The free-space diagram Fréchet distance

free-space diagram of red , blue curve. in contrast definition in text, uses parameter interval [0,1] both curves, curves parameterized arc length in example.

an important tool calculating fréchet distance of 2 curves free-space diagram, introduced alt , godau. free-space diagram between 2 curves given distance threshold ε two-dimensional region in parameter space consist of point pairs on 2 curves @ distance @ ε:

d

ε


(
a
,
b
)
:=
{

(
α
,
β
)
∈
[
0
,
1

]

2


∣
d
(
a
(
α
)
,
b
(
β
)
)
≤
ε

}


{\displaystyle d_{\varepsilon }(a,b):=\{\,(\alpha ,\beta )\in [0,1]^{2}\mid d(a(\alpha ),b(\beta ))\leq \varepsilon \,\}}

the fréchet distance



f
(
a
,
b
)


{\displaystyle f(a,b)}

@ ε if , if free-space diagram




d

ε


(
a
,
b
)


{\displaystyle d_{\varepsilon }(a,b)}

contains path lower left corner upper right corner, monotone both in horizontal , in vertical direction.