Optimal time computation of the tangent of a discrete curve

D⊂Z2
∃(a, b, µ, ω)∈Z4∀M= (x, y)∈D, µ ≤ax −by < µ +ωa
b
µ ω
ω=sup(|a|,|b|)
c0, ..., cnn∈N
0≤i≤n, 0≤j≤n cicj

cici−kcici+k+p
ci−k−qcici+kci−k−1cici+k+1 ci−kcici+k+p+1
ci−k−q−1cici+k

PiRiMiLiGi
Ri
Pi+1 Ri+1 Mi+1 Li+1 Gi+1 Pi+1
Mi+1 Pi+1
Mi+1 Ri+1 Gi+1 Li+1
Gi+1 Li
Gi+1 =LiLi+1
Pi+1 Ri+1
Li+1
Pi+1 Ri+1 Li+1
Ri+1 ˆ Ri+1 Ri
ˆ
Ri+1 Li+1
ˆ
Li+1,ˆ Ri+1]
Ri+1 ˆ Li+1 =Li+1
ˆ
Ri+1 6=Ri+1 Li+1 ˆ Li+1 =Li+1,ˆ Ri+1]
]ˆ Li+1 =Li+1,ˆ Ri+1]⊂]ˆ Li+1 =Li+1,ˆ Ri+1 =Ri+1]
O(1)
O(1)
O(n)