Triangulations, carquois et théorème de Ptolémée

Z
An

(E, M)E M
E
E M
E M
M M
M M
Rn
(x1, x2, . . . , xn),(y1, y2, . . . , yn)∈Rnd((x1, x2, . . . , xn),(y1, y2, . . . , yn))
d((x1, x2, . . . , xn),(y1, y2, . . . , yn)) = v
u
u
t
n
X
i=1
(xi−yi)2.
X, Y f :X−→ Y
O Y
f−1(O) = {x∈X|f(x)∈O}
X
E E0
f:E→E0f−1
f

S
f: [0,1] −→ S
t7−→ f(t).
SRn
x∈Rnr > 0s∈ S
d(x, s)< r
S
S S
[a, b]×[c, d]
(x, c)∼(b−x, d)x a ≤x≤b
A
A
S S
S2=v∈R3| ||v|| = 1

n n
R3
S2n
R2
S D1, D2, . . . , Db
C1C2. . . CbS R =S\(D1∪. . .∪Db)
∂R =C1∪. . . ∪Cb
S2\(D1∪D2)

S2\D1
S M ={x1, . . . , xn}xi∈S
xi∈δS R R = (S, M)
M⊆δS
R= (S, M)γ, δ
γ: [0,1] →S
t7→ γ(t)
δ: [0,1] →S
t7→ δ(t).
γ(0) = δ(0) γ(1) = δ(1)
h: [0,1] ×[0,1] →S
h(0, t) = γ(t)h(1, t) = δ(t)t.
γ δ
(R, M) (R, M )
αi∈R αi(0) ∈M
αi(1) ∈M
M
R2
R M
R∪∂R T
M