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**MPI-I-2005-4-003**. March** **2005, 42 pages. | Status:** **available - back from printing | Next --> Entry | Previous <-- Entry

Abstract in LaTeX format:

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.

Acknowledgement:** **

References to related material:

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