδεση γεωμετρικών και αναλυτικών (= αριθμητικών) μεθόδων.

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Συστήματα αναφοράς και χρόνου E ισαγωγικές έννοιες

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2

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Descartes (1596-1650):

(= )

:

.. ,

(, , , )


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Descartes (1596-1650):

(= )

:

.. ,

(, , , )

:

.. y(x) (= x y)


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( )

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.

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:

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:

,

(= + ).


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:

( )

=

=

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( )

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B

P

N

:

: P n n q1(P), q2(P), ..., qn(P)

: .

,

( ) .

:

1=(P1)=2=(P2)=90,1=(P1)2=(P2)

1=2 ( )

1=2 = -90 1 2 1=2 ( ).


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q1, q2 = .

q1, q3 = .

q2, q3 = .

q1

=

.

( : ).

: q1(P), q2(P), q3(P)


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: q1(P), q2(P), q3(P)

:

.

.. (, , ):

P f(P)

:

f(q1,q2,q3),

(q1,q2,q3 = P).


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:

P (= + + )

(.. , , .)

( )

:

( P)

( ).

3

(.. 2 , 1 ).

3


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| |,

> 0 < 0

=

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()

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() ()

P q1(P), q2(P), q3(P)3

P

3 3 ( v1, v2, v3)


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()

:

() ()

P q1(P), q2(P), q3(P)3

P

3 3 ( v1, v2, v3)

:

() () P

q1, q2, q3


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()

:

() ()

P q1(P), q2(P), q3(P)3

P

3 3 ( v1, v2, v3)

:

() () P

q1, q2, q3

:

=

=

+

() x1, x2,x3

=

()


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: E .

( )

: 5

= .

(Lobatsevsky, Riemman) !

Riemman = .

: ( ).

: .

=

=


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: E .

( )

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5

( )

.

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( ).


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P: , O P.

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A

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T , !

C ( = A, B)

A B !

( C O).

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3 () x1, Ox2, Ox3,


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3 () x1, Ox2, Ox3,

1 : x2Ox3

P1: Ox1 1

x1= OP1 (+ , )


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3 () x1, Ox2, Ox3,

1 : x2Ox3

P1: Ox1 1

x1= OP1 (+ , )

2 : x1Ox3

P2: Ox2 2

x2= OP2 (+ , )


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3 () x1, Ox2, Ox3,

1 : x2Ox3

P1: Ox1 1

x1= OP1 (+ , )

2 : x1Ox3

P2: Ox2 2

x2= OP2 (+ , )

3 : x1Ox2

P3: Ox3 3

x3= OP3 (+ , )


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, , .

A ,

:

O Ox1, Ox2,Ox3 , , ( 1).


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E


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.

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=


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:

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:

|AB| A, B, :

:


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(): = 90 (cos = cos90= 0)

: (= 1)

E o =

= .

( ).

( ) .


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:

K :

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M :


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E :

E :

=


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E :

M:

:


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()

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( )

A 33 W :

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:

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Q ( Q-1 = QT , | Q | = 1 ) :

() S :


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3 3

O ( ) S: S

= ,

=

( S )

= S :

a, b, c=

E :


E

:

P

:

P :

( O O' P

:


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P ( ):


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. = k

=

R= (R-1 = RT, |R| = 1)

H :


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:

:

:


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: RT = R-1. : RT = R-1 | R | = 1

(2,3), (3,1) (1,2):

,( ).

( ) :

() () (2,3)

:

: .

33 R Ek:R' = REk R'' = EkR=

(): (| R | = 1 ).

(| R | = -1)

.


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: RT = R-1. : RT = R-1 | R | = 1

(2,3), (3,1) (1,2):

,( ).

( ) :

() () (2,3)

:

: .

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.

3

R=

R = R(a,b,c) 3 .

9 33 R 6 RRT = I

9 6 = 3 .

:6 9 9 RRT,

3 = 3 .

:

M

3 , 1, 2, 3.

R1(1), R2(2), R3(3) = 1, 2, 3.

:

: .


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.

p2 sin

p2 cos

p1 cos

p1 sin

p1, p2 = ,

p1, p2 =

( )


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R() :


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3

( ):


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.

3

( ):


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( ):

.

3


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.

3

1

2

3


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.

. Cardan.

E 1-2-3.

:

Cardan ( 1, 2, 3 ) = :

1, 2, 3 = ( R )


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.

. Cardan.

Cardan


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.

. Cardan.

Cardan


E

.

. Cardan.

Cardan


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.

. Cardan.

Cardan


E

.

. Cardan.

Cardan


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.

. Cardan.

Cardan


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.

. Cardan.

Cardan


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.

. Cardan.

Cardan


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Euler

.

. Euler.

Euler = :

, , = ( R )


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.

I


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.

I

:

!


E

:

=

=

:


E


E

:


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12 (6 Cardan 6 Euler)

4:

( ) 3 :

2 ( ) 3 (3) (3).

H 3 1 2 .

, :

(1,2,3) (2,3,1) (3,1,2)


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(1) Cardan

1, 3 1, 3.

3 2,3

( )

3, 1 3, 1.


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(2) Cardan

2, 3 2, 3.

3 1,3

3, 2 3, 2


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(3) Euler

3, 1 3, 3

3 3

3, 3 3, 1


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(4) Euler

3, 2 3, 3

3 3

3, 3 3, 2

, ,

(3):


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