Industrial Geometry

This project aims at the development of new methods in Industrial Geometry by applying techniques of classical geometry in combination with methods from approximation theory, numerical analysis and geometry processing. Those geometric concepts, which are known under the term higher geometries, include the various differential geometries (elementary Euclidean, affine and projective), and the representation of groups by their action on varieties of geometric objects, like Laguerre sphere geometry, line geometry, or kinematic spaces. An important aspect of the project is also the extension of these classical concepts to meet the needs in applications. Let us point here to the highlights of our work.

Motivated by applications in architectural design, we could define and study a new class of quadrilateral meshes - conical meshes - which discretize the network of principal curvature lines on a smooth surface. These meshes have planar faces and offset meshes at constant face/face distance, which turned out to have important applications for the beam layout in the actual construction of architectural freeform structures (Fig. 1). Conical meshes are an entity of Laguerre sphere geometry. The same holds for the new class of edge-offset meshes, whose definition comes from practical requirements on the supporting beam layout and whose study and computation exhibits relations to concepts such as circle patterns, Koebe polyhedra and Laguerre-isothermic surfaces. Shape limitations of quadrilateral edge-offset meshes motivated us to develop algorithms for computing approximate edge offsets and associated supporting beam layouts. Using ideas of relative differential geometry, we could introduce a wide class of practically interesting meshes with planar faces (relative principal meshes) for which an adapted discrete curvature theory could be developed. As a result of these studies, we found discrete differential geometry for architectural design to be a wide area for future research. Some of the arising problems are currently investigated in two additional projects (funded by FWF and FFG/Waagner-Biro Stahlbau AG, respectively).

Fig. 1: Architectural design based on a conical mesh; part of this image appeared on the front cover of the SIGGRAPH 06 Proceedings issue of ACM Transactions on Graphics

Another focus of research has been geometric computing in shape spaces. We presented a novel framework to treat shapes in the setting of Riemannian geometry. Shapes are considered as points in a shape space. We showed how to equip shape space with useful semi-Riemannian metrics and how to efficiently compute its geodesics. Working in shape space, various problems from geometric modeling and geometry processing can be treated in a consistent and unified way by linking them to geometric concepts such as parallel transport or the exponential map. These applications include shape morphing, deformation transfer and shape exploration.

As an extension of line geometry and Laguerre geometry, we studied the Euclidean geometry of line elements. This led to extensions of classical results and to algorithms for shape recognition and reconstruction. The geometry of surface elements and an associate feature sensitive metric on surfaces have been effectively used for feature sensitive geometry processing, such as mesh decimation or surface fitting, and for the detection, classification and editing of features.

(a) (b)
Fig. 2:(a) Reconstruction of axis and exact spiral surface approximating the shell of a snal using line element geometry.
(b): The feature sensitive metric is a tool for feature sensitive geometry processing and feature classification. The image shows prongs (pink), valleys (blue), and ridges (orange).

Robust local shape descriptors related to curvatures on multiple scales (integral invariants) have been studied from a theoretical and computational perspective and could be successfully applied to shape matching problems such as registration or the automatic reassembly of fractured artefacts. Dynamic registration of rigid and articulated objects could be achieved in a space-time model exploiting kinematical properties. Shapes which can be partially matched onto themselves exhibit symmetries. Combined processing in a space of transformations between local neighborhoods and in the actual object space forms the basis of an algorithm for symmetrization, i.e., minimal deformation of a nearly symmetric shape toward a fully symmetric one.

Further topics addressed in this project include the approximation of a solid by a union of balls, applications in computational anatomy, Voronoi diagrams for oriented spheres, approximation in kinematic spaces, the computation of Minkowski sums and convolution surfaces, and studies of relations between Laguerre minimal surfaces and linear elasticity as a preparation for future research tasks.

Results have been obtained in the following areas:

• Geometry of shape manifolds
• Approximation in kinematic spaces and shape spaces
• Computational line and sphere geometry
• Computational differential geometry