Graphics Reference
In-Depth Information
4.1
GEODESIC DISTANCES ON SURFACES
In general, surfaces are defined as 2-dimensional manifolds with or without boundary. is gen-
eral definition can be translated into a parametric formulation to ease the formalization of geodesic
and curvature expressions.
A surface is called
regular
if in a neighborhood of each of its points it can be expressed as
(
regular parametrization
)
D.u;v/Dx.u;v/;y.u;v/;z.u;v/
where
.u;v/
is a regular (i.e., a sufficient number of times differentiable) vector function satis-
fying
0
@
1
A
0
@
1
A
¤0:
@x
@u
@y
@u
@z
@u
@x
@v
@y
@v
@z
@v
u
v
D
Any regular surface can be considered as a metric space with its own
intrinsic metric
, with
the
surface element
defined as
ds
2
Dd
2
DE.u;v/du
2
C2F.u;v/dudvCG.u;v/dv
2
where
E.u;v/Dh
u
;
u
i
,
F.u;v/Dh
u
;
v
i
,
G.u;v/Dh
v
;
v
i
. e length of a curve
de-
fined on the surface by the equations
uDu.t/
,
vDv.t/
,
t20;1
, can be computed as
Z
p
1
0
2
C2Fu
0
2
dt:
l./D
Eu
0
v
0
CGv
0
en, the
geodesic distance
between two points on the surface is defined as the infimum of the
lengths of all curves on the surface connecting the two points (cf. sections
3.1
and
3.2
).
e expression
IDds
2
DE.u;v/du
2
C2F.u;v/dudvCG.u;v/dv
2
is called the
first fundamental form
. e
second fundamental form
is defined as
IIDLdu
2
C2MdudvCNdv
2
with
LD
uu
n
,
MD
uv
n
,
ND
vv
n
, and
n
the normal vector
nD
u
v
j
u
j
:
Figure
4.1
shows that the distance between two fingers measured along the hand is always
the same, independently of the posture. e use of geodesic distances proved to be effective in a
number of studies, and paved the road to a number of tools for intrinsic non-rigid shape analysis.
v
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