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Curve subdivision schemes on manifolds and in Lie groups are constructed from linear subdivision schemes by first representing the rules of affinely invariant linear schemes in terms of repeated affine averages, and then replacing the operation of affine average either by a geodesic average (in the Riemannian sense or in a certain Lie group sense), or by(More)
Differential invariants of curves and surfaces such as curvatures and their derivatives play a central role in Geometry Processing. However, they are sensitive to noise or minor perturbations and cannot directly be computed on multiple scales. Integrals of geometric functions, taken over local neighborhoods constructed via spheres, eliminate these drawbacks(More)
The extraction of curvature information for surfaces is a basic problem of Geometry Processing. Recently an integral invariant solution of this problem was presented, which is based on principal component analysis of local neighbourhoods defined by kernel balls of various sizes. It is not only robust to noise, but also adjusts to the level of detail(More)
In architectural freeform design, the relation between shape and fabrication poses new challenges and requires more sophistication from the underlying geometry. The new concept of conical meshes satisfies central requirements for this application: They are quadrilateral meshes with planar faces, and therefore particularly suitable for the design of freeform(More)
Linear stationary subdivision rules take a sequence of input data and produce ever denser sequences of subdivided data from it. They are employed in multiresolution modeling and have intimate connections with wavelet and more general pyramid transforms. Data which naturally do not live in a vector space, but in a nonlinear geometry like a surface, symmetric(More)
We study the combined problem of approximating a surface by a quad mesh (or quad-dominant mesh) which on the one hand has planar faces, and which on the other hand is aesthetically pleasing and has evenly spaced vertices. This work is motivated by applications in freeform architecture and leads to a discussion of fields of conjugate directions in surfaces,(More)
This paper presents a new method for the recognition and reconstruction of surfaces from 3D data. Line element geometry, which generalizes both line geometry and the La-guerre geometry of oriented planes, enables us to recognize a wide class of surfaces (spiral surfaces, cones, heli-cal surfaces, rotational surfaces, cylinders, etc.) by fitting linear(More)
We solve the form-finding problem for polyhedral meshes in a way which combines form, function and fabrication; taking care of user-specified constraints like boundary interpolation, planarity of faces, statics, panel size and shape, enclosed volume, and cost. Our main application is the interactive modeling of meshes for architectural and industrial(More)
We introduce a computational framework for discovering regular or repeated geometric structures in 3D shapes. We describe and classify possible regular structures and present an effective algorithm for detecting such repeated geometric patterns in point- or meshbased models. Our method assumes no prior knowledge of the geometry or spatial location of the(More)
By its dual representation, a developable surface can be viewed as a curve of dual projective 3-space. After introducing an appropriate metric in the dual space and restricting ourselves to special surface classes, we derive linear approximation algorithms for developable NURBS surfaces, including multiscale approximations. Special attention is paid to(More)