Geometry of smooth or discrete surfaces can be described either by local properties or global ones. Local dif-ferential properties are the tangent plane (at the first order), the principal directions and curvatures (at the second order, see Fig. 1.3), or higher order coefficients. In the smooth case, all this information is encoded in the Taylor expansion of the function whose graph locally defines the surface in a given coordinate system. We call a jet such a Taylor expansion and Fig. 1.1 illustrates this local approximation. Global differential properties usually refer to loci of points having a prescribed differential property. Examples such loci are lines of curvature, parabolic lines (where the Gauss curvature vanishes Fig. 1.2), ridges (lines of extremal curvature) or the medial axis (centers of maximal spheres included in the complement of the surface inR3). Hence local information is required to be able to generate global information.
In the present work, we first investigate estimation of local differential properties of any order. Then we study a global differential object on surfaces : the set of lines of extremal curvature, called ridges.
Figure 1.1: The graph of jets around some vertices of a mesh are local approximation of the surface (see chap. 4).
Figure 1.2: The parabolic curves on the Apollo of Belvedere drawn by Felix Klein (from [Koe90]).
1.2.1 Estimation of local differential quantities
While local differential quantities are well defined and easy to compute on smooth surfaces, they are not well defined for discrete surfaces. When defining a method to estimate differential quantities on a discrete surface, a way to evaluate the method is to compare the results obtained on some discretizations of a given smooth surface and the actual values for this smooth surface. The sensitivity of the method with respect to the properties and the
1.2. ESTIMATION GEOMETRIC PROPERTIES : LOCAL AND GLOBAL ASPECTS 17
Figure 1.3: Michelangelo’s David: principal directions associated with kmaxscaled by kmin(see chap. 4).
quality of the discretizations can be analyzed. The convergence of the estimated values to the correct ones can also be specified for some sequence of discretizations. The development of algorithms providing such guarantees has been subject to intense research [Pet01], and recent advances provide guarantees either point-wise (see chapter 4) or in the geometric measure theory sense [CSM03]. It is worth noting that some widely used methods such as the angular defect for the Gauss curvature do not provide convergent estimations as demonstrated in [BCM03].
1.2.2 Estimation of global differential properties, the example of ridges
Estimating global differential loci needs reliable point-wise estimates, but in addition, imposes to respect (global) topological constraints. These difficulties are tangible from a practical perspective, and only few algorithms are able to report global differential patterns with some guarantee. For example, reporting the homotopy type of the medial axis has only been addressed quite recently [CL05], but problems involving homeomorphy or isotopy are more demanding.
We focused in our work on lines of extremal curvature on a surface, called ridges. In terms of topological guarantees, we wish to report isotopic approximations. To get acquainted with extrema of curvature, first consider the case of plane curves. Points where the curvature is extremal are called vertices, the set of centers of osculating circles is the focal curve and, the centers of circles tangent in two places to the curve is called the symmetry set.
These objects are related : the border points of the symmetry set (centers of circles for which the two tangent points coincide) are the singularities of the focal curve, and the circles centered at these points touch the curve
18 CHAPTER 1. THESIS OVERVIEW at vertices. For example, Fig. 1.4 shows the focal curve of an ellipse which has four cusps corresponding to the four vertices. For surfaces, one can define a focal surface for each principal curvature and the same properties hold. The equivalent of vertices of a curve are lines on the surface corresponding to contact points with spheres centered on the singularities of the focal surfaces. These lines called ridges of a surface also has an alternative characterization : they consists of the points where one of the principal curvatures has an extremum along its curvature line. Denoting k1and k2the principal curvatures —we shall always assume that k1≥k2, a ridge is called blue (red) if k1(k2) has an extremum. Moreover, a ridge is called elliptic if it corresponds to a maximum of k1or a minimum of k2, and is called hyperbolic otherwise. Ridges on an ellipsoid are displayed on Fig. 1.5 and 1.6. Fig.
1.7, displaying a subset of the ridges on the David’s head, illustrates how these lines enhance the sharpest features of a model. Ridges witness extrema of principal curvatures and their definition involves derivatives of curvatures, whence third order differential quantities. Moreover, the classification of ridges as elliptic or hyperbolic involves fourth order differential quantities, so that the precise definition of ridges requires C4differentiable surfaces.
Ridges were mentioned in 1904 by A. Gullstrand, Nobel Prize for Physiology and Medicine, for his work in optics where fourth order differential quantities were necessary to explain the accommodation of the eye lens [Por01]. More recently, singularity theory allowed a precise setting to describe ridges and umbilics as special points on these lines.
Figure 1.4: Focal curve (red) of an ellipse
Figure 1.5: Umbilics, ridges, and principal blue fo-liation on the ellipsoid (see chap. 3).
Figure 1.6: Schematic view of the umbilics and the ridges (see chap. 3).
1.2.3 Applications
For many applications, estimating first and second order differential quantities, that is the tangent plane and curvature-related quantities, is sufficient. In computer graphics, shading algorithms require the normal vector field. Gauss and mean curvatures are commonly used for surface segmentation, the mean curvature vector can be used for smoothing or denoising of surfaces. However, higher order local properties and global ones are also more and more frequent. The lines of curvature are used for surface remeshing with quad elements [ACSD+03]. The topology of vector and tensor fields helps scientific visualization [DH94]. The medial axis or skeleton is used for surface reconstruction [AB99, BC01]. The extraction of ridges is applied to the registration of medical images