Geometry for Didactics
146
ΕΥΣΤΑΘΙΟΥ Ε. ΒΑΣΙΛΕΙΟΥ ΓΕΩΜΕΤΡΙΑ ΓΙΑ ΤΗ ΔΙΔΑΚΤΙΚΗ ΣΗΜΕΙΩΣΕΙΣ ΑΘΗΝΑ 2012 ΠΑΝΕΠΙΣΤΗΜΙΟ ΑΘΗΝΩΝ
Transcript of Geometry for Didactics
-
.
2012
-
GEOMETRY FOR DIDACTICS
AMS (2000) Subject Classification:
01A20, 01A55, 01A60,
51A05, 51A10, 51E15,
53A04, 53A05
COPYRIGHT 2012 by Efstathios E. Vassiliou
All rights reserved
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1 1
1.1 . . . . . . . . . . . . . . . . . . . . . 2
1.2 . . . . . . . . . . . . . . . . . . . . 5
1.3 . . . . . . . . . . . . . . . . 8
1.4 Hilbert . . . . . . . . . . . . . . . . . . 11
1.5 . . . . . . . . . . . . . . . . . . 16
1.6 . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.7 . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2 27
2.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.1 . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.4 . . . . . . . . . . . . . . . . . . . . . . . . 46
2.5 . . . . . . . . . . . . 52
2.6 . . . . . . . . . . . . . . . . . . . . 59
2.7 P2 . . . . . . . . . . . . . . . . . . . . . . . . . 67
3 73
3.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.1 . . . . . . . . . . . . . . . . . . . . . . . . . 74
vii
-
viii
3.2 . . . . . . . . . . . . . . . . . . . . . . . 79
3.3 . . . . . . . . . . . . . . . . . . . . . . . . 81
3.4 . . . . . . . . . . . . . . 91
3.5 Frenet . . . . . . . . . . . . . . . 93
3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4 101
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
103
107
111
-
1
( )
180o -
, ( )
,
...
, ,
, -
K. F. Gauss (1824) [10, . 109]
(). , 2012.
-
2 1.
-
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,
( 13).
( 4)
() ,
, Hilbert
( 5).
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19 , 6 7 ,
.
1.1
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Kongruenz, congruence),
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1.
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, (, , ) 2 = 2 + 2. ,
() ,
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-
1.2. 5
, -
Rene Descartes
[] (15961650) Pierre de Fermat(16011665). ,
, -
.
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( ),
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. , , L. Kronecker
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[1].
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1.2
300 ..
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6 1.
,
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2- ,
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19 , .
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Ba-
ruch Spinoza (16321677). Immanuel Kant (17241804),
,
,
.
, a priori.
Karl Friedrich Gauss
(17751855), ( ) -
,
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[26].
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[36].
1. .
2.
.
3. -
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4. .
5. , ,
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1.3. 9
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28 .
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19 , [ 1868
Eugenio Beltrami (18351900 )].
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28 -
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. 5
.
1.
.
2. .
3. .
4. .
5. () , -
.
6. ,
.
7. ,
.
1 John Playfair (17481819) .
-
10 1.
5 , .
3 John Wallis (16161703).
5
(410485 ..), Giovanni Gerolamo Saccheri (16671733) (, -
,
) Adrien-Marie Legendre (1752 1833).
5 :
) -
.
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1.4. Hilbert 11
(. 2),
( )
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(5 .. ), ,
. 16
. ,
.
1.4 -
D. Hilbert
, ,
,
, ,
,
.
, , , 1 ,
AB.
A
AB B, .
1.1. .
; ,
. ,
-
12 1.
.
: (-)
, (.
.. 4) ,
(
,
.).
, ,
( ). ,
,
,
. , ,
, -
.
, Julius Wilhelm
Richard Dedekind (18311916), Giuseppe Peano (18581932), David Hilbert
(18621943) George David Birkhoff (18841944).
-
Hilbert (. [17]), (
) .
Hilbert
:
, (), , () ,
, / (Kongruenz, congruence)
, , :
I: ( ).
II: .
III: (Kongruenz, congruence).
IV: .
V: .
, , Hilbert (.
[10], [17], [36]).
.
.
,
, .
.1 A, B
. ( : .)
-
1.4. Hilbert 13
.2 A, B .
A, B, ,
= AB = BA
.3 . -
() .
.4 A, B, C, ,
.
.5 A, B ,
.
.6 ,
.
.7
.
.
,
,
.
.1 B A C, A, B, C
B C A.
.2 A, C,
B, A C.
.3 A, B, C ,
.
.4 A, B, C
, .
() AB,
AC, BC.
Pasch ( -
Moritz Pasch (18431931)).
: AB
A B. AB BA.
. (congruence)
, ,
.
.1 A B k A
, k,
A B, AB
AB.
-
14 1.
: AB AB( AB AB, AB = AB). : ( ) A B
C AB, A B C.
, A (
A), A. A
A.
.2 AB AB AB AB, AB AB. (congruence) -
. ( ) AB AB
.3 A, B k A, B
( k = k , ). C A, B, C
A, B, AC AC CB CB, AB AB.
:
h k () O,
, h, k,
O. (h, k). h P k Q ( h = OP k = OQ),
(h, k) POQ.
( ) -
, ,
.
( ),
, ..,
. , -
, (
).
, A, B, C -
AB, BC AC. ABC.
.4 (h, k) ( O)
( ).
. h
O, k ( O),
(h, k) (h, k) ( : (h, k) (h, k)) (h, k) .
.5 ABC ABC AB AB, AC AC BAC BAC, ABC ABC.
-
1.4. Hilbert 15
IV.
IV.1 ( , Playfair)
P ( P )
k .
V.
V.1 ( ) AB CD -
, AB A1,
A2, . . . , A, AA1 A1A2 A2A3 A1A CD B A1 A.
V.2 ( ) ,
, (. -
)
IIII V.1.
V.1 ( ) V.2 ( ) -
() Dedekind :
,
,
.
-
,
[17].
Hilbert ,
,
. -
,
, , .
, (
) -
( ).
2.
-
, , Henri
Poincare (1854-1912) : (. [33, . 161]):
... , , .,
.
, -
.
-
16 1.
, -
,
, ,
, .
... , Hilbert, ,
,
,
, , .
... ,
,
,
.
, Stanley Jevons,
.
:
... Les expressions "etre situe sur, passer par", etc., ne sont pas de-
stinees a evoquer des images; elles sont simplement des synonymes du
mot determiner. Les mots "point, droite et plan" eux-memes ne doivent
provoquer dans l esprit aucune representation sensible. Ils pourraient
indifferemment designer des objets d une nature quelconque, pourvu
qu on put etablir entre ces objets une correspondance telle qu a tout
systeme de deux objets appeles points correspondit un des objets ap-
peles droites et un seul.
... Ainsi M. Hilbert a, pour ainsi dire, cherche a mettre les axiomes sous
une forme telle qu ils puissent etre appliquees par quelqu un qui n en
comprendrait pas le sens, parce qu il n aurait jamais vu ni points, ni
droite, ni plan.
... On pourra ainsi construire toute la geometrie, je ne dirais pas precise-
ment sans y rien comprendre, puisqu on saisira l enchainement logique
des propositions, mais tout au moins sans y rien voir. On pourrait confier
les axiomes a une machine a raisoner, par example au "piano raisoneur"
de Stanley Jevons, et on en verrair sortir toute la geometrie.
1.5
,
:
-
1.5. 17
( , - , .).
( , , .). ( , Hilbert). ( ). , . (
, ).
, :
1. (consistent), -
. .
;
2. (independent),
() . .
3. (complete), , -
,
. , ,
.
, -
. ,
[ Kurt Godel(19061978)] -
,
.
() . , () -
,
.
.
. , -
( ) .
.
, , -
.
, , , -
. ,
, ( -
)
.
( ) :
-
18 1.
X S. S S, X , - M S. S M, X . , X , S, M, M X . ,
.
,
2 .
1.6
,
, 19 -
. ,
5 ( -
). ,
, .
I. Kant 1808, -
,
.
1.3,
5 1868 E. Beltrami, -
(. ).
:
S S 5 . S () , - , S, , , 5 .
( )
, -
.
(. 10) ,
5 ,
:
,
.
. -
( 5 ,
-
1.6. 19
).
( -
) ,
, .
Janos W. Bol-
yai (18021860) Nikolai I. Lobachewsky (17931856), -
( ) 1829
1832 . K. F. Gauss (17751855)
,
, , -
. Gauss
, ,
,
,
,
. -
.. [7], [10], [14] [19].
-
Gerolamo Saccheri
(16671773), . -
5
! , (
,
)
. , , -
(
), ,
, . Saccheri
, -
, , ,
.
-
E. Beltrami 1868.
, 1.2 .
Gauss, -
, ( -
[6]).
-
,
. -
() . ,
-
20 1.
, -
,
.
P (. ).
P, ,
.
.
1.2. ( [10]).
, 2, , ,
.
,
.
(. [6].
C ,
.
, -
. Hilbert
. -
Beltrami
.
Klein-Beltrami, -
. -
-
1.6. 21
, (
) 1871.
1.3. Klein-Beltrami ( [5])
-
() ,
. , P
AB ( A, B).
-
() ,
.
2
, ,
, .
(
180o).
Poincare
1.4. Poincare ( [10])
( ),
.
-
22 1.
.
. -
Poincare
, [10, . 164170].
1.7
Georg Friedrich Bernhard Riemann (18261866).
10,
. , 1, 2, 3, 4
5
: () ,
, 2 ,
! ,
, 1, 3, 4
, 5 ,
2
2:
. ,
.
,
! ,
-
S1 S2,
:
S1: .
S2: () .
,
1, 2, 3, 4, S1 -
S2 ,
1, 2, 3, 4, S2
S1 -
.
, (
).
, , 1.
-
1.7. 23
( )
. . (
1.5)
1.5. ( [9])
-
( R3),
,
. , -
.
. ,
. ,
.
, ,
1.6,
() .
(.
), .
1.5. ( [9])
,
-
24 1.
. , :
() -
() -
()
()
-
-
, ,
-
.
-
() . ,
Riemann
( Gttingen 1854) "Uber die Hypothesen
welche der Geometrie zu Grunde liegen (:
).
Poincare (. .
15), Hilbert,
( ..
,
Poincare
.).
Riemann,
, , (. -
) . , Riemann
(manifold), -
. Albert
Einstein (18781955) Riemann
. ,
( )
( ),
.
-
1.8. 25
( ) ( -
). ,
(gauge theory), -
(string theory) . ,
,
.
, -
, .
, , -
, : J. N. Cederberg [9], R. L. Faber [14], .
[38] (
) M. Spivak [35] [ , ,
Riemann (:
),
] H. Poincare [33] ( ,
). , ,
, , -
E. T. Bell [7], S. Hollingdale [18], L. Mlodinov [25], .
[27], J. Pierpont [32], D. J. Struik [39], I. M. Yaglom [41].
.
[7], [9], [10] [36], .
1.8
1. Saccheri (. [10, . 104107]).
2.
Poincare (. [10, . 164170]).
3. (. [10, . 138139
143162]).
-
2
-
17 -
. . . , -
(-
) Desargues 1639,
-
.
E. T. Bell [7, . 158]
,
1 2 Hilbert -
.
27
-
28 2.
,
, .
3 . ,
, ,
.
( 4) -
( , ) -
.
5 .
, ( )
, . ,
-
.
, 6 -
. ,
.
2.0
, -
.
,
. , .
-
, ().
-
, ,
Filippo Brunelleschi (13771446), Paolo
Uccello (13791475), Leone Battista Alberti (14041472), Pierro de la Francesca
(14161492), Sandro Botticelli (14451510), Leonardo da Vinci (14521519), Al-
brecht Durer (14711528), Michelangelo Buonarroti (14751564), Raffaello Santi
Sanzio (14831520) .. -
-
,
.
, ,
. , [ ( ) ] () (E), . P (E) P , OP ( O
-
2.0. 29
) ( ). R R .
2.1
, , P
(E) ( ), P
AB (. 2.2), ( ) , O
(E). AB ,
( ).
2.2
-
30 2.
, [ (E) ], ,
AB (. ).
2.3
, ,
(
CD, AB
,
).
, 2.3, -
:
.
, AB.
. , Gio-
vanni Antonio Canal, Canaletto (16971768),
. -
.
, .
, Andrea Mantegna (1431;1506) -
. , -
, ( ) . ,
,
, .
-
2.0. 31
-
32 2.
-
: O ,
( ) , , - , ( )
(E). [
( )]. 2.4 , ,
.
2.4
2.5
, ( )
, ,
.
17
. 18 ,
, -
.
19 .
Girard Desargues (15911661), Blaise Pascal
(16231662), Philippe de la Hire (16401718) ..,
Gaspar Monge (1746 1818), Jean-
Victor Poncelet (1788 1867), Charles Brianchon (17851864), August Ferdinand
Mobius (17901868), Jacob Steiner (17961863), Julius Plucker (18011868)
Karl Georg Christian von Staudt (17981867).
: . -
, .
. -
, ,
-
2.1. 33
,
.
, -
. Sophus Lie (18421899), Henrie Poi-
ncare (18541912) , , Christian Felix Klein (18491925)
.
2.1
, -
, , , .
, -
, ,
, () -
. , ,
. -
() .
D. Hilbert [17] :
P , , (points)
A, B, C, . . . , P, Q, . . . , ,, . . . , X, Y.
L , , (lines)
a, b, c . . . , k, , . . . , , , . . . , x, y.
I P L, I P L, (incidence).
.
.
,
.
P L = .
, (P, k) P L, - (P, k) I (P, k) < I.
-
34 2.
(P, k) I () :
P ( ) k,
P k,
k P.
. ,
P k. (),
.
,
(. 2.6.8, ,
).
.
2.1.1 . (P,L,I) . Pi P (collinear), k L,
(Pi , k) I, i = 1, 2, 3, . . .
, ki L ( ) (concurrent lines) P P,
(P, ki) I, i = 1, 2, 3, . . .
, k, L (parallel), - k//, :
k = ,
k , P P : (P, k) I (P, ).2.1.2 . (affine plane)
(P,L,I), :( 1) (
). ,
:
[ (P, Q) P P : P , Q ] [ ! k L : (P, k) I (Q, k) ].
( 2) ( ).
. ,
[ (P, k) P L : (P, k) < I ] [ ! L : (P, ) I, k// ].
( 3) ,
.
-
2.1. 35
( 2), , J. Playfair
5 () .
, [36].
2.1.3 . 1) () -
. ,
,
I , (). ,
(P, ) I P .
2)
P := {A, B, C, D},L := {{A, B}, {A, C}, {A, D}, {B, C}, {B, D}, {C, D}}
I , (P,L,I) - .
, .
3) () Klein-Beltrami, 20.
2.1.4 . k ( 1)
(join) P, Q
k = P Q = Q P
( 1).
(P,L,I)
2.1.5 . A, B, C
,
= A B = A C = B C.
. 2.1.1
(A, ) I (B, ) I.
A , B, ( 1) A B
(A, A B) I (B, A B).
, ( 1), = A B. .
-
36 2.
2.1.6 . k, ,
P (P, k) I (P, ) I.
. k, l (k//), ( k , ) P, .
k //, P (
). Q,
(Q, k) I (Q, ). P = Q. , P , Q, ( 1) P Q. , P, Q k , ( 1) ,
k = P Q = ,
k , . , , -
P = Q.
2.1.7 . k ,
( 2.1.6) (intersect-
ion) k,
P = k = k
2.1.8 . -
.
. -
. , k// //m.
k = m, . k , m, k//m,
P (. P = k m),
(P, k) I, k//,(P, m) I, m//.
(P, ) < I ( //k //m). , P, - , ( k m).
( 2), k//m.
2.1.9 . -
, .
. ( 3) A, B, C,
, AB B C . , (C, AB) < I. , ( 2), L (C, ) I //A B. m L, (A, m) I m//B C.
-
2.1. 37
, m. , = m, A (
m) , , //AB. , //m. , //m, //A B, m//A B, . ,
A m A B. , m , D = m (. 2.1.6 2.1.7).
, , (-
) .
2.5
, D -
A, B, C .
D , A (
B, C). , D = A,
A B A = D ()., D ,
A B. , 2.1.5, A, B, D
A B. , D A B (, ). D
, .
2.1.10 . 1) 2.1.9 -
4.
4
2.1.3(2).
2) , , -
( ) (. [3]). ,
( ), , ,
.
2.1.11 .
1. 2.1.3(1) 2.1.3(2).
-
38 2.
2.
.
3. -
.
4. k, , m k//.
m , .
2.2
, (P,L,I), P L , P L = , I P L . P L , , .
2.1.2.
2.2.1 . (projective plane)
(P,L,I), :( 1) P, Q P, P , Q, L, (P, ) I (Q, ).( 2) k, L, k , , P P, (P, k) I (P, ).( 3) ,
.
2.2.2 . 1) ( 1) ,
( , ) . ,
( 1) ( 1). , ( 2)
. :
.
2) ,
( 1) P, Q
= P Q = Q P.
3) , 2.2.4, -
P ( 2) . k,
P = k = k.
-
2.2. 39
2.2.3 . 1) :
P := {Ai |i = 1, . . . , 7},L := {1 = {A1, A2, A3}, 2 = {A1, A4, A6}, 3 = {A1, A5, A7},
4 = {A2, A4, A7}, 5 = {A2, A5, A6}, 6 = {A3, A4, A5}, 7 = {A3, A6, A7}}
I . (P,L, ) -
.
(
).
[. 2.4.9(2)]
7. 7
.
( !)
.
2.6
( )
,
,
.
, .
,
( )
.
-
40 2.
1 2 3 4 5 6 7
A1 A2 A3 A4 A5 A6 A7
2) () R3 :
P := {P U R3 : dimU = 1},
L := { V R3 : dimV = 2}, U R3 U R3 ( V R3). , I P L
(P, ) I U V, P = U = V.
(P,L,I) , 2 P2.
P2 R3,
0 (0, 0, 0), P2 R
3, 0.
2.7
( , ).
( ) -
(/ , R3 .),
( 1) ( 3).
-
(P,L,I).
-
2.2. 41
2.2.4 . P ( 2) .
. , Q , P
(Q, k) I (Q, ). , P, Q k , ( 1)
k = P Q = , , k , .
2.2.5 . A, B, C ,
, ,
= A B = B C = A C.
. 2.1.5.
2.2.6 .
.
. .
( 3),
. A, B, C, D,
. A B, A C A D, [: ,
()].
[: , .., A B = , (A, ) I ()].
2.8
, ( )
X = (A B), Y = (A C), Z = (A D).
X , Y , Z , X . , , .., X = Y ,
A B = A X = A Y = A C,
(. 2.2.5) , A B , A C. A, B, C, D ,
,
.
-
42 2.
2.2.7 .
, .
. A, B, C, D ( 3).
A B, B C, C D D A. [: A, B, C,
D ()].
. -
, , , O P A B, B C C D. , A B B C, B = O. , B C C D, C = O, B = O = C.
2.2.8 . O -
.
. , k, ,
m, n,
.
2.9
O.
P = k , Q = km R = kn (. 2.9), [: ,
()].
, O [: , ,
O = P, O k ()]. ,
x = O P, y = O Q, z = O R.
-
2.2. 43
x , y , z , x: , .., x = y,
P = k x = k y = Q,
, P , Q.
O k, , m,
n, ,
.
2.2.9 .
1. (P,L,I) :( 1) P, Q P P , Q, k L, (P, k) I (Q, k).( 2) k, L k , , P I, (P, k) I (Q, k).( 3) = ( 3).
) k ( 1) .
) (P,L,I);
2. P :
(P, ) < I. ;
3. . P (P, ) - . , ,
(x, y, z) R3 , [x, y, z] - (P,L, ): [x, y, z] , .
2.7.1 . P2 := (P,L, ) , P2.
. ( 1) [a1, b1, c1] [a2, b2, c2] P [a1, b1, c1] , [a2, b2, c2] (, R, (a2, b2, c2) = (a1, b1, c1)). () . ,
{a1x + b1y + c1z = 0
a2x + b2y + c2z = 0(2.7.8)
, () ,
1 (. [2, . 135], [4, . 249], [13, . 84]). ,
(x, y, z) , , , (x , y, z) = (x, y, z) R. (2.7.8) ,
( 1).
( 2)
L, [x, y, z].
{1x + 1y + 1z = 0
2x + 2y + 2z = 0
(2.7.8). , ,
() [x, y, z] ., [1,0, 0], [0,1,0], [0,0,1] [1,1, 1].
.
. , [1,0,0], [0,1,0] [1,1,1] , (2.7.7) (x, y, z) = (0,0,0), [. 2.7.2(4) ].
( 3).
(P,L, ) , - () P2 = (P,L,),
-
70 2.
R3, : P P : L L
P {t(a, b, c) | t R} [a, b, c] P,(2.7.9)
L (, , ) L.(2.7.10)
. (,) . , P ,
, P2 P ,
P = {t(a, b, c) | t R} = (, , ),
(a, b, c) (, , ) R3 .
P {t(a, b, c) | t R} (, , ),
t(a, b, c) t , 0 (2.7.7),
(ta) + (tb) + (tc) = 0 a + b + c = 0,
[a, b, c] . , ,
(P) = [a, b, c] = ((, , )),
(,) . , 1 1 (
), (,) P2 P.
: 2.7.1, P2 P2, P2 . P2 () :
2.2.3(2) [ R3],
, . -
.
2.7.2 P2.
(1) .
, R, (1, 0,1) = (0, 1,2). , [x, y, z] ,
{0x + 1y + 2z = 0
1x + 0y + 1z = 0
-
2.7. P2 71
x = z y = 2z.
[x, y, z] = [z,2z, z] = [z(1,2,1)].
z , 0 [ (x, y, z) = (0, 0,0), ],
= [1,2,1].
(2) = .
[x, y, z] ,
[x, y, z] 0x + 1y + 2z = 0 y = 2z,
[x, y, z] = [x,2z, z]. (x, y, z) , (0,0,0), x , 0, z , 0. x , 0,
[x, y, z] = [x,2z, z] =[x
(1,2z
x,z
x
)]=
[1,2z
x,z
x
],
, =z
x,
[x, y, z] = [1,2, ], R.(2.7.11)
z , 0,
[x, y, z] = [1,2z, z] =[z
(x
z, 2,1
)]=
[x
z, 2,1
],
, = x/z,
[x, y, z] = [,2,1], R.(2.7.12)
(2.7.11) (2.7.12) : = 0 -
[1,0,0], , 0
[x, y, z] =[
(1
,2, 1
)]=
[1
,2,1
],
() (2.7.12).
, [1,0,0] [,2, 1], R.(3) [0, 0,1] [1,1,1].
[0,0,1] , [1,1,1], [0, 0,1] [1,1,1]. ,
z = 0, x + y + z = 0,
-
72 2.
= =
( x , 0). ,
[0,0,1] [1, 1,1] = .
(4) .
< x, y, z >,
[ai , bi , ci], i = 1, 2, 3.
aix + biy + ciz = 0; i = 1, 2, 3,(2.7.13)
.
(2.7.13) . ,
(2.7.14)
a1 b1 c1a2 b2 c2a3 b3 c3
= 0.
2.7.3 .
1. (2.7.3), (2.7.6) -
. (2.7.7)
.
2. 2.7.1 (. 3),
, 11 .
3.
) [2, 1,3] [1,0,1],) [1,0,2] [2, 0,4].
4.
) ,
) .
5. .
-
3
-
-
, -
-
...
, 17 18 , -
, ,
[]
E. T. Bell [7, . 353]
, 1 -
.
3, -
.
,
73
-
74 3.
, Frenet-Serret. -
4 5.
3.0
, , -
. ,
, , , ..
( )
, -
.
, .
()
. -
.
. -
R. Descartes, P. Fermat C. Huygens,
3.
,
.
-
, . -
I. Newton, G. W. Leibniz,
L. Euler, G. Monge, J. Bernoulli, A. C. Clairaut, F. Frenet, J. A. Serret, Ch. Dupin,
J. Bertrand, .
, ,
, ,
, ..
.
3.1
, X (
) . , -
-
(, , .), f : R R, , ...
X ,
-
3.1. 75
, . ,
:
3.1.1 . (parametrized curve),
(space curve) : I R3, I R .
, ( -
) (-
) t I. ., t I, (t) R3,
(3.1.1) = (1, 2, 3).
( )
i : = ui ; i = 1,2,3,
ui : R3 R i-, .
i.
(I) R3, (plane curve). ,
(I) R2. ( ),
, X = (I). - (I).
: R R2 : t 7 (
2
2t,
2
2t),
: R R2 : t 7 (t, t) : R R2 : t 7 (t3, t3)
: R R2 : t 7(t, t) : t 0(t2, t2) : t > 0,
(R) = (R) = (R) = (R) = R, R2
(. 3.1 ).
,
, (t) = 1, t R, , (t) , 0, t R, , 0, (0) = (0,0), ( , -
0).
-
76 3.
x
D
y
3.1
: (1) R -
() (2)
( )
R.
, -
,
.
: I R3 to - (to) , 0. (tangent line)
x
y
( )ota
( )ota
3.2
-
3.1. 77
(to),
(3.1.2) (s) = (to) + s(to) , s R.
(to) to . -
, .
(regular),
,
( ).
-
( 3), -. , , -
11. , ,
, Cr-, r 3.
,
(3.1.3) (t) = (1(t), 2(t),
3(t))
(tangent vector) (velocity)
(t),
v(t) := (t) = (1(t)2 + 2(t)2 + 3(t)2)1/2
(speed) t.
,
(3.1.4) (t) = (1 (t), 2 (t),
3 (t))
(acceleration) t.
: I R3, (length)
(3.1.5) L() :=
I
(t)dt.
, : I R3, < , >
(3.1.6) : I R : t 7
(3.1.7) : I R3 : t 7 (t) (t).
-
78 3.
(3.1.6) (3.1.7) , -
R3.
:
< , > (t) =< (t), (t) > + < (t), (t) >,(3.1.8)
( )(t) = ((t) (t)) + ((t) (t)).(3.1.9)
: -
[. (3.1.1)]
,
< (t), (t) >= 1(t)1(t) + 2(t)2(t) + 3(t)3(t),
(t) (t) =(2(t)3(t) 3(t)2(t), 3(t)1(t) 1(t)3(t), 1(t)2(t) 2(t)1(t)
),
.
(I),
3.1.2 . (1) : I R3 . (t) = 0, t I , .
,
(t) = 0 (t) = (t) = t + .
.
(2) 0. (to) 0 (to) , 0, (to) (to).
, t I - (t) 0,
: I R : t 7 (t)2 = 1(t)2 + 2(t)2 + 3(t)2 =< (t), (t) > .
to, (to) = 0, , (3.1.8),
0 = (to) =< (to), (to) > + <
(to), (to) >= 2 < (to), (to) >,
(to) (to).(3) : I R2 , (t) , 0, t I.
(I) 0, (t) (t), t I.
-
3.2. 79
, (t) 0
: I R : t 7< (t), (t) > .
, = 0.
(t) = < , > (t) =< (t), (t) > + < (t), (t) >= 2 < (t), (t) > .
= 0 (t) (t), t I, .
(4) P Q -
P, Q.
p q P Q ,
P, Q
: [0,1] R3 : t 7 (t) := p + t(q p)
[. (3.1.5)]
L() = 1
0
(t)dt = 1
0
q pdt = q p.
: [x, y] R3 (x) = p (y) = q. u R3 u = 1,
< a, u > (t) = < (t), u > + < (t), u > = < (t), u >
(t)u = (t),
L() = y
x
(t)dt y
x
< (t), u > dt = < (t), u > |yx
= < q, u > < p, u > = < q p, u >,
u R3 u = 1. u = (q p)/q p,
L() / q p = q p = L().
3.2
3.2.1 . : I R3 , : J R3 - (reparametrization) ,
h : J I, = h (. ).
-
80 3.
, t s, t = h(s).
I - X R3
J
h
6
-
3.1
h h
1 1 .
, h
1 1 h(s) , 0, s J . .
,
.
X := (I) = (J) X .
3.2.2 . , L() = L().
. : [a, b] R3 : [a, b] R3 = h, h : [a, b] [a, b] .
L() = b
a(s)ds =
b
a( h)(s)ds
=
b
a(h(s)) |h(s)|ds.
h > 0, h ,
L() = b
a(h(s))h(s)ds =
h(b)
h(a)(h(s))dh(s)
=
b
a
(t)dt = L().
h < 0, h ,
L() = b
a(h(s))h(s)ds =
h(b)
h(a)(h(s))dh(s)
= a
b
(t)dt = L(),
.
-
3.3. 81
.
, .
,
, X = (I). , ,
, 1.
3.2.3 . : I R3 : J R3 (s) = 1, s J .
. I := [a, b]. ( )
s : [a, b] [0, L()] : t 7 t
a
(u)du.
,
s(t) = (t) = v(t) > 0,
(. ),
h : [0, L()] [a, b]
(3.2.1) h(s) = 1/s(h(s)), s [0, L()].
:= h,
(s) = (h(s)) |h(s)| = (h(s))/|s(h(s))| = 1,
.
3.2.4 . : J R3 (s) = 1, s J , . ,
,
()
. .
3.3
-
X = (I), . , X
-
82 3.
. , -
, .
(s), s J , .
(3.3.1) T (s) := (s) s J.
, T (s) (s). T (s) = 1, s J , T (s) R3.
s J
(3.3.1) T : J s 7 T (s) = (s) R3.
3.3.1 . (s), s J , ,
T T,
T (s) T (s), s J.
. T (s) , T (s)2 =< T (s), T (s) > ,
< T, T > (s) = 2 < T (s), T (s) >= 0, s J,
.
T (s) = (s) () 1, T (s) =(s) () T (s).
T (s)
(3.3.2) k(s) := T (s)
(s). k(s) - . k(s) = 0, s 1. ()
(3.3.2) k : J s 7 k(s) [0, ).
3.3.4, -
.
1,
. k(s) , 0, ,
(3.3.3) (s) :=1
k(s)
-
3.3. 83
. , s J ,
(3.3.4) N(s) :=1
k(s)T (s)
(normal vector)
(s). ( 3.3.1) .
(3.3.4) N : J s 7 N(s) R3.
, s J ,
(3.3.5) B(s) := T (s) N(s)
(3.3.5) B : J s 7 B(s) R3.
B(s) (binormal vector) (s). - T (s), N(s) .
, s J ,
{T (s), N(s), B(s)}
x
y
z
( )T s
( )B s
( )N s
( )T s
( )B s
( )N s
( )sb
3.3
-
84 3.
R3, (moving
frame) Frenet (Frenet frame) (s) . ,
{T, N, B}
Frenet .
,
, -
(s), (s) . ,
. .
3.3.
, -
, . , s J , T (s) N(s) E. () (s) E ( T (s), N(s)) (osculating plane) (s). , (s) N(s) B(s) (normal plane) (s), (s) T (s) B(s) (rectifying plane) (s).
( )T s
( )B s
( )N s
3.4
, ( (s)) B(s), T (s)
-
3.3. 85
N(s). - ,
.
3.3 3.4 .
{T (s), N(s), B(s)} (s), T (s), N(s)) ...
, ,
. < N(s), B(s) >= 0, s J , < N,B >= 0 [. (3.1.6), (3.1.8)]
< N (s), B(s) > + < N(s), B(s) >= 0.
(3.3.6) (s) := < N(s), B(s) >=< N (s), B(s) >
(torsion) (s).
(3.3.6) : J s 7 (s) R.
(3.3.6) , B(s), ,
.
,
, .
( ) 3.3.5.
Frenet
, -
.
3.3.2 . u, v R3 u = 1 u v,
(u v) u = v.
. u = (a, b, c) v = (x, y, z).
u v =
e1 e2 e3a b c
x y z
= (bz cy)e1 (az cx)e2 + (ay bx)e3
= (bz cy, cx az, ay bx).
-
86 3.
,
(u v) u =
e1 e2 e3bz cy cx az ay bxa b c
= [c(cx az) b(ay bx)]e1 [c(bz cy) a(ay bx)]e2+
+ [b(bz cy) a(cx az)]= (c2x acz aby + b2x)e1 (cbz c2y a2y + abx)e2+
+ (b2z bcy acx + a2z)e3 =
= ((1 a2)x acz aby, (1 b2)y abx cbz,(1 c2)z bcy acx)
= (x a(ax + by + cz), y b(ax + by + cz),z c(ax + by + cz))
= (x a < u, v >, y b < u, v >, z c < u, v >)= (x, y, z) = v,
.
3.3.3 . :
T (s) N(s) = B(s),N(s) B(s) = T (s),B(s) T (s) = N(s).
. (3.3.5), B(s).
B(s) T (S) = (T (s) N(s)) T (s) = N(s).
, :
N(s) B(s) = N(s) (T (S) N(s)) == (T (S) N(s)) N(s) = (N(S) T (s)) N(s) = T (s).
() Frenet-Serret,
T (s), N (s), B(s) - .
() F. Frenet J. Serret.
3.3.4 . : J R3 , k > 0 {T, N, B} Frenet . ( Frenet-Serret):
(F . 1) T = kN
(F . 2) N = kT + B(F . 3) B = N .
-
3.3. 87
. (F. 1) N(s) [. (3.3.4)]. (F. 2) : , s J ,
T (s), N(s) B(s) , - a, b, c : J R,
(3.3.7) N (s) = a(s)T (s) + b(s)N(s) + c(s)B(s), s J.
a, b, c -
(3.3.7) T (s), N(s), B(s).
< N (s), T (s) > = a(s)< T (s), T (s) > + b(s)< N(s), T (s) >
+ c(s)< B(s), T (s) >
= a(s)1 + b(s)0 + c(s)0 = a(s).
, < T, N >= 0
< T, N > =< T , N > + < T,N >= 0,
, (F. 1),
< T,N >= < T , N >= < kN,N >= k 1 = k.
a(s) = k(s),
(3.3.7)
(3.3.8) N (s) = k(s)T (s) + b(s)N(s) + c(s)B(s), s J.
"" N(s)
< N (s), N(s) > = k(s)< T (s), N(s) > + b(s)< N(s), N(s) >+ c(s)< B(s), N(s) >
= k(s)0 + b(s)1 + c(s)0 = b(s).
< N,N >= 1
< N,N > = 2 < N,N >= 0,
b(s) = 0,
(3.3.8)
(3.3.9) N (s) = k(s)T (s) + 0N(s) + c(s)B(s), s J.
-
88 3.
,
< N (s), B(s) > = k(s)< T (s), B(s) > + 0< N(s), B(s) >+ c(s)< B(s), B(s) >
= k(s)0 + 0 + c(s)1 = c(s).
< B,N >= 0
< B, N > =< B, N > + < B,N >= 0,
, (3.3.6),
< B(s), N (s) >= < B(s), N(s) >= (s),
c(s) = (s),
(3.3.9)
N (s) = k(s)T (s) + (s)B(s); s J,
(F. 2).
(F. 3) B = T N , (F. 1), (F. 2) 3.3.3. ,
B = T N + T N
= kN N + T (kT + B)= k 0 + (kT T + T B)= k 0 + T B= N.
Frenet-Serret :
(3.3.10)
T
N
B
=
0 k 0
k 0 0 0
.
T
N
B
.
, -
(F. 1) (F. 3):
( ), 2 2 ,
.
-
.
3.3.5 . : J R3 . :
(i) k = 0 .
(ii) k > 0, = 0 .
-
3.3. 89
. (i) k = 0 T = = 0 (s) = (s) = s + [. 3.1.2(1)]. , ,
= 1.(ii) B = N = < N,B > = 0
B = 0, B = T N , so J ,
B(s) = B(so), s J.
= 0 B(s) = B(so) s J T (s) B(so), s J < T (s), B(so) >=< (s), B(so) >= 0, s J < (s), B(so) > = 0, s J < (s), B(so) >= , s J < (s), B(so) >=< (so), B(so) >, s J < (s) (so), B(so) >= 0, s J (s) (so) B(so), s J.
(s) (so) Eo T (so) N(so), (s) Eo + (so), s J .
, (s) E, E, Eo ( R2, 2) E 0.
,
E = (s) + Eo, s J. so J ,
(s) (so) + Eo, s J.
Eo {u, v},
(3.3.11) (s) = (so) + (s)u + (s)v; s J,
, : J R . , : (3.3.11)
(s) (so) = (s)u + (s)v; s J,
, u
< (s) (so), u > = < (s)u, u > + < (s)v, u >= (s)1 + (s)0 = (s),
-
90 3.
=< (so), u >, . - .
(3.3.11)
T (s) = (s) = (s)u + (s)v Eo; s J,
N(s) =1
k(s)T (s) =
1
k(s)((s)u + (s)v) Eo, s J.
, T (s) N(s) Eo B(s) - Eo. ,
B(s) (3.3.6), = 0.
3.3.6 . : J R3 . k > 0.
. (xo, yo) r,
(s) = r(cos
s
r, sin
s
r
)+ (xo, yo); s J [0,2],
T (s) = (s) =( sin s
r, cos
s
r
),
T (s) = (s) =1
r
( cossr, sins
r
)
k(s) = T (s) = 1r
.
, k > 0 .
(s) := (s) +1
kN(s).
(s) = (s) +1
kN (s)
= T (s) +1
k(kT (s) + B(s))
= T (s) T (s) + 0 = 0,
. a := (s) R3 ,
(s) a = 1kN(s) = 1
k= r,
. (s) a r. , .
-
3.4. 91
3.3.7 . 1)
[. (3.3.3)],
(s) + 1kN(s).
2) 3.3.5 3.3.6
,
.
3.4
Frenet, -
.
: I R3 , -
. , .. ,
3.2.3 ,
.
,
X = (I) (-) . , .
3.4.1 .
(. 3.2.3). {T , N , B}, k, Frenet, , T, N, B, k,
T (t) := T (s(t)),( i )
N(t) := N(s(t)),( ii )
B(t) := B(s(t)),( iii )
k(t) := k(s(t)),( iv )
(t) := (s(t)).( v )
3.4.2 . : I R3 () ,
k = 3
.
. = h , h = s1 s . = s, , t I,
(t) = ( s)(t) = s(t)(s(t)) = s(t)T (s(t)),(3.4.1)(t) = s(t)T (s(t)) + s(t)2(T (s(t))
= s(t)T (s(t)) + s(t)2k(s(t))N (s(t))(3.4.2)
-
92 3.
(t) (t) = s(t)T (s(t)) [s(t)T (s(t))++ s(t)2k(s(t))N (s(t))],
(3.4.3)
(t) (t) = s(t)s(t)[T (s(t)) T (s(t))]++ s(t)3k(s(t))[T (s(t)) N(s(t))]
= 0 + s(t)3k(s(t))B(s(t))
= (t)3k(s(t))B(s(t)).
(t) (t) = (t)3k(s(t)),
k(t) := k(s(t)) =(t) (t)(t)3
,
.
3.4.3 . k > 0,
=< , > 2 =
[] 2
.
. ,
. (3.4.2)
(t) = s(t)T (s(t)) + s(t)s(t)T (s(t)) +
+[s(t)2k(s(t))]N(s(t)) + s(t)3k(s(t))N (s(t))
= s(t)T (s(t)) + s(t)s(t)k(s(t))N (s(t)) +
+[s(t)2k(s(t))]N(s(t)) s(t)3k(s(t))2T (s(t)) ++s(t)3k(s(t))(s(t))B(s(t))
= X (t)T (s(t)) + Y (t)N (s(t)) + Z (t)B(s(t)),
X (t) = s(t) s(t)3k(s(t))2,Y (t) = s(t)s(t)k(s(t)) + [s(t)2k(s(t))],
Z (t) = s(t)3k(s(t))(s(t)).
-
3.5. Frenet 93
(3.4.3) B(s(t)) T (s(t)) N(s(t)), :
< (t) (t) , (t) > = s(t)3k(s(t)) < B(s(t)), (t) >= s(t)3k(s(t))X (t) < B(s(t)), T (s(t)) > +
+ s(t)3k(s(t))Y (t) < B(s(t)), N (s(t)) > +
+ s(t)3k(s(t))Z (t) < B(s(t)), B(s(t)) >
= 0 + 0 + s(t)6k(s(t))2 (s(t))
= (t) (t)2 (s(t)),
(t) := (s(t)) =< (t) (t), (t) >(t) (t)2
,
.
3.5 Frenet
T (t), N(t) B(t) (. 3.4.1), , .
3.5.1 . : I R3 , - , , {T (t), N(t), B(t)} Frenet.
T (t) =(t)(t)
,(3.5.1)
N(t) =
((t) (t)) (t)(t) (t) (t)
,(3.5.2)
B(t) =(t) (t)(t) (t)
.(3.5.3)
. = h (. - 33.2.2), h = s1 s .
T (t) = T (s(t)) = (s(t)) = ( h)(s(t))
= (h(s(t)))h(s(t)) = (t) 1s(t)
=(t)(t)
.
-
94 3.
, T = T s, T = T h,
N(t) = N(s(t)) =T (s(t))
k(s(t))=
(T h)(s(t))k(t)
=T (h(s(t)))h(s(t))
k(t)=
T (t)k(t)s(t)
=T (t)s(t)
(t)3
(t) (t)
= T (t)(t)2
(t) (t).
, T (t) N(t) . T (t) N(t), T (t) T (t). T (t) = 1. , 3.3.2 u = T (t) v = T (t),
T (t) = (T (t) T (t)) T (t).
T (t) =(t)s(t)
[. (3.5.1)]
T (t) =(t)s(t) (t)s(t)
s(t)2,
T (t) =
((t)s(t)
(t)s(t) (t)s(t)
s(t)2
)
(t)s(t)
=1
s(t)4[((t) (t)s(t) (t) (t)s(t)] (t)
=1
s(t)3[((t) (t)) (t)],
N(t) =((t) (t)) (t)(t) (t) (t)
.
,
B(t) = B(s(t)) = T (s(t)) N(s(t)) = T (t) N(t)
=(t)(t)
((t) (t)) (t)(t) (t) (t)
=(t)(t)
((t) (t)(t) (t)
(t)(t)
)
=(t) (t)(t) (t)
,
-
3.5. Frenet 95
, 3.3.2,
u =(t)(t) v =
(t) (t)(t) (t)
.
3.5.2 . : I R3 , - , , {T, N, B} Frenet ( ). () Frenet-Serret:
(F . 1) T = kvN ,
(F . 2) N = kvT + vB,(F . 3) B = vN , v(t) := (t) t I.
. : J R3 Frenet {T , N, B} . t I,
T (t) = (T s)(t) = T (s(t))s(t) = k(s(t))N (s(t))s(t)= k(t)v(t)N(t),
N (t) = (N s)(t) = N (s(t))s(t)= k(s(t))s(t)T (s(t)) + (s(t))s(t)B(s(t))= k(t)v(t)T (t) + (t)v(t)B(t),
B(t) = (B s)(t) = B(s(t))s(t) = (s(t))N (s(t))s(t)= (t)v(t)N(t),
.
, ,
, .
, -
, .
, : -
(translation) ( ) h R3
h : R3 R3 : u 7 h(u) := u + h.
(rotation) R3
, f : R3 R3 ,
< f (u), f (v) >=< u, v >; u, v R3,
. , f
( ).
-
96 3.
: h h, -
, .
f c = f (c) f.
(rigid mo-
tion).
-
, .
3.5.3 . k(s) > 0 (s), s J = [0, c], - . : J R3 k . ,
.
x
y
z
0( )T s
0( )B s
0( )N s
0( )T s
0( )B s
0( )N s
0( )sa
1e
2e0
3e
a
b
3.5
,
, [6].
3.6
1. : (i) ,
() P Q R3. (ii) . (iii)
-
3.6. 97
. (iv)
.
2. (0,0) r
: [0,2] R2 : t 7 (r cost, r sint).
: (i) . (ii) -
. (iii)
. (iv) . (v)
k .
3. y = x2
. (t) = (t3, t6), t R;
4. f : R R. - T , N , B
.
5. ,
: [0, 2] R3 : t 7 ( 12
cost, sint,1
2cost
).
6.
(t) = (a cost, b sint), t [0,2], a, b > 0.
;
7. ()
: [0,2] R3 : t 7 (r cost, r sint, bt); r > 0, b R.
(circular helix),
.
(i) . (ii)
. (iii)
zOz . (iv)
zOz .
.
-
98 3.
2 bp
z
x
t
y
3.5
8.
(t) :=
(t,t2
2
), 0 t 1.
9.
: R R2 : t 7 (t2, t3).
; .
10. a(0) = (0,1), .
11. : I R3 v R3,
(0) v (t) v, t I.
(t) v, t I.
12.
: R R3 : t 7 (3t, 3t2, 2t3)
-
3.6. 99
y = 0, x = z.
13. r (0,0) x Ox . (i) A,
(0,0). (ii)
t = 0, t = t = 2. (iii)
, t [0, 2]. -
.
14. (s) . (i) - w (: Darboux) : T = T , N = N B = B. (ii) T T = k2. [ , , s].
15. : J R3 : (s) P. .
16. a : J R3 .
P.
17. a : J R3 , a .
18. : J R3 . 0 , u R3 .
19. -
.
(t) = (a sin2 t, a sint cost, a cost)
(0, 0,0).
20. -
. : (t) = (cost, sint, t), t R. () (t) (to),
to =
2.
21.
. : (t) =(t,t2
2,t3
3
), t R.
.
-
100 3.
(t) to = 1.
22. : I R3 , k(s) > 0 s I, (s). (s) := T (s), T (s) . (i) . (ii)
( so), (s) = k(s),
s I. (iii) k .. (spherical indicatrix)
() T . N B.
R3 0.
23. : I R2 k(s) < 1, s I.
(s) := (s) + N(s), s I.
( ), N(s) . k
k =k
1 k .
24. (s) (s) , 0. k B(s).
25.
, .
-
4
, , ( -
) , -
(), ,
.
Euler Gauss. Egregium Gauss -
,
. -
( ) Riemann,
() .
Riemann
Einstein -
(.
). , ,
.
( )
101
-
[1] . : , ,
1985.
[2] . : . , 1980.
[3] . . : ,
, 1985.
[4] . .. ( ): ,
, , , 2003.
[5] . : , , ,
2009.
[6] . . : -
, , 2009.
[7] E. T. Bell: The Development of Mathematics, Dover, New York, 1992.
[8] J. W. Blattner: Projective plane geometry, Holden-Day, San Francisco, 1968.
[9] J. N. Cederberg: A Course in Modern Geometries, Springer, New York, 1989.
[10] D. D. Davis: , -
, , 2005.
[11] M. P. Do Carmo: Differential geometry of Curves and Surfaces, Prentice-Hall,
Englegood Cliffs, New Jersey , 1976.
103
-
104
[12] P. Dombrowski: 150 Years after Gauss "disquisitiones generales circa super-
ficies curvas", Asterisque 62, Soc. Math. de France, Paris, 1979.
[13] N. V. Efimov E. R. Rozendorn: Linear Algebra and Multidimensional Geo-
metry, MIR Publishers, Moscow, 1975
[14] R. L. Faber: Foundations of Euclidean and non-Euclidean Geometry, Marcel
Dekker, New York, 1983.
[15] G. H. Hardy: A MAthematicians Apology, Cambridge Univ. Press, Canto edi-
tion, Cambridge, 2002.
[16] M. Henle: Modern Geometry: The Analytic Approach, Prentice Hall, New Jer-
sey, 1997.
[17] D. Hilbert: Foundations of Geometry (Grunlangen der Geometrie), Open
Court, Illinois, 1971. : , -
, , 1995.
[18] S. Hollingdale: Makers of Mathematics, Penguin, London, 1991.
[19] M. Kline: Mathematics in Western Culture, Oxford Univ. Press, 1953. -
: ( -), ,
, 2002.
[20] W. Klingenberg: A Course in Differential Geometry, Springer, New York,
1978.
[21] . : , Leader Books, ,
2006.
[22] M. Lipschutz: Differential Geometry, Schaums Outline Series, McGraw Hill,
New York, 1974. : , , . -
, , 1981.
[23] J. McCleary: Geometry from a Differential Point of View, Cambridge Univ.
Press, Cambridge, 1994.
[24] R. J. Mihalek: Projective Geometry and Algebraic Structures, Academic
Press, New York, 1972.
[25] L. Mlodinov: Euclids Window. The Story of Geometry from Parallel Lines to
Hyperspace, Allen Lane, The Penguin Press, London, 2002.
[26] S. NegrepontisD. Lamprinidis: The Platonic Anthyphairetic Interpretation of
Pappus account of analysis and synthesis, History and Epistemology in Ma-
thematics Education, Proceedings of the 5th European Summer University,
Prague, 2007, pp. 501511
-
105
[27] . : , , 1964.
[28] B. ONeil: Elementary Differential Geometry, Academic Press, New York,
1997. : , -
, 2002.
[29] J. Oprea: Differential Geometry and its Applications, Prentice Hall, Upper
Saddle River, New Jersey, 1997.
[30] . : I: , , 1996.
[31] A. Pressley: Elementary Differential Geometry, Springer, London, 2001.
[32] J. Pierpont: The history of mathematics in the nineteenth century, Bull.
Amer. Math. Soc. 37 (2000), 38 [reprinted from Bull. Amer. Math. Soc. 11
(1905), 238246].
[33] H. Poincare: Dernieres Pensees, Flammarion, Paris, 1913.
[34] C. Reid: , , , 2007.
[35] M. Spivak: Differential Geometry, Vol. II, Publish or Perish, Wilmington, 1979.
[36] . : , ( IIV),
. , , 1952.
[37] F. W. Stevenson: Projetive Planes, W. H. Freeman, San Francisco, 1972.
[38] . :
( ), , , 1987.
[39] D. J. Struik: , . ,
, 1982.
[40] M. B. W. Tent: , , -
, , 2007.
[41] I. M. Yaglom: Felix Klein and Sophus Lie, Birkhauser, Boston, 1988.
-
, 7
, 83
, 79
, 81
, 126
, 7
2, 22
, 15
, 22
, 9
(), 15
, 18
Dedekind, 15
Pasch, 13
Playfair, 9
, 17
, 17
, 17
, 47
, 8
, 8
, 7
, 9
, 56
, 12
, 44, 45
, 8
, 19
, 9
, 10, 22
, 9
, 10, 18
Klein, 66
, 47
, 47
Darboux, 99
, 83
, 77, 82
, 83
, 83
, 77, 82
, 66
, 60
107
-
108
Klein-Beltrami, 20
Poincare, 21
, 44
, 50
, 37
(), 97
, 19
, 10
, 22
, 22
, 35, 38
, 84
, 84
, 84
, 38
, 58
, 34
, 77
, 76
, 77
, 33, 38
, 76
, 53
, 53
, 53
, 34
, 7
, 96
, 61
, 60
, 59
, 61
, 48
, 48
, 83
, 83
, 83
(), 97, 125, 130
(), 99, 130
, 75
, 126
, 77
, 75
, 77
, 81
, 77
, 100
, 75
, 75
, 82, 91
, 47
, 7
, 99
, 95
, 77
, 77
, 17
, 59
, 60
11, 61
, 61
, 7
, 9
, 81
, 81
, 53
, 7
, 38
, 44
, 49
2, 40
-
109
, 39
, 34
, 33, 38
, 77
, 52
, 82
, 52
, 36
, 52
, 47
, 96
, 47
, 85, 91
, 95
, 66
,
, 100
, 33, 38
, 51
, 36, 38
Frenet, 84, 91, 93
, 84
, 84
, 84
Frenet-Serret, 86
, 95
, 10
, 18, 19
-
2
( )
2.6.10(2). .
2.1.9, 4 A,B, C, D, 3
. ( ) A,B, C, D
, .
:
i) .
AB, AC AD, (. 2.2.6). ,
, AD. , AB,A C X Y . X , Y , A B = A X = A Y = A C ().
(). , 2012.
-
112
ii) A,B, C, D , A.
Z , A ,
.
2.6.10(3). : P .. -
, P. , ( 3)
3 A, B, C. P
,
. P ,
3 2 A,B, C,
.
,
, P. 2.6.10(2), -
, X Y .
X , P , Y , P X P Y . , ( 2), k, P.
k, PX, PY 3 P.
: 4 A,B, C, D 2.1.9
, P A, B, C, D.
i) P , .. A.
PB, PC PD , [ , , P B = P C, P = A B,C ()].
ii) P , 2
-
113
A,B, C, D, .
, , P = (A C) (B D), C k//B D D //A C. k , [ k = C ()]. k // [
B D// ()]. E = k . E , P, P E. P E, A C B D , [, .., P E = A C, E A C ()].
-
114
iii) P 2
, A B (. ). P C P D A B. , P C = P D
C = (P C) (C D) = (P D) (C D) = D,
. , A B = P C, , A,B, C , . A B, P C, P D .
iv) P A, B, C, D,
P A, P B, P C, .
2.6.10(4). S = k m m ,
S , ( 2).
2.2.9(2). ( 3) [
( 3)].
2.2.9(3). , , 4 ( 3), -
2 . P, 2
, 4 , P.
2.2.9(4). S = k , A k, A , S, B , B , S. A , B, A B. 2.2.6,
-
115
A B C A B. C . ,
, (C, k) I, k = A C = A B, (B, k) I (). :
i) k , S = k .
k A , S [. 2.6.10(2)],
A () //. O , A.
O .
ii) k . k
A, B A, B. 4
, .
4, X .
X k, , . X ,
, k, A A B A, X B.
O .
-
116
X .
2.2.9(5). .
1 1 P2. ,
R3, 0,
. ,
, 0. ,
0 .
2.3.4(1). 7 .
2.3.4(2). . :
( 1),
.
( 2),
[ ,
, ( ),
( 1)].
2.6.10(3)
3 , (: -
).
2.3.4(3). , ( 3)
( 1) ( 3). 2.2.4 2.2.7.
2.4.10(1).
2.4.4.
2.4.10(2). 2.4.7.
2.4.10(3). O . , -
2.4.6, |J(A)| = |J(O)| = |J()|. 2.4.6 O m .
2.4.10(4). -
4 ( 7 ).
, 32 + 3 + 1 = 13.
2.4.10(6). O m,
-
2.4.4 2.4.6 2.4.7.
, 2.4.6
|J(m)| = |J(O)| 1. [ 2.2.9(4) ], |J(k)| = |J()|, 2.4.7.
2.4.10(7). 2.4.8:
O . n O Ai, Ai (i = 1, . . . , n)
-
117
. O m .
n1 O. , O, (n + 1)(n 1) + 1 = n2 . : Q
, O Q, m, O Ai. Q n2 .
2.4.10(8). Ai (i = 1, . . . , n) . P
, , [. 2.4.10(6)
] |J(P)| = |J()| + 1 = n + 1.
P , Ai.
P = A1. O .
P , PO kj //OAj (j = 2, . . . , n), n + 1 (. ).
.
2.5.8(3). {A,B} {C,D} ,
k = {A,B} = {C,D}.
, {A,D} {B,C}
l = {A,D} = {B,C},
{A,C}, {B,D}
m = {A,C} = {B,D}.
, A,B, C, D
k, l, m, 7 .
, ,
{A,B, k}, {C,D, k}, {A,D, l}, {B,C, l}, {A,C,m}, {B,D,m}, = {k, l, m}.
-
118
.
2.5.8(4). , {A1, A5, A7} - 4 , -
.
2.6.10(1). (P) (). X , (X ) (). X , P, X P, (X P) = (X ) (P) = ()., 1 1 X P = ().
1 1,
, Q P k L, Q < k (Q) (k). , 1 1, k, m, k , m (k) = (m). S = km m Q , S. Q < k. , (,) , Q m (Q) (m) = (k), .
2.6.10(2).
[: P < , (P) < () ()].
2.6.10(3). (P , ) P L P . P P L, (P) = P () = . (P) (). - 2.6.10(2), P ,, , 1(P ) 1(), .2.6.10(4). :
, A, B. () = (AB) = (A)(B)., , C, D ( A, C,
-
119
7 , 3 )
= A B = C D, () = (C) (D) = (A) (B). .
2.6.10(5). . -
, () = (A) (B) = (), A,B . L, = .2.6.10(7). (P, ) P1 L1 P , , .
(P, ) I1(1, 1) - (1(P), 1()
) I2
(2(1(P)), 2(1())
) I3
(2, 2)
?(2 1, 2 1)
-
2.6.10(8). () := (A) (B), A,B - . : ,
C D,
( A,C, 3 ), -
A, B, C, D
(A), (B), (C), (D). , () = (A)(B) = (C)(D), () .
(,) - : (P, ) P , = A B, , , (P) (A) (B) = ().
1 1 , (,) (. 2.1.6)., 2.6.10(5).
2.6.10(9). ,
2.6.10(7).
(1P, 1L). (, ) Aut(P) ,
(,)1 := (1, 1)
[. 2.6.10(3)].
, .
2.7.3(3). ) [2, 1,3] [1,0,1] =.) [2,0, 4] = [2(1,0,2)] = [1,0,2], - .
2.7.3(4). ) . ) = [0,1,2].
2.7.3(5). [t, 1,0], t R.
-
120
3
( )
3.6(1). (i) p, q P, Q,
PQ
: [0, 1] R3 : t 7 (t) := p + t(q p).
(ii)
(t) = q p , 0, ,
L() = 1
0
(u)du = 1
0
q pdu = q p.
, (t) = 0, 1.4.2 k(t) = 0, t [0, 1].(iii)
s(t) = t
0
(u)du = t
0
q pdu = q pt,
s : [0,1] [0, q p] : t 7 q pt
h := s1 : [0, q p] [0,1] : s 7 sq p,
: [0, q p] R3
(s) := ( h)(s) = (
s
q p
)= p +
s
q p (q p).
(iv) (s) = q pq p ,
L() = qp
0
(u)du = qp
0
du = q p.
, k T (s) = (s) =q pq p , T
(s) = 0
[ (3.3.2)] k(s) = 0.
3.6(2). (i)
(t) = (r sint, r cost),
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121
||(t)|| = r , 0, (t) = r , 0, .
(ii) t ,
(t) (t). ,
< (t), (t) >= r2 cost sint + r2 cost sint = 0,
(t) (t).(iii) (t) = (r cost,r sint) = (t).(iv) (t) = r,
L() = 2
0
(t)dt = 2
0
rdt = 2r.
(v) (t) = r , 0.
s : [0,2] R : t 7 s(t) := t
0
(u)du = rt.
s : [0,2] [0, 2r] ,
h := s1 : [0, 2r] [0,2] : s sr
,
: [0,2r] R2
(s) = ( h)(s) = (s
r
)=
(r cos
s
r, r sin
s
r
).
T (s) = (s) =( sin s
r, cos
s
r
),
T (s) =(1r
coss
r,1r
sins
r
),
k(s) = T (s) = 1r
3.6(3). ,
(t) = (t, t2); t R,
. ,
(t) = (1, 2t) , 0, t R.
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122
,
3.4.2.
(t) = (1,2t) (1,2t, 0),(t) = (0, 2) (0,2,0),(t) (t) = (0,2,0)
k(t) = 2(1 + 4t2
)3/2.
,
, (t) = (3t2,6t5), t = 0 .
3.6(4). (t) = (t, f (t)), t R, f .
(t) = (1, f (t)) (1, f (t), 0),(t) = (0, f (t)) (0, f (t), 0),(t) = (0, f (t)) (0, f (t), 0),(t) (t) = (0, 0, f (t)).
(t) = (1 + f (t)2)1/2 , 0; t R, , . ,
3.5.1, :
T (t) =1
(1 + f (t)2
)1/2 .(1, f (t), 0
) 1(1 + f (t)2
)1/2 .(1, f (t)
),
N(t) =f (t)
|f (t)| (1 + f (t)2)1/2.( f (t), 1, 0)
f(t)
|f (t)| (1 + f (t)2)1/2.( f (t), 1),
B(t) =
(0,0,
f
|f |
)= (0, 0,1) = e3.
, 3.4.2,
k(t) =|f (t)|
(1 + f (t)2
)3/2 .
B(t) = e3 (). . [6]. , (t) , = 0 (. 3.3.5).
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123
3.6(5).
T (t) = (t) =
( 1
2sint, cost, 1
2sint
),
(t) =(1
2sin2 t + cos2 t +
1
2sin2 t
)1/2= 1,
,
L() = 2
0
(t)dt = 2
0
1dt = 2.
T (t) = (t) =
( 1
2cost, sint, 1
2cost
),
k(t) = T (t) =(1
2cos2 t + sin2 t +
1
2cos2 t
)1/2= 1.
B(t) = T (t) N(t) = (t) (t) = 12
(e1 + e3) = c
B(t) = 0, (t) = < N(t), B(t) >= 0,
, , .
3.6(6). x = a cost y = b sint,
x2
a2+y2
b2= 1,
.
:
(t) = (a sint, b cost) (a sint, b cost, 0),
(t) = (a2 sin2 t + b2 cos2 t)1/2.
(t) = (a cost,b sint) (a cost,b sint, 0)
(t) (t) = abe3,
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124
(t) (t) = ab,
k(t) =(t) (t)(t)3 =
ab
(a2 sin2 t + b2 cos2 t)3/2.
3.6(7). (i)
(t) = (r sint, r cost, b),
(t) = (r2 + b2)1/2 =: c > 0.
s : [0,2] R : t 7 t
0
(u)du = tc
h : [0,2c] [0,2] : s 7 sc
.
: [0, 2c] R3 : s 7 (s) = ( h)(s) =(r cos
s
c, r sin
s
c,b
cs).
(ii)
T (s) = (s) =( rc
sins
c,r
ccos
s
c,b
c
),
T (s) =( rc2
coss
c, r
c2sin
s
c, 0
)
k(s) = T (s) = rc2, 0.
, :
N(s) =1
k(s)T (s) =
( coss
c, sin s
c, 0
),
B(s) =
e1 e2 e3T1 T2 T3N1 N2 N3
= (
b
csin
s
c, b
ccos
s
c,r
c),
B(s) =( bc2
coss
c,b
c2sin
s
c, 0
)
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125
(s) = < N(s), B(s) >= bc2
.
(iii)
cos =< (t), e3 >(t) e3
=b
c.
cos =< T (s), e3 >T (s) e3
= < T (s), e3 > =b
c.
(iv) (s)
s(t) = (s) + tB(s); t R,
(s) B(s). ,
cos =< B(s), e3 >B(s) e3
= < B(s), e3 > =r
c.
3.6(8). :
(t) = (1, t) (1, t, 0),(t) = (1 + t2)1/2 > 0,(t) = (0,1) (0,1,0),
(t) (t) =
e1 e2 e31 t 0
0 1 0
= e3,
(t) (t) = 1.
,
k(t) =(t) (t)(t)3 =
1
(1 + t2)3/2.
3.6(9).
(t) = (2t,3t2)
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126
(0) = (0, 0) R. (, 0) (0,+). t , 0
(t) = (4t2 + 9t4)1/2,(t) = (2,6t),
(t) (t) =
e1 e2 e32t 3t2 0
2 6t 0
= 6t2e3 = (0,0,6t
2),
(t) (t) = 6t2,
k(t) =(t) (t)(t)3 =
6t2
(4t2 + 9t4)3/2.
3.6(10). (t) = (cost, sint) (0) = (1, 0).
(t) = (cos(t + /2), sin(t + /2)) , (0) = (0,1).
,
(t) = (t) = (cos(/2 t), sin(/2 t)),
(0) = (0) = (0,1).
3.6(11). < (0), v >= 0, < (t), v > ,, < (t), v >= 0. ,
< (t), v > < , v > (t) = < (t), v > + < (t), v >= 0+ < (t), 0 >= 0.
3.6(12). (t)
t(s) = (t) + s(t); t R,
(t) = (3,6t,6t2) ( s ). y = 0
x = z v = (1, 0,1). ,
cos =< (t), v >(t) v =
3 + 6t2
9 + 36t2 + 36t4
2=
2
2= cos(
4).
3.6(13). (i) A, (0,0), A = (x, y), K = (0, r) a
-
127
K = (a, r).
x
y
K K
r
A
AA
B
q
rp 2 rpO
x Ox B AB a.
AKB a = r. A A A
K B KK .
x = OB AA = r r sin = r( sin),y = OK AA = r r cos = r(1 cos),
() =(r( sin), r(1 cos)).
(ii)
() =(r(1 cos), r sin),
(0) = (0,0), () = (2r,0), (2) = (0,0).
, = 0 = 2, , =
(s) = () + s()
=(r( sin), r(1 cos)) + s(r(1 cos), r sin)
= (r,2r) + s(2r,0) = (r + 2rs, 2r).
(iii) ,
()2 = r2(1 + cos2 2 cos + sin2 )
= 2r2(1 cos) = 2r2(1 cos2
2+ sin2
2
)
= 4r2 sin2
2,
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128
() = 2r sin
2
= 2r sin
2, 0 2.
,
L() = 2
0
()d = 2r 2
0
sin
2d
= 2r
0
2 sind = 4r( cos
0
)
= 4r( cos + cos0) = 8r.
-
(2k,2(k + 1)), k Z. .
3.6(14) (i) = xT + yN + zB,
x, y, z.
T = xT T + yN T + zB T= x0 yB + zN = yB + zN
, T = T ,
yB + zN = T = kN,
y = 0 z = k. ,
N = xT N + yN N + zB N= xB + y0 + zT = xB zT
, N = N ,
xB zT = N = kT + B
x = z = k. , ,
= T + kB.
:
B = (T + kB) B = (T B) + kB B = N + 0 = B.
( ) .
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129
(ii) T = kN
T = kN + kN = kN + k(kT + B) = k2T + kN + kB.
,
T T = kN (k2T + kN + kB)= k3(N T ) + kk(N N) + k2(N B)= k3B + 0 + k2T = k2(kB + T ) = k2.
3.6(15). (s) s(t) = (s) + t(s), t R. , ts R, s(ts) = p, p R3 P,
(s) + ts(s) = p.
, .. (s), , s(t) = (s) + t(s) ts,
(s) + ts(s) = p,
s J . , s J , (s) R
( ) (s) + (s)(s) = p,
.
s 7 (s) ,
( ) (s)(s) = p (s) < (s)(s), (s) >=< p (s), (s) > (s) < T (s), T (s) >=< p (s), (s) > (s) =< p (s), (s) >.
( ) , s J :
( ) (s) + (s)(s) + (s)(s) = 0 T (s) + (s)T (s) + (s)T (s) = 0 (1 + (s))T (s) + (s)k(s)N(s) = 0 1 + (s) = (s)k(s) = 0 (s) = 1 (s)k(s) = 0 (s) = c s (c s)k(s) = 0 k(s) = 0,
-
130
. c s , 0, (s) = 0 (). , (s) = 0 (s) = p, s J , ( ).
3.6(16). (s)
s(t) = (s) + tN(s); t R,
(s) N(s). , , s J , (s) R
( * ) (s) + (s)N(s) = p,
(: p P). : ( ) -
(s) =< (s)N(s), N(s) >=< p (s), N(s) >.
( ) :
() (s) + (s)N(s) + (s)N (s) = 0 T (s) + (s)N(s) + (s)(k(s)T (s) + (s)B(s)) = 0 (1 (s)k(s))T (s) + (s)N(s) + (s)(s)B(s) = 0 1 (s)k(s) = (s) = (s)(s) = 0 = c (), k(s) = 1 (s) = 0.
, 0, = 0,
, k(s) = 1/ = . : P .
3.6(17). (s) [. 3.6(7)] (s) B(s),
s(t) = (s) + tB(s); t R.
, ,
s J , (s) R
( ) (s) + (s)B(s) = p.
(s) =< (s)B(s), B(s) >=< P (s), B(s) >,
-
131
, ( ),
(s) + (s)B(s) + (s)B(s) = 0,
, ,
T (s) (s)(s)N(s) + (s)B(s) = 0,
1 = (s)(s) = (s) = 0,
.
3.6(18). () : .() : , s J , (s) R
( ) (s)T (s) = u = .
(s) =< (s)T (s), T (s) >=< u, T (s) >,
(s) . ( ) :
( ) (s)T (s) + (s)T (s) = 0 (s)T (s) + (s)k(s)N(s) = 0 (s) = (s)k(s) = 0.
(s) = 0 (s) = c. c = 0, u = c T = 0, . c , 0, c k(s) = 0 k = 0, .
3.6(19). (i) (s) . (x, y, z) R3 (so) (x, y, z)(so) N(so) B(so), T (so) = (so). (x, y, z) (so)
( ) < (x, y, z) (so), T (so) >= 0,
, ,
< (x, y, z) (so), (so) >= 0.(ii) (t), - . (x, y, z) (to) ()
< (x, y, z) (to), T (to) >= 0.
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132
(3.5.1),
< (x, y, z) (to),(to)(to)
>= 0,
, ,
< (x, y, z) (to), (to) >= 0.
. , (0,0,0) (t),
< (0,0, 0) (t), (t) >=< (t), (t) >= 0, t R.
(t) =(2a sint cost, a(cos2 t sin2 t),a sint).
,
< (t), (t) >=
< (a sin2 t, a sint cost, a cost), (2a sint cost, a(cos2 t sin2 t),a sint) >=2a2 sin3 t cost + a2 sint cos3 t a2 sin3 t cost a2 sint cost =
a2 sin3 t cost + a2 sint cos3 t a2 sint cost =a2 sint cost(sin2 t + cos2 t) a2 sint cost = 0.
3.6(20). (i) (s) . (x, y, z) R3 (so) (x, y, z) (so) T (so) N(so), B(so). (x, y, z) (so)
( ) < (x, y, z) (so), B(so) >= 0.
B(so) = T (so) N(so) = (so) T (so)k(so)
= (so) (so)k(so)
,
( )
< (x, y, z) (so), (so) (so) >= 0.
(ii) (t), . (x, y, z) (to) ( )
< (x, y, z) (to), B(to) >= 0.
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133
[. (3.5.3)]
B(to) =(to) (to)(to) (to)
< (x, y, z) (to),(to) (to)(to) (to)
>= 0,
, ,
< (x, y, z) (to), (to) (to) >= 0.
. -
< (x, y, z) (2
),
(2
)
(2
)>= 0.
:
(2
)=
(cos
2, sin
2,
2
)=
(0, 1,
2
)
(t) = ( sint, cost, 1)
(2
)=
( sin
2, cos
2,1
)= (1, 0,1)
(t) = ( cost, sint, 0)
(2
)=
( cos
2, sin
2,0
)= (0,1, 0)
< (x, y, z) (0,1, 2
), (1,0,1) (0,1, 0) >= 0
< (x, y 1, z 2
), (1,0,1) (0,1,0) >= 0
x y 1 z 2
1 0 10 1 0
= 0
x + z =
2.
3.6(21). (i) (s) . (x, y, z) R3 (so) (x, y, z) (so) T (so) B(so), N(so). (x, y, z) (so)
( ) < (x, y, z) (so), N(so) >= 0.
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134
N(so) =T (so)k(so)
=(so)k(so)
,
( )
< (x, y, z) (so),(so)k(so)
>= 0,
, ,
< (x, y, z) (so), (so) >= 0.(ii) (t), . (x, y, z) (to) ( )
< (x, y, z) (to), N(to) >= 0. [. (3.5.2)]
N(to) =
((to) (to)
) (to)(to) (to) (to)
,
< (x, y, z) (to),((to) (to)
) (to)(to) (to) (to)
>= 0,
, ,
( ) < (x, y, z) (to), ((to) (to)) (to) >= 0.
. ( ), -
< (x, y, z) (1), ((1) (1)) (1) >= 0.
:
(1) =(1,
1
2,1
3
)
(x, y, z) (1) = (x 1, y 12, z 1
3
)
(t) =(1, t, t2
)
(1) = (1, 1,1)
(t) = (0,1,2t)
(1) = (0, 1,2)
(1) (1) =
e1 e2 e31 1 1
0 1 2
= (1,2, 1)
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135
< (x 1, y 1/2, z 1/3), (1,2, 1) (1,1,1) >=
e1 e2 e31 2 11 1 1
= 0,
x z = 23
.
3.6(22). (i) (s) = T (s) = k(s)N(s). , (s) = k(s) , 0, .
(ii)
(s) = s
so
(u)du,
(s) = (s) = T (s) = k(s)N(s) = k(s).
(iii) (
k = 1, ),
1.4.2 1.4.3 -
.
( s):
= T = kN,
= kN + kN
= kN + k(kT + B)= k2T + kN + kB,
= kN (k2 + kN + kB)= k3N T + kkN N + k2N B= k2T + 0 + k3B = k2T + 0N + k3B,
2 = k6 + k42 = k4(k2 + 2).
,
k = 3 =
k2(k2 + 2
)1/2
k3=
(k2 + 2
)1/2
k.
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136
.
, , :
= 2kkT k2T + kN + kN + kB + kB + kB
= 2kkT k3N + kN + k(kT + B) + (k + k)B + k(N)= 3kkT + (k3 + k k2)N + (2k + k)B.
,
=< , > 2 =
k3(k k)k4(k2 + 2)
=k kk(k2 + 2)
.
3.6(23). () :
(s) = (s) + N (s) = T (s) k(s) T (s) = (1 k(s)) T (s) , 0.
() , -
3.4.2. s,
= (1 k)T = |1 k| = 1 k
= (1 k)T + (1 k)T = (1 k)T + (1 k)kN
= (1 k)T ((1 k)T + (1 k)kN= (1 k)(1 k)T T + (1 k)2kT N= k(1 k)B
= k(1 k)2
k =k(1 k)2(1 k)3 =
k
1 k .
3.6(24). Frenet-Serret B,
N(s) = B(s)(s)
,
:
k(s) = T (s) = (N(s) B(s))
= ( B(s)(s)
B(s))
= ( B(s)(s)
) B(s) + ( B(s)(s)
) B(s)
= B(s)(s) + B(s)(s)
(s)2 B(s) + 0
=1
(s)2((s)B(s) (s)B(s)) B(s).
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137
3.6(25). , -
. 1518,
(*) p = (s) + (s)(N(s) + B(s)),
p (s) . (*) , s,
T + (N + B) + (kT + B N) = 0,
, ,
(1 k)T + ( )N + ( + )B = 0,
1 k = 0, = 0, + = 0. = 0, = c =
1 ck = 0, c = 0.
c , 0 k = 1/c , = 0.
c.
