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Abstract:
Accurate estimations of geometric properties of a surface (a curve) from
its discrete approximation are important for many computer graphics and
computer vision applications.
To assess and improve the quality of such an approximation we assume
that the
smooth surface (curve) is known in general form. Then we can represent the
surface (curve) by a Taylor series expansion
and compare its geometric properties with the corresponding discrete
approximations. In turn
we can either prove convergence of these approximations towards the true
properties
as the edge lengths tend to zero, or we can get hints how
to eliminate the error.
In this report we propose and study discrete schemes for estimating
the curvature and torsion of a smooth 3D curve approximated by a polyline.
Thereby we make some interesting findings about connections between
(smooth) classical curves
and certain estimation schemes for polylines.
Furthermore, we consider several popular schemes for estimating the
surface normal
of a dense triangle mesh interpolating a smooth surface,
and analyze their asymptotic properties.
Special attention is paid to the mean curvature vector, that
approximates both,
normal direction and mean curvature. We evaluate a common discrete
approximation and
show how asymptotic analysis can be used to improve it.
It turns out that the integral formulation of the mean curvature
\begin{equation*}
H = \frac{1}{2 \pi} \int_0^{2 \pi} \kappa(\phi) d\phi,
\end{equation*}
can be computed by an exact quadrature formula.
The same is true for the integral formulations of Gaussian curvature and
the Taubin tensor.
The exact quadratures are then used to obtain reliable estimates
of the curvature tensor of a smooth surface approximated by a dense triangle
mesh. The proposed method is fast and often demonstrates a better
performance
than conventional curvature tensor estimation approaches. We also show
that the curvature tensor approximated by
our approach converges towards the true curvature tensor as the edge
lengths tend to zero.
