![](bg6.png)
This approach allows to adjust the network of the re-
constructed curves running on the surface and to get
a “coherent” mesh interpolating the animated surface
by an updating process from the previous time posi-
tion. The method has been tested on an animated an-
alytical surface.
Then, in order to validate the method in the gen-
eral case of a physical animated surface, a testing
framework surface is under development, so that, per-
formance comparisons with real physical data have
not been realized yet. Anyway, the first experiments
have highlighted some practical difficulties concern-
ing the implementation of the acquisition process. Es-
sentially, we have to ensure a permanent smooth con-
tact between the ribbon and the animated surface. So-
lutions (i.e., smoother and more flexible ribbons,...)
are on progress.
REFERENCES
Coons, S. (1964). Surfaces for computer aided design.
Technical Report, M.I.T.,1964, Available as AD 663
504 from the National Technical Information service,
Springfield, VA 22161.
Coons, S. (1974). Surface patches and b-spline curves.
Computer Aided Geometric Design, In R. Barnhill
and R. Riesenfeld editors, Academic Press.
Farin, G. (2002). Curves and Surfaces for CAGD - Fifth
Edition. Academic Press.
Fontaine, D., David, D., and Caritu, Y. (2003). Sourceless
human body motion capture. In Proc. Smart Objects
Conference (SOC’03).
Hoschek, J. (1983). Dual b
´
ezier curves and surfaces. Com-
puter Aided Geometric Design, North Holland, pages
147–156.
Hoshi, T. and Shinoda, H. (2008). 3d shape measuring
sheet utilizing gravitational and geomagnetic fields. In
Proceeding of the SICE Annual Conference 2008. The
University Electro-Communications, Japan.
Nielson, G. (2004). ν-quaternion splines for the smooth
interpolation of orientations. IEEE Transactions on
visualization and computer graphics, 10(2):224–229.
Peters, J. (1990). Local smooth surface interpolation: A
classification. Computer Aided Geometric Design,
7:191–195.
Sarraga, R. F. (1987). G1 interpolation of generally unre-
stricted cubic b
´
ezier curves. Computer Aided Geomet-
ric Design, 4:23–39.
Shirman, L. A. and S
´
equin, C. H. (1987). Local surface in-
terpolation with b
´
ezier patches. Computer Aided Ge-
ometric Design, 4:279–295.
Sprynski, N., Lacolle, B., David, D., and Biard, L. (2007).
Curve reconstruction via a ribbon of sensors. In Pro-
ceeding of the 14th IEEE International Conference on
Electronics Circuits and Systems. ICECS, Marrakech,
Maroc.
APPENDIX
A: Initial Reconstructed Curves. A 3D curve
C(s) = (x(s), y(s), z(s)), parameterized with respect to
its arc-length s satisfy |C
′
(s)| ≡ 1, so that the deriva-
tive curve C
′
(s) is a curve lying on the unit sphere.
Initial data are unit tangential vectors at points with
assigned arc length parameters. The methodology is
thus as follows.
• First, we interpolate data using cubic splines on
the sphere, leading to the derivative curve C
′
(s).
• Then, by integration we get a solution for C(s).
Cubic splines on the unit sphere (see (Nielson, 2004))
are an extension of the usual B-splines in the euclid-
ian space. The main differences are the following.
⋄ The evaluation of the control polygon of cubic
splines on the spherical space requires to solve a non
linear system through an iterative algorithm.
⋄ The usual De Casteljau algorithm, based on linear
interpolations, has to be replaced by the spherical in-
terpolation
Sler p(a, b, t) =
sin((1 −t)θ)a + sin(tθ)b
sin(θ)
,
where a and b are two unit vectors, θ the angle be-
tween vectors a and b, and t ∈ [0, 1].
It is proved in (Sprynski et al., 2007) that this con-
struction is invariant under rotations and scaling, and
that these spherical splines minimize a combination
of the curvature κ
1
, the torsion κ
2
, and the variations
of the curvature, precisely
min
∫
(κ
′2
1
+ κ
2
1
(κ
′2
1
+ κ
2
2
)),
which gives physical sense to the reconstruction.
B: Cubic Hermite Interpolation. Given spatial
points p
0
and p
1
associated with tangent vectors t
0
and t
1
, together with two parameters α
0
and α
1
(α
0
<
α
1
), there exists a unique cubic spatial parametric
curve r(t) such that
r(α
0
) = p
0
, r(α
1
) = p
1
, r
′
(α
0
) = t
0
, r
′
(α
1
) = t
1
.
Precisely, r(t) is defined by
r(t) = H
0
(
ˆ
t)p
0
+ H
1
(
ˆ
t)p
1
+ (α
1
− α
0
)H
2
(
ˆ
t)t
0
+ (α
1
− α
0
)H
3
(
ˆ
t)t
1
,
with
ˆ
t =
t−α
0
α
1
−α
0
and where functions φ
j
are the cu-
bic Hermite polynomials (Farin, 2002) on the interval
[0, 1], and r(t) and is denoted shortly by
r(t) = H[p
0
, p
1
, t
0
, t
1
;α
0
, α
1
](t).
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