MGNE_U3_contenido - Wendy Flores

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ℍ
𝔇
𝔇
𝔇 𝔇
𝔇. Siendo al igual que el modelo de 
Beltrami-Klein, la 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
P
P
PQ
m PQ P
n P  m
P
n S n

P
 
PS
 
Teorema 3.1. 
Corolario 3.1.1. 
 
 
 
 
∎ 
 
 
O OR
X OX OR
A B
AB )(A B AB
A B
 
P P
A B
)(A B
P
P AB * *A P B
P
m n
P
 
m n
A B
A B
 A B A B
A B
 
 
AB A B
A B
 AB
C D
C D
 A B )(C D
 AB )(C D
 )(C D A B
 
A B
∎ 
* *A B C A B C
* *C B A
 * *A B C B AC
 
* *A B C * *C B A
 AC
D E
 A B C )(D E
* *A B C * *C B A
∎ 
 
m n
P
A B
m
n PA PB
A B
P m n
 
 O
d
 
 
P
A
B C B A C
* *A B C 𝐸 𝐹 𝐺
F E G
OF OE OG
 
m n P
1 2
P m n
 
m
 
P
)(A B
m n P
m n P
AB A B
 
)(A B
m
n
 
 
O
P
Q 'O
m Q 'P
 
P
 
 
m n m
n
 m n m n
m n
 
 m n
m  P m
m
1t 2t
m  P m
1t 2t m
m n
n m
 
m n n
m m n
m n k
k
k  P m  P n k
   P m P n
 
r O P O
'P P
'P OP 2'OPO rP 
OP 'OP OP 'OP
'P P
Teorema 3.2. O r
 'P P P
 P 'P P
'P
 
 
'P P 'OP OP
 
2
2'OP OP OP r  OP r
P P OP r
2' 'OP OP r OP r  'OP r P 'P
OP 'P P P
OP r 2 ' 'OP r Or O PP   'r OP
'P
P
'P ''P
'P 2'OPO rP  2'' 'OPO rP  
''OP OP ''OP OP P ''P OP
''P P
Teorema 3.3. P TU
OP 'P P
TU
T U
 T OP
'P
 
T OP
 Δ𝑂𝑃𝑇 Δ𝑂𝑇𝑃′ ∢𝑇𝑂𝑃
∢𝑂𝑃𝑇 ∢𝑃′𝑇𝑂
∢𝑶𝑷𝑻 ∢𝑷′𝑻𝑶
 OT r
'
OP OT
OT OP

'
OP r
r OP
 2'OP OP r
'P P
 OP T
U T
U 'P
 
T U OP
 'P TU
∎ 
Teorema 3.4. P Q
OP O OQ QP
T U PT PU
'P P TU OP
 
T U
 
T U
 ∢𝑂𝑇𝑃 ∢𝑂𝑈𝑃
∢𝑶𝑻𝑷 ∢𝑶𝑼𝑷
 TP UP P
TU
 P 'P 'P
TU OP
 
'P P
 'P P
Teorema 3.5. T U
P TU PT PU ΔPTU ≅ ∆PUT
OP TU P PT PU
T U
 P TU ∢𝑂𝑃𝑇
∢𝑂𝑈𝑃 Δ𝑂𝑇𝑃 Δ𝑂𝑈𝑃
PT PU
 
PT PU PT PU
 Δ𝑃𝑇𝑈 ∢𝑃𝑇𝑈 ∢𝑃𝑈𝑇
OP TU
∢𝑷𝑻𝑼 ∢𝑷𝑼𝑻
 TP UP
T U P
PT PU
𝑻�̂�
 
∎
Lema 3.1. O
 m n O
 1 2,P P  1 2,Q Q 1 2 1 2O P QP OQO O  
O
 O t T
O  
2
OT O
 
∢𝑃2𝑃1𝑄2 ≅ ∢𝑃2𝑄1𝑄2 ∢𝑃1𝑄2𝑄1 ≅
∢𝑃2𝑃1𝑄1
 
∢𝑂𝑃1𝑄2 ∢𝑂𝑄1𝑃2
1
21
2
OP OQ
OQ OP

1 2 1 2O P QP OQO O  
 
 C OC
1P 2P 1 2* * *O P C P
1P 2P 1 2* * *O P C P
      
2 2 2
OT CT OC 
     
2 2 2
1 2
1 2
OT OC CT
OC CT OC CT
OC CP OC CP
OP OP
 
        
      

 

∎
Teorema 3.6. P P
O P C
'P P
 'P
 C 'PP
 
'PP
 CO CP O
 T
OT
OT
  
2
2'OT OP OP r 
T
 
 
T U O
 
T U O
 OP Q
  
2
2 OOT Qr OP  
'Q P 'P P
'P P
∎
O P Q O
 
P Q O
'P P
'P P
P Q 'P
PQ 'PP
C
D C
 
P Q
A A A
 
A O
A
O
'A A
C m 'AA
A C
n A
C CA m n 
A
 
A B P Q
A B
 ,AB PQ
 ,
AP
AB P
AQ
Q
BQ
BP



AB AB
A B P Q
A B
AB
   , ln ,d A B AB PQ
P Q  ,AB PQ x  
1
,AB QP
x

 ln 1/ ln lnx x x      , ,AB PQ AB QP
 ,d A B A B
AB CD AB
CD    d AB d CD
 
A
P Q B
A B * * *Q A B P
* * *Q A B P
 ,AB PQ 1 
AP
AQ
BP BQ PQ B A
B Q
CD
E CD    d AB d CE
 ln ln lnxy x y 
A B C
* *A B C      d AC d CB d AB 
P Q
A B C
* * * *Q A B C P AP BP BQ AQ
 
 , 1
BQ BQ
A
AP BP
AB PQ
BP BPQ BQ
 
 
  
 ,AC PQ  ,BC PQ 1
       
   
ln , ln ,
ln , ,
d AC d CB AC PQ BC PQ
AC PQ BC PQ
  
   
     , , ,
CQ BQ BQ
AQ CQ A
AP CP AP
AC PQ BC PQ AB PQ
CP BP B QP
    
     
 

  

     d AC d CB d AB 
O k
O k O
P O
*P OP  *OP k OP
O k
Lema 3.2. C s
O k
* *C
 
ks Q * *Q
Q
 O
 ,x y
 ,kx ky
 ,x y
 ax by c 
ax by ck 
CQ
* *C Q Q *Q
 
ax by c 
    
2 2 2
1 2x h y h s   
   
2 2 2 2
1 2x kh y kh k s   
   
2 2 2
1 2x h y h s   
∎
 
Teorema 3.7. O r
C s O p O
2r
k
p

' ks *C C
O k P 'P
't ' 'P
'PP P
 O OP Q
P P Q
OP
 
2' 'OP OP OP r
pOQ OQ OP

'P Q O
2r
k
p

*
O k
' * '
 
*D '
 't ' 'P
u Q
't u
 t P
t u R ∢𝑅𝑄𝑃 ≅ ∢𝑅𝑃𝑄
t 't S ∢𝑆𝑃ʹ𝑃 ≅ ∢𝑆𝑃𝑃ʹ
∢𝑺𝑷ʹ𝑷 ∢𝑺𝑷𝑷ʹ
 
 ΔPSPʹ
S 'PP
't t
*PP
't t *PP
∎ 
Corolario 3.7.1. 
P
'p POP O  1k  ' '
2p r 'P P
Lema 3.3. O P Q
O 'P 'Q P Q
 
 
ΔP𝑂𝑄 ΔQʹOPʹ
ΔP𝑂𝑄 ΔQʹOPʹ ∢𝑃𝑂𝑄
2' 'OP OQOP r OQ   ∎ 
𝚫𝐏𝐎𝐐 𝚫𝑸ʹ𝑶𝑷ʹ
Teorema 3.8. O
O
O
 A O P
'A 'P A P
 
'A 'P A P
 Δ𝑂𝐴𝑃 Δ𝑂𝑃ʹAʹ
 
∢𝑂𝑃ʹAʹ 'P
'OA
𝚫𝑶𝑨𝑷ʹ 𝚫𝑶𝑷ʹ𝐀ʹ
 'P
O OP P
'P P
∎ 
 
Teorema 3.9. O
O
O O
 'A O A
'A OA
A
OA A
 
O
O
∎ 
Teorema 3.10. 
 
 
P m
P
 'P P ' '

'm 'P ' '
 
 'm
m 'PP
P 'P
 
 
∎ 
Teorema 3.11. O , , ,A B P Q
', ', ', 'A B P Q
   , ' ', ' 'AB PQ A B P Q
 
' ' ' '
' '
y
AP A P AQ A Q
OA OP OA OQ
 
 
' ' '
' ' '
AP AP OA OQ A P
AQ OA AQ OP A Q
  
 
' ' ' '
' '
y
BP B P BQ B Q
OB OP OB OQ
 
 
' ' ' '
' ' '
BQ BQ OB B Q O P
BP OB BP OQ B P
  
 
 
 
' ' ' ' ' ' '
,
' ' ' ' ' '
' ' ' '
' ', ' '
' ' ' '
AP BQ OQ A P B Q O P
AB PQ
AQ BP OP A Q OQ B P
A P B Q
A B P Q
A Q B P
 
   

 
    

 


 
∎ 
Teorema 3.12. 
 C s
O r P
'P
 
 CP Q 'Q
2
' '
CQ CP
s
CQ CP
  P Q 'Q 'CQ CP CQ 
2 2 2
'
s s s
CQ CP CQ
  ' 'CQ CP CQ 
'P 'Q Q 'P
 
'
' O
 O C
O
 
' C '
 
'
 A B P Q
A B
P Q 'P 'Q
'A 'B
    ' 'd AB d A B
 
B A D A B D
     d AD d AB d BD 
r O A B C
A B
A C
'A 'A
A 'A s
2'' sAA OA   ' 'AA A O AO 
 
     
2 2
2 2' ' ' ' '' 'A O A Os AA A O AO AOA O A O A O r       
O
A
ΔABC
ΔOBʹCʹ
𝚫𝐀𝐁𝐂 𝚫𝐎𝐁ʹ𝐂ʹ
O
ΔABC ΔXYZ ∢𝐴 ≅ ∢𝑋    d AC d XZ
   d AB d XY ΔABC ΔXYZ
ΔOBʹCʹ Δ 𝑌ʹZʹ
 
Lema 3.4.  d OB d
 1
1
d
d
r e
OB
e



e
r
P Q B
* * *Q O B P    ln ,d d OB OB PQ 
 ,d
OP BQ BQ r OB
e OB PQ
OQ BP BP r OB

  

 
OB
 1
1
d
d
r e
OB
e



∎ 
A X O 
OB OY OC OZ ∆𝐵𝑂𝐶 ≅ ∆𝑌𝑂𝑍
O
Δ𝑂𝐵𝐶 Δ𝑂𝑌𝑍
B C
Y Z
Δ𝑂𝐵𝐶 Δ𝑂𝑌𝑍
 
Teorema 3.13. 
 d
d
180

Teorema 3.14. 
 
tan
2
d de
  
 
 

 d  d PQ P
 d
P PQ
 
Q P
 
PQ
 P

P R
Q P
R P
 ΔRPΣ ΔRΣP
  d 
 
∢𝑅𝑃𝑄 ∢𝑃𝑅𝑄 ΔPRΣ
 
ΔPRQ 2 2
2 2
 
   
 
 
 PQ tanr 
1 tan
1 tan
de





 
4 2
 
  
1 tan
2
tan
4 2
1 tan
2

 

 
  
     
     
 
1 tan 1 tan1 tan
2 22
1
1 tan 1 tan 1 tan
1 tan 4 2 2 2
1 tan
1 tan 1 tan 1 tan 1 tan
4 2 2 2 2
1
1 tan 1 t
2
de
 
   

    

       
          
        
     
                 
           
               
          
 
  
 
an
2
2
1 tan
12
2 tan tan
2 2
1 tan
2


 

 
 
 
 
  
  
   
   
   
 
  
 
 
 
tan
2
d
de
  
 
 

∎ 
 
 
http://www.amazon.com/s/ref=ntt_athr_dp_sr_1?_encoding=UTF8&field-author=Richard%20Courant&search-alias=books&sort=relevancerank
http://www.amazon.com/s/ref=ntt_athr_dp_sr_2?_encoding=UTF8&field-author=Herbert%20Robbins&search-alias=books&sort=relevancerank
http://www.amazon.com/s/ref=ntt_athr_dp_sr_3?_encoding=UTF8&field-author=Ian%20Stewart&search-alias=books&sort=relevancerank
http://www.amazon.com/Robin-Hartshorne/e/B001IQZAEQ/ref=ntt_athr_dp_pel_1
http://www.ms.uky.edu/~droyster/courses/spring08/math6118/Classnotes/Chapter09.pdf
http://www.ms.uky.edu/~droyster/courses/spring08/math6118/Classnotes/Chapter09.pdf
http://www.math.cornell.edu/~mec/Winter2009/Mihai/section7.html
http://mathworld.wolfram.com/Klein-BeltramiModel.html