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Géométrie dans un espace affine euclidien

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www.touscours.net, Groupes,Permutations,Anneaux,Arithmétique dans Z,Corps commutatif,Les polynômes formels à une indéterminée à coefficients dans un corps K,Fonctions polynomiales,racines,Espaces vectoriels,K-algèbres,Espaces vectoriels de type fini,Matrices,Déterminants,Fractions rationnelles,Produit scalaire sur un R-ev,Espace vectoriel euclidien,R-ev euclidien orienté de dimension 2,R-ev euclidien orienté de dimension 3,Espaces affines,Géométrie dans un espace affine euclidien

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Géométrie dans un espace affine euclidien

  1. 1. ! " # ε $ % & ' # "(= ) % * • ="( " % +# "("( →×εε , $ + , =⇔= -"( , "("( = , "("("( +≤ • . ε " ε " { }∈= ""("( • . &" / ε " { }&&&& """("( ∈∈= 0 1 # " # "2 3 , % 4/ . &" / &" " # 5"( & # 5"( & % ' 3 6 , # ε∈ -≠ " # { }+ ∈+ λλ "% % +# / , # % touscours.net
  2. 2. & 7 8 . , ε % . ε∈ % + 8 = # 8 9 # 3 3 ⊥ ( : ,$, 3 ⊥ ∈ ) % ; + 8 # 3 3 "("( = % < 8 % ∈ " &&& += % ≥ " = = % 4 ∈ " ( ∈ = % > + = • +#= ε∈ $ { }-?∈ " # { }⊥∈= "ε • @ ' 3 A + # ⋅ →ε " ) ε∈ { }-?∈ ∈ " # ⋅ " # { }=⋅∈= "ε % < "( 3 ∈ % 4 " 3 & % = % " ε∈ -=⋅⇔ ⋅=⋅⇔ =⋅⇔∈ #= $ % • B= / . ε∈" " % . { } { }=∈==∈= ""("(" εε . C"D % &((( && && ⋅=+⋅−=−=− -=⋅⇔= touscours.net
  3. 3. * { }-" =⋅∈= ε " E #= $ % < # #= % • # $ = % . = $ % . ε∈ F ="( ) 8 % %λ += % & %λ=⋅ % ⋅ == & %λ G + . εε → % ε ⇔ " : ,$, "(("(("" =∈∀ ε + ε / ε ε $ ( ( : ,$, • . (∈ϕ " " ε∈" " #" # "((:::":( ==== ϕ • . 3 % < 3 # % . +=ϕ % H 3 ϕ ( : ,$, 3 (∈ϕ . ∈ " ε∈ " += % = ===== ::::(( ϕϕ ( I ! ε = ε " : ,$, $ ( % ! ε = ε " : ,$, $ (?( % (J ε " ε " ( , (ε " # (ε ε , % 4/ + = (J ε % touscours.net
  4. 4. 0 4 " = $ , " " = $ ( ) % 4 " 3 ⊥ = " 3 = $ F = $ " 3 # ( . =( " =( " " = $ " 3 − " % + / ( = $ = % 4 / ε " / / 3 " # / #= #= % (! + & + 4 . " +=ϕ ( " (ε∈ (∈ϕ % < 3 ϕ " # % • ϕ # " # # % " % ' 3 " (ε % • / ϕ # θ ( π& % ' % . ε∈ " % . ε∈ % (:(( ϕ+=+= % :(J( :(:(/ =−⇔ =−⇔=⇔ ϕ ϕϕ < { } { }-("JK ( ==∈=− ϕϕ ϕ 8 " 8 ( / :(J( − −=⇔ ϕ % < / Ω % ($ :(J( − −+=Ω ϕ H (((" : Ω+Ω=Ω+Ω=∈∀ ϕε " (: Ω=Ω ϕ % touscours.net
  5. 5. 7 < 3 Ω # θ % J " " # 3 Ω Ω (∈θρ % ' 3 3 (:= $ =ΩΩ Ω=Ω C&D:5"( : πθ (ε 4 1 ∅ ; ε J ; ' { }Ω ' # Ω & / . / " & / & % . &= " εJ& = % . &≠ &LL " =& " ) &&= . Ω θ & Ω # / # θ& )θ # 5"( & (J 8 # J " L " / L Ω 3 / / L % " / / % ; / " M -" " &" * / % 4 . (J ε∈ % . " 3 # - & / % . / 3 3 " (ε∈ " - & / " − = * / % touscours.net
  6. 6. > * + . (?(J εε∈ % +=ϕ % (?(∈ϕ % ϕ / " (N= % / % " ε∈ " " (((: ϕ+=+== (: ϕ= / % J " / (?(J εε & # / % . ε∈ " (:= % < 3 % " ϕϕ =J % = / : = % H " : # ⊥ ∈∈ +=: % == : % ( ( < = % ϕϕ == J+ % H / ( " / &+= % / LL: &+ ( " = " ) / @ A $ / % (:= : ( touscours.net
  7. 7. O ! / % < = / % 4 / 1 ∅ ; ε εJ { }Ω ' # Ω ' / % J ∅ ' / ( : ,$, ) / { }-?(∈ G = 3 & . ""(= " % ! 3 =+ " ) -"-("( ≠ % ! $ " − % # $ . - - - " =+ % < . ∈ % && -- && --- - (( "( + −+ = + −+− = ⋅ = % &&&& =+ =+ θ θ +# θ " # # 5"( & % & & & & && && & & ++ + = ⋅ =θ 43 . " & & & % +#3 - & (( & (( -- & & & & = + −−+ + −− % touscours.net
  8. 8. + . εε → % ⇔ "(%("(("""P =∈∀∈∃ + ε % " . " "(%("(("""PQ =∈∀∈∃ + ε % % . εε → " -> % < 3 ⇔ # # % ⇔ # ϕ% ) (∈ϕ ⇔ % " . % . ψ % (∈= ψϕ % • . ϕ ( " 3 " 3 % • " & ( • . " " % +# ε , "(( ε % & 4 / < ) ""-( % R 3 • ; ) ∈ % !! + • .= ( / $ #/ !! touscours.net
  9. 9. • B ∈α !! %α % B -! ∈α %( -- !!!! −+α ( ---- %( α+=+ • ' # ∈θ !"! θ% ' -! # ∈θ ( - % - !!"!! −+ θ + + / = !! +% " ) ×∈ P"( % • . ×∈ P"( % # θ α % %" " ) P∈α ∈θ % !! +% !"! θ% ( α+ ( 4 %α ( # # # " % • J " % 4 # # ( : ,$, # " = !! +% 3 # = !! +% % 4 . !! +% % -! / !!! =−⇔=+⇔ --- (% . = -≠ " #= / " !! + % . = -= " # % . ≠ " / -! % !! += %( " !! += -- % % (( -- !!!! −=− % # θ α % %" ) P∈α ∈θ " 3 " " # -! α -! # θ % J 3 !! +% " ) ×∈ P"( % touscours.net
  10. 10. - 4 " # !! +% " # !! +% " % J " " !! " " " # +% " = " !! +% % / 4 1 ∅ ; J { }-! . $ " : ,$, ( # -! # θ # -! α ( % % ≠θ α " (' . C"D C:":D / ( / % / 3 C"D C:":D ( " / " " " % . ∈βα" % . βα +!! % % =+ =+ : : % % !! !! βα βα " : ,$, −= −=− !! !!!! % %( : :: αβ α % < 3 P∈α " ∈β % =:: % . " $ :: # ::"( !! !! − − = :: α " !! !! :::: = − − =α C&D::"( "(::"( ((( :: π α !!!! = −= −−−= touscours.net
  11. 11. . ε % . ""(= ε % . ε∈ & "( ∈θρ % < 3 "( θρ = 3 (% θρ= " ) (θ %% θθ + " : ,$, 3 C&D(5"( πθθ = % J 3 # 8 = " • + = / "-( θρ = = % • ! ≠ / = ∈++−=∈+= "&"(""&"( ππαρπαρ " ) α # "( % 43 " / • + #3 *=ρ ( $ # ε∈ 3 # = "( θρ *=ρ % " # # ε∈ 3 # / ∈θ 3 (%* θ= " # = *% • #3 θρ = " + ∈θ ( ∈θ π& π &Lπ &L*π J " ε *" # """(= % (+ * + 4 . (ε∈ " ϕ % touscours.net
  12. 12. & (∈ϕ % # " #/ "( ω # θ % • . C&D- πθ = " J=ϕ " % • . C&D- πθ ≠ , . # / # " 3 % ε∈ " ( 3 (: ϕ= % . " 8 % ∈ " ∈ " ϕ " ( =: % (: ϕ= % < 3 #/ "( ω # θ ω θ # / % J " = % , . # / . ε∈ " % : = % ϕϕ =J % # #/ "( ω # θ ) " θ # ϕ " : = ω θ H : : ⊥ ∈∈ +=: % 4 = % . $ % ( / ϕ ⊥ : ,$, ( M " $ " $ 3 ϕ # # % < / L % / % # #/ ":( ω # θ ) # % touscours.net
  13. 13. * = " ) #/ ":( ω # θ " % ω θ # M $ #( -≠ " #= / % < 3 ( #/ ":( ω " # θ % ( 4 1 ∅ J ; ε J εJ θρ " -≠θ #/ "( ω ) (= ' #/ "( ω # θ ∅ θρ " -≠θ #/ "( ω N #/ "( ω # θ " ) " ∈≠ - 3 " / ' E : : : : !! ) += −= −= &: &: >: !! ( # # # 3 & & > : # − − !! : " : ,$, # ϕ − − -- -- -- ""( % 4 " 3 (:((" ϕε +=+=∈∀ . − − += !! -- -- -- & & > : : : touscours.net
  14. 14. 0 (& J # # π ! ( = $ ! " #/ ! # ( =ϕ (∈ϕ % J≠ϕ " " #/ % (* ' ! = = = ⇔ += −= −= ⇔ -& * & & > !! < # " # % (0 #/ " # . #/ "( ω # -≠θ -≠ % { }$:"ε∈= " # 3 : 3 % $" − − & && &> : % = = ⇔∈ * (N: #/ ! - * % / / , . &LL " =& " ) &&= ( # &= & & , . α =∩ & " & # α& % ' 3 ! 3 / " / & ( # " " " % touscours.net
  15. 15. 7 G = 3 * 43 #! =++ " ) -"-"-(""( ≠# % ! $ # < 3 " ε∈ -=⋅⇔⋅=⋅⇔=⋅⇔∈ <) 3 %λ= & =λ % J / &&&&& !# !# =++ =++ < # " & & & & # % . -& =∧ " &LL % . -& ≠=∧ " &∩ " % 4 ⊥ " (∈ " &⊥ ( &∈ % (( &∩=∈ ( -≠ % # $ . $ % % ="( " %λ= % += % && %-% λλ =+=⋅ " & ⋅ = " &&& # #! ++ −++ = ⋅ = % # $ ="( % < ∧=∧+∧=∧ % =∧ ( ⊥ % ∧ =="( % touscours.net
  16. 16. > = 3 3 = 3 . ε∈ " ( ""( ! % < = = 3 ""( !$ θ * ""(""( !$$! θθ= % " " 3 ∈θ " %%( θθθ += " " 3 ε∈ " % 8 " 3 ""( !$ θ = = 3 $ ⇔+=⇔ !$ %(% θ "( θ$ = % ""( ! S % 4 ε = = 3 • . !∈ " = ""-( !θ " ∈θ 3 3 % • . !∉ " # # / $ = " 5"( %=θ ( "( " ! S % 4 "&"( !$ πθ + "&"( !$ ππθ ++− ∈ 3 3 % ' 3 / $ θ ( !∉ " && $ += " $=θ " $=θ % θ & 3 . ε∈ " ( ""( ! % < = 3 ""( ϕθ$ * ϕθ$= " ϕθ$= " ϕ$! = % 4 . ε∈ " ""( ! % < 8 % 8 % 3 ""( ϕθ$ = 3 $ 3 ( "( ϕθ$ = % $ ""( θ$! = % touscours.net
  17. 17. O , . = " ( 3 -== θθ $$ " 3 -=$ % ' 3 " ""( ϕθ$ 3 -=$ = 3 % , . ≠ " !∈ " ( 3 θ$! = -=θ % ' 3 " ""( ϕθ$ " : ,$, = "&"( ϕπ! "&"( ϕππ! +− ( ∈ϕ 3 3 = 3 % , 4 " !∉ "( αρ = % ""( " ( 3 +=−== === C&D C&D ππαϕρθθ παϕρθθ $!$ $!$ ' 3 " ""( ϕθ$ = 3 % J 3 ε = 3 % ' 3 . ""( ϕθ$ = 3 " 8 & & $ = 4 &&&&&&& & ( $$$ =++= θϕϕθ ' # = 3 !∉ % . ε∈ " 3 !∉ 8 % 8 " ""( ! % $ = % . [ ]πθ "-∈ # "( θ$! =⋅= % " & && !% += " θθ &&&& & ( $$% =−= " -≥$ -≥θ ( [ ]πθ "-∈ " θ$% = . ϕ # "( "( % %%($% ϕϕθ += 4 ""( ϕθ$ = 3 % ϕ θ touscours.net
  18. 18. ' 3 3 !∉ " ->$ ] [πθ "-∈ % ϕθ""$ &&& !$ ++= " = $ ! θ " θ ϕ $ = " θ ϕ $ = % " = 3 3 ] [ [ ]ππϕπθ """-"- −∈∈>$ # # " = 3 :&"&"( :&"&"( :&"&"( :&"&"( ππϕππθ πϕππθ ππϕπθ πϕπθ $ $ $ $ ++++−− +++− +++− ++ 4 3 3 ε " 3 # #/ ! " = 3 ""( ϕθ$ -≥$ " [ ]πθ "-∈ C"D ππϕ −∈ % H $ = " 3 3 ( / % touscours.net

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