Differential Geometry and Shape Analysis - Mathematical Tools for Shape Analysis and Description - page 31

forms:

KD LN M 2

EGF 2 :
Finally, the integral of the Gaussian curvature over the surface region is called the total
curvature . e total curvature is closely related to the Euler characteristic of the surface (see
section 7.2 ), as stated in the Gauss-Bonnet theorem, which provides an important link between
local geometric properties and global topological properties.

4.2.1 COMPUTING CURVATURE ON MESHES
From the computational point of view, computing curvature on triangle meshes is not trivial, be-
cause a triangular mesh is parametrized by a piecewise continuous function whose second deriva-
tives are, almost everywhere, null. In other words, the curvature on a triangulation is concentrated
along edges and at vertices, since every other point has a neighborhood homeomorphic to a planar
Euclidean domain whose Gaussian curvature is null. e methods proposed in the literature for
curvature evaluation can be divided into two main groups: continuity-based and property-based
algorithms. e first ones transform the discrete case to the continuous one by using a local fit-
ting of the surface which enables us to apply standard definitions. For instance, the quadratic
fitting technique [ 137 ] approximates the surface with an interpolating quadratic surface and the
curvatures of the quadratic surface are taken as the approximated curvatures of the triangle mesh.
e second class of algorithms defines equivalent descriptors starting from basic properties
of continuous operators but directly applied to the discrete settings, guaranteeing the validity of
differential properties. For example, using the angle excess [ 164 ] the discrete Gaussian curvature
at a vertex p of a triangular mesh can be evaluated by

P

K.v/D 2

i i

A

P

where

i i is the sum of the angles at p in Star.p/ (imagine locally cutting Star.p/ along
any of the edges incident in p , and developing Star.p/ onto the plane without shrinking the
surface) and A is the area of Star.p/ or of some subregion of it containing p (see Figure 4.5 ).
is result is consistent with the intrinsic nature of the Gaussian curvature: since this formula
only takes angles into account, its value does not change under isometric deformations, that is, if
the mesh is deformed preserving the length of edges (i.e., the distance between points). However,
this approach is sensitive to noise and requires smoothness conditions on the input mesh.
See [ 86 , 137 , 177 ] and the references therein for a survey on methods for curvature esti-
mation in the discrete setting.

4.2.2 CONCEPTS IN ACTION
Segmentation based on curvature e automatic detection of surface features is a fundamental
task in shape analysis, as it facilitates operations such as editing, matching, texturing, morphing,