Repère projectif et birapports

−→
E3
P
e
P P P
P(−→ E3)P−→ E3P
e
M
∼
=PP(−→
E3)
−→
∆( −→ ∆(
e
M)
P(−→ E3)
P(−→ E3)
P
e
P
/{0}
∼
=∼
=
p:−→ E/{0} → P(E)
~u 7→ p(~u) = −→ ∆(~u)
•−→ ED−→ E2
D
D ∞DD
e
D−→ E2e D
−→
E3
−→
E2
•−→ E3P−→ E3
e
P P ∞
−→
E3
e
Me Pe D
P D
D P ϕ= 1 ϕ
P D
e
Pe D

−→
E
−→
E2(−→ ∆1,−→ ∆2,−→ ∆0)
(~u1, ~u2, ~u0)~u0
~u0=k1~u1+k2~u2k1, k2
~e1=k1~u1, ~e2=k2~u2~e0=~u0~e1+~e2
−→
∆~u −→ ∆(~u)
~u B(~e1, ~e2) : ~u = T ~e1+ Z ~e2
−→
∆6=−→ ∆1
(−→ ∆1,−→ ∆2,−→ ∆0,−→ ∆) B(~e1, ~e2)Rp
Φ∗~u1, ~u2
½Φ∗(~u0, ~u2) = k1Φ∗(~u1, ~u2),Φ∗(~u, ~u2)=Tk1Φ∗(~u1, ~u2)
Φ∗(~u0, ~u1) = k2Φ∗(~u2, ~u1),Φ∗(~u, ~u1) = Z k2Φ∗(~u2, ~u1)
Z6= 0 (−→ ∆6=−→ ∆1)
T
Z=Φ∗(~u0, ~u1) Φ∗(~u0, ~u2).Φ∗(~u, ~u1) Φ∗(~u, ~u2)
(−→ ∆1,−→ ∆2,−→ ∆0)−→ E2
−→
∆−→ E2∼
=
~e1, ~e2, ~e0=~e1+~e2X ~e1+Z~e2
∆−→ E
−→
∆0
−→
E2(−→ ∆i(~ui))1,2,3,4
~ui
−→
∆i
[−→ ∆1,−→ ∆2,−→ ∆3,−→ ∆4] = Φ∗(~u3, ~u1)
Φ∗(~u3, ~u2)ÁΦ∗(~u4, ~u1) Φ∗(~u4, ~u2)
Φ∗
−→
∆6=−→ ∆1[−→ ∆1(~e1),−→ ∆2(~e2),−→ ∆3(~e1+~e0) : −→ ∆4(k~e1+~e0) ] = k
e
M1,e M2,e M3,e M4
(−→ ∆( e M1),−→ ∆( e M2),−→ ∆( e M3),−→ ∆( e M4))

−→
E2
[e M1,e M2,e M3,e M4]
e
D
RbDe Mi
αi, βi
δi,j =¯¯¯¯αiαj
βiβj¯¯¯¯[e
M1,e M2,e M3,e M4] = δ3,1
δ3,2.δ4,1 δ4,2
D
D ∞DRb
(1,−1) δ1,i =αi+βi= 1
[∞D,M2,M3,M4] = δ4,2
δ3,2
D
ti i Ra(−→ AB,A) (αi, βi) = (ti, ti)
δi,j =−(ti−tj)
[ M1,M2,M3,M4] = t3−t1
t3−t2.t4−t1 t4−t2
[t1, t2, t3, t4]
[ M1,M2,M3,M4] = M3M1
M3M2.M4M1 M4M1
De M1∞Dδ1,j = 1
[∞D,M2,M3,M4] = t4−t2
t3−t2
t1→ ∞
tRb[∞D,A,B,M ] = [ ∞,0,1, t ] = t
e
D −1
−→
E2
D
~
i(a+b)(c+d) = 2(ab +cd)
2/b = 1/c + 1/d b c, d
[ ] a=−b a2=bc |a|
(c, d)
D2b=c+d b (c, d)
e
D
[−→ ∆1,−→ ∆2,−→ ∆0]e D
D D
Ω~ i=−→ OΩ [−→ ∆1(
~
i),−→ ∆2(O),−→ ∆0(Ω)]
De D Ra(
~
i, O)
−→
∆(M) Dt
~
i+ O ∼
=T
~
i+ Z O

Ra
e
D D
Rb
e
D−→
∆( e M) e M∈e DC = A/2+B/2⇒M = T A/2 + Z B/2∼
=αA + βB
Rb
Rp
D(α0, β0)
Rb(α, β)e M∼
=(α/α0, β/β0)
−→
∆1−→ ∆2D−→ ∆0D∼
=(−α, β)
−→
∆( e M)
f−→ E2
−→
E2
f
•f−→ E2−→ ∆
∼
=f
h(−→ ∆) h−→ E2
h(−→ ∆(~u)) = −→ ∆(f(~u))
∼
=−→
E2
−→
E2
•h
−→
∆1(~e1),−→ ∆2(~e2),−→ ∆0(~e1+~e2),−→ ∆1(f(~e1)),−→ ∆2(f(~e2))
−→
∆0(f(~e1) + f(~e2))
•−→
∆(k~e1+~e2)−→ ∆(kf(~e1) + f(~e2)).
−→
E2e D
h
−→
E2he Me D
−→
∆(f(e M) h(−→ ∆( e M) h
f h e D D
hD D −→ ∆(fD
ϕ(f(M)) 6= 0 ϕ◦f
Dϕ(f(M)) −ϕ(f(N)) = ϕ(f(−−→ NM)) = 0
hD Ra(
~
i, +t
~
i a =ϕ(f(0)) 6= 0
f(
~
i) = k~ i f ϕ ~ i
h(M) = h(O) + (k/a)t~ iD
h(M) = M0, h(N) = N0OM
ON =OM0
ON0
hD D ϕ(f( = 0 f(
D Ra(
~
i, B(
~
i, −→ E2

fΣh
Σ = µa b
c d ¶⇒f(t
~
i+ O) = (at +b) + (ct +d) O , ϕ(f(t
~
i+ O) = ct +b
c h h
−d/c t t0Ra
ehe
D
h:t7→ t0=at +b
ct +d
Rp(−→ ∆1,−→ ∆2,−→ ∆0)R0
p(−→
∆0
1,−→
∆0
2,−→
∆0
0)
B B0B B0Σ
B−→ e0
1
−→
e0
2
RpR0
p
∼
=h
RpR0
p~u1, ~u2, ~u0
(−→ ∆0
1,−→
∆0
2,−→
∆0
0)k1, k2, k0k0~u0=k1~u1+k2~u2
U1,U2,U0~u1, ~u2, ~u0BΣ1
¡U1U2¢k1, k2k1, k2, k0
Σ1K = k0U0K = k0Σ−1
1U0∼
=Σ−1
1U0
Rp
R0
pDKk1, k2Σ = DKΣ1
D~ i
Rp(−→ ∆1(
~
i),−→ ∆2(O),−→ ∆0(
~
i+ O)) R0
p(−→
∆0
1(A),−→
∆0
2(B),−→
∆0
0(C))
BA = a
~
i+ O,B = b
~
i+ O,C = c
~
i+ O
Σ1=µa b
1 1 ¶,U0=µc 1¶⇒K∼
=µ1−b
−1a¶µ c 1¶=µc−b a−c¶
Σ = µc−b0
0a−c¶µ a b 1 1 ¶=µa(c−b)b(a−c) c−b a −c¶
ΣRpR0
p
µt
1¶=µa(c−b)b(a−c) c−b a −c¶µ t0 1¶e:t=a(c−b)t0+b(a−c)
(c−b)t0+ (a−c)
et=h(t0) Σ−1t0=h−1(t)
µt0
1¶∼
=µa−c−b(a−c)
−(c−b)a(c−b)¶µ t 1¶:t0=(c−a)(t−b)
(c−b)(t−a)
h(
~
i(direction de ∞D)7→ a, O7→ b,~ i+ O 7→ c)h(∞) = a, h(0) = b, h(1) =
c, h(t0) = t t0= [∞,0,1, t0]h
h(t0) = [ h(∞), h(0), h(1), h(t) ] = [ a, b, c, t0]t= [ a, b, c, t0]
et7→ t0a, b, c ∞,0,1