Helmut Pottmann - Profile on Academia.edu (original) (raw)

Papers by Helmut Pottmann

ACM Transactions on Graphics, 2008

Motivated by applications in architecture and manufacturing, we discuss the problem of covering a... more Motivated by applications in architecture and manufacturing, we discuss the problem of covering a freeform surface by single curved panels. This leads to the new concept of semi-discrete surface representation, which constitutes a link between smooth and discrete surfaces. The basic entity we are working with is the developable strip model. It is the semi-discrete equivalent of a quad mesh with planar faces, or a conjugate parametrization of a smooth surface. We present a B-spline based optimization framework for efficient computing with D-strip models. In particular we study conical and circular models, which semi-discretize the network of principal curvature lines, and which enjoy elegant geometric properties. Together with geodesic models and cylindrical models they offer a rich source of solutions for surface panelization problems.

Proceedings of the 2008 ACM symposium on Solid and physical modeling - SPM '08, 2008

This article shows to which extent a particular field of mathematics, namely discrete differentia... more This article shows to which extent a particular field of mathematics, namely discrete differential geometry, has recently become relevant in architectural design. It is very interesting that new mathematics has emerged from this cooperation with a branch of knowledge hitherto not known for its use of mathematical methods.

Handbook of Computer Aided Geometric Design, 2002

Chapter ?? describes the fundamental geometric setting for 3D modeling and addresses Euclidean, a... more Chapter ?? describes the fundamental geometric setting for 3D modeling and addresses Euclidean, affine and projective geometry, as well as differential geometry. In the present chapter, the discussions will be continued with a focus on geometric concepts which are less widely known. These are projective differential geometric methods, sphere geometries, line geometry, and non-Euclidean geometries. In all cases, we outline and illustrate applications of the respective geometries in geometric modeling.

In the reconstruction process of geometric objects from several three-dimensional images (clouds ... more In the reconstruction process of geometric objects from several three-dimensional images (clouds of measurement points) it is crucial to align the point sets of the different views, such that errors in the overlapping regions are minimized. We present an iterative algorithm which simultaneously registers all 3D image views. It can also be used for the solution of related positioning problems

Mathematics and Visualization, 2003

We investigate the geometry of that function in the plane or 3-space, which associates to each po... more We investigate the geometry of that function in the plane or 3-space, which associates to each point the square of the shortest distance to a given curve or surface. Particular emphasis is put on second order Taylor approximants and other local quadratic approximants. Their key role in a variety of geometric optimization algorithms is illustrated at hand of registration in Computer Vision and surface approximation.

Computer-aided Design, 2000

Given a solid 3 S R ⊂ with a piecewise smooth boundary, we compute an approximation of the bounda... more Given a solid 3 S R ⊂ with a piecewise smooth boundary, we compute an approximation of the boundary surface of the volume which is swept by S under a smooth one-parameter motion. Using knowledge from kinematical and elementary differential geometry, the algorithm computes a set of points plus surface normals from the envelope surface. A study of the evolution

Lecture Notes in Computer Science, 2004

We understand and reconstruct special surfaces from 3D data with line geometry methods. Based on ... more We understand and reconstruct special surfaces from 3D data with line geometry methods. Based on estimated surface normals we use approximation techniques in line space to recognize and reconstruct rotational, helical, developable and other surfaces, which are characterized by the configuration of locally intersecting surface normals. For the computational solution we use a modified version of the Klein model of line space. Obvious applications of these methods lie in Reverse Engineering. We have tested our algorithms on real world data obtained from objects as antique pottery, gear wheels, and a surface of the ankle joint.

Fascinating and elegant shapes may be folded from a single planar sheet of material without stret... more Fascinating and elegant shapes may be folded from a single planar sheet of material without stretching, tearing or cutting, if one incor- porates curved folds into the design. We present an optimization- based computational framework for design and digital reconstruc- tion of surfaces which can be produced by curved folding. Our work not only contributes to applications in architecture and

Lecture Notes in Computer Science, 2004

We introduce the isophotic metric, a new metric on surfaces, in which the length of a surface cur... more We introduce the isophotic metric, a new metric on surfaces, in which the length of a surface curve is not just dependent on the curve itself, but also on the variation of the surface normals along it. A weak variation of the normals brings the isophotic length of a curve close to its Euclidean length, whereas a strong normal variation increases the isophotic length. We actually have a whole family of metrics, with a parameter that controls the amount by which the normals influence the metric. We are interested here in surfaces with features such as smoothed edges, which are characterized by a significant deviation of the two principal curvatures. The isophotic metric is sensitive to those features: paths along features are close to geodesics in the isophotic metric, paths across features have high isophotic length. This shape effect makes the isophotic metric useful for a number of applications. We address feature sensitive image processing with mathematical morphology on surfaces, feature sensitive geometric design on surfaces, and feature sensitive local neighborhood definition and region growing as an aid in the segmentation process for reverse engineering of geometric objects.

European Workshop on Computational Geometry, 2000

We compute a set of balls that approximates a given 3D object, and we derive small additive bound... more We compute a set of balls that approximates a given 3D object, and we derive small additive bounds for the overhead in balls with respect to the minimal so- lution with the same quality. The algorithm has been implemented and tested using the CGAL library (7).

Many applications, such as contour machining, rapid prototyping, and reverse engineering by laser... more Many applications, such as contour machining, rapid prototyping, and reverse engineering by laser scanner or coordinate measuring machine, involve sampling of free-from surfaces along section cuts by a family of parallel planes with equidistant spacing ∆ and common normal N. To ensure that such planar sections provide faithful descriptions of the shape of a surface, it is desirable to choose the relative orientation that maximizes, over the entire surface, the minimum angle between N and the local surface normal n. We address this optimization problem by computing the (symmetrized) Gauss map for the surface, projecting it stereographically onto a plane, and invoking the medial axis transform for the complement of its image to identify the orientation N that is "most distant" from the symmetrized Gauss map boundary. Using a Gauss map algorithm described elsewhere, the method is implemented in the context of bicubic Bézier surfaces, and applied to the problem of minimizing the greatest scallop height incurred in contour machining of surfaces using a 3-axis milling machine with a ball-end cutter. 

This article presents a brief introduction to the classical geometry of ruled surfaces with empha... more This article presents a brief introduction to the classical geometry of ruled surfaces with emphasis on the Klein image and studies aspects which arise in connection with a computational treatment of these surfaces. As ruled surfaces are one parameter families of lines, one can apply curve theory and algorithms to the Klein image, when handling these surfaces. We study representations of rational ruled surfaces and get efficient algorithms for computation of planar intersections and contour outlines. Further, low degree boundary curves, useful for tensor product representations, are studied and illustrated at hand of several examples. Finally, we show how to compute efficiently low degree rational G 1 ruled surfaces. ᭧

The extraction of curvature information for surfaces is a basic problem of Geometry Processing. R... 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 required. In the present paper we show an asymptotic analysis of the moments of inertia and the principal directions which are used in this approach. We also address implementation and, briefly, robustness issues and applications.

Special issue on geometric modeling and processing

Most approaches to least squares fitting of a B-spline surface to measurement data require a para... more Most approaches to least squares fitting of a B-spline surface to measurement data require a parametrization of the data point set and the choice of suitable knot vectors. We propose to use uniform knots in connection with a feature sensitive parametrization. This parametrization allocates more parameter space to highly curved feature regions and thus automatically provides more control points where they are needed.

Industrial Geometry aims at unifying existing and developing new methods and algorithms for a var... more Industrial Geometry aims at unifying existing and developing new methods and algorithms for a variety of application areas with a strong geometric component. These include CAD, CAM, Geometric Modelling, Robotics, Computer Vision and Image Processing, Computer Graphics and Scientific Visualization. In this paper, Industrial Geometry is illustrated via the fruitful interplay of the areas indicated above in the context of novel solutions of CAD related, geometric optimization problems involving distance functions: approximation with general B-spline curves and surfaces or with subdivision surfaces, approximation with special surfaces for applications in architecture or manufacturing, approximate conversion from implicit to parametric (NURBS) representation, and registration problems for industrial inspection and 3D model generation from measurement data. Moreover, we describe a 'feature sensitive' metric on surfaces, whose definition relies on the concept of an image manifold, introduced into Computer Vision and Image Processing by Kimmel, Malladi and Sochen. This metric is sensitive to features such as smoothed edges, which are characterized by a significant deviation of the two principal curvatures. We illustrate its applications at hand of feature sensitive curve design on surfaces and local neighborhood definition and region growing as an aid in the segmentation process for reverse engineering of geometric objects.

The present paper studies Plücker coordinates for line elements in Euclidean three-space. The wel... more The present paper studies Plücker coordinates for line elements in Euclidean three-space. The well known relation between line geometry and kinematics is generalized to equiform motions and the geometry of line elements. We consider bundles and linear complexes of line elements and survey their properties.

Proceedings Geometric Modeling and Processing 2000. Theory and Applications, 2000

A geometric approach to the computation of precise or well approximated tolerance zones for CAD c... more A geometric approach to the computation of precise or well approximated tolerance zones for CAD constructions is given. We continue a previous study of linear constructions and freeform curve and surface schemes under the assumption of convex tolerance regions for points. The computation of the boundaries of the tolerance zones for curves / surfaces is discussed. We also study congruence transformations in the presence of errors and families of circles arising in metric constructions under the assumption of tolerances in the input. The classical cyclographic mapping as well as ideas from convexity and classical differential geometry appear as central geometric tools.

Developable Surfaces

Mathematics and Visualization, 2009

Developable surfaces are surfaces in Euclidean space which ‘can be made of a piece of paper’, i.e... more Developable surfaces are surfaces in Euclidean space which ‘can be made of a piece of paper’, i.e., are isometric to part of the Euclidean plane, at least locally. If we do not assume sufficient smoothness, the class of such surfaces is too large to be useful — if includes all possible aways of arranging crumpled paper in space. For C 2 surfaces, however, developability is characterized by vanishing Gaussian curvature, and by being made of pieces of torsal ruled surfaces. We will here use ‘developable’ and ‘torsal ruled’ as synonyms, because we are most interested in the ruled surface which carries a developable surface patch. We first have a look at the Euclidean differential geometry of developable surfaces, and then study developables as envelopes of their tangent planes. This view-point identifies the curves in dual projective space with the torsal ruled surfaces. We describe some fields of applications where the concept of developable surface arises naturally and knowledge of the theory leads to geometric insights. These include developables of constant slope, the cyclographic mapping, medial axis computations, geometrical optics, rational curves with rational offsets, geometric tolerancing, and the use of developable surfaces in industrial processes.

ACM Transactions on Graphics, 2008

Motivated by applications in architecture and manufacturing, we discuss the problem of covering a... more Motivated by applications in architecture and manufacturing, we discuss the problem of covering a freeform surface by single curved panels. This leads to the new concept of semi-discrete surface representation, which constitutes a link between smooth and discrete surfaces. The basic entity we are working with is the developable strip model. It is the semi-discrete equivalent of a quad mesh with planar faces, or a conjugate parametrization of a smooth surface. We present a B-spline based optimization framework for efficient computing with D-strip models. In particular we study conical and circular models, which semi-discretize the network of principal curvature lines, and which enjoy elegant geometric properties. Together with geodesic models and cylindrical models they offer a rich source of solutions for surface panelization problems.

Proceedings of the 2008 ACM symposium on Solid and physical modeling - SPM '08, 2008

This article shows to which extent a particular field of mathematics, namely discrete differentia... more This article shows to which extent a particular field of mathematics, namely discrete differential geometry, has recently become relevant in architectural design. It is very interesting that new mathematics has emerged from this cooperation with a branch of knowledge hitherto not known for its use of mathematical methods.

Handbook of Computer Aided Geometric Design, 2002

Chapter ?? describes the fundamental geometric setting for 3D modeling and addresses Euclidean, a... more Chapter ?? describes the fundamental geometric setting for 3D modeling and addresses Euclidean, affine and projective geometry, as well as differential geometry. In the present chapter, the discussions will be continued with a focus on geometric concepts which are less widely known. These are projective differential geometric methods, sphere geometries, line geometry, and non-Euclidean geometries. In all cases, we outline and illustrate applications of the respective geometries in geometric modeling.

In the reconstruction process of geometric objects from several three-dimensional images (clouds ... more In the reconstruction process of geometric objects from several three-dimensional images (clouds of measurement points) it is crucial to align the point sets of the different views, such that errors in the overlapping regions are minimized. We present an iterative algorithm which simultaneously registers all 3D image views. It can also be used for the solution of related positioning problems

Mathematics and Visualization, 2003

We investigate the geometry of that function in the plane or 3-space, which associates to each po... more We investigate the geometry of that function in the plane or 3-space, which associates to each point the square of the shortest distance to a given curve or surface. Particular emphasis is put on second order Taylor approximants and other local quadratic approximants. Their key role in a variety of geometric optimization algorithms is illustrated at hand of registration in Computer Vision and surface approximation.

Computer-aided Design, 2000

Given a solid 3 S R ⊂ with a piecewise smooth boundary, we compute an approximation of the bounda... more Given a solid 3 S R ⊂ with a piecewise smooth boundary, we compute an approximation of the boundary surface of the volume which is swept by S under a smooth one-parameter motion. Using knowledge from kinematical and elementary differential geometry, the algorithm computes a set of points plus surface normals from the envelope surface. A study of the evolution

Lecture Notes in Computer Science, 2004

We understand and reconstruct special surfaces from 3D data with line geometry methods. Based on ... more We understand and reconstruct special surfaces from 3D data with line geometry methods. Based on estimated surface normals we use approximation techniques in line space to recognize and reconstruct rotational, helical, developable and other surfaces, which are characterized by the configuration of locally intersecting surface normals. For the computational solution we use a modified version of the Klein model of line space. Obvious applications of these methods lie in Reverse Engineering. We have tested our algorithms on real world data obtained from objects as antique pottery, gear wheels, and a surface of the ankle joint.

Fascinating and elegant shapes may be folded from a single planar sheet of material without stret... more Fascinating and elegant shapes may be folded from a single planar sheet of material without stretching, tearing or cutting, if one incor- porates curved folds into the design. We present an optimization- based computational framework for design and digital reconstruc- tion of surfaces which can be produced by curved folding. Our work not only contributes to applications in architecture and

Lecture Notes in Computer Science, 2004

We introduce the isophotic metric, a new metric on surfaces, in which the length of a surface cur... more We introduce the isophotic metric, a new metric on surfaces, in which the length of a surface curve is not just dependent on the curve itself, but also on the variation of the surface normals along it. A weak variation of the normals brings the isophotic length of a curve close to its Euclidean length, whereas a strong normal variation increases the isophotic length. We actually have a whole family of metrics, with a parameter that controls the amount by which the normals influence the metric. We are interested here in surfaces with features such as smoothed edges, which are characterized by a significant deviation of the two principal curvatures. The isophotic metric is sensitive to those features: paths along features are close to geodesics in the isophotic metric, paths across features have high isophotic length. This shape effect makes the isophotic metric useful for a number of applications. We address feature sensitive image processing with mathematical morphology on surfaces, feature sensitive geometric design on surfaces, and feature sensitive local neighborhood definition and region growing as an aid in the segmentation process for reverse engineering of geometric objects.

European Workshop on Computational Geometry, 2000

We compute a set of balls that approximates a given 3D object, and we derive small additive bound... more We compute a set of balls that approximates a given 3D object, and we derive small additive bounds for the overhead in balls with respect to the minimal so- lution with the same quality. The algorithm has been implemented and tested using the CGAL library (7).

Many applications, such as contour machining, rapid prototyping, and reverse engineering by laser... more Many applications, such as contour machining, rapid prototyping, and reverse engineering by laser scanner or coordinate measuring machine, involve sampling of free-from surfaces along section cuts by a family of parallel planes with equidistant spacing ∆ and common normal N. To ensure that such planar sections provide faithful descriptions of the shape of a surface, it is desirable to choose the relative orientation that maximizes, over the entire surface, the minimum angle between N and the local surface normal n. We address this optimization problem by computing the (symmetrized) Gauss map for the surface, projecting it stereographically onto a plane, and invoking the medial axis transform for the complement of its image to identify the orientation N that is "most distant" from the symmetrized Gauss map boundary. Using a Gauss map algorithm described elsewhere, the method is implemented in the context of bicubic Bézier surfaces, and applied to the problem of minimizing the greatest scallop height incurred in contour machining of surfaces using a 3-axis milling machine with a ball-end cutter. 

This article presents a brief introduction to the classical geometry of ruled surfaces with empha... more This article presents a brief introduction to the classical geometry of ruled surfaces with emphasis on the Klein image and studies aspects which arise in connection with a computational treatment of these surfaces. As ruled surfaces are one parameter families of lines, one can apply curve theory and algorithms to the Klein image, when handling these surfaces. We study representations of rational ruled surfaces and get efficient algorithms for computation of planar intersections and contour outlines. Further, low degree boundary curves, useful for tensor product representations, are studied and illustrated at hand of several examples. Finally, we show how to compute efficiently low degree rational G 1 ruled surfaces. ᭧

The extraction of curvature information for surfaces is a basic problem of Geometry Processing. R... 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 required. In the present paper we show an asymptotic analysis of the moments of inertia and the principal directions which are used in this approach. We also address implementation and, briefly, robustness issues and applications.

Special issue on geometric modeling and processing

Most approaches to least squares fitting of a B-spline surface to measurement data require a para... more Most approaches to least squares fitting of a B-spline surface to measurement data require a parametrization of the data point set and the choice of suitable knot vectors. We propose to use uniform knots in connection with a feature sensitive parametrization. This parametrization allocates more parameter space to highly curved feature regions and thus automatically provides more control points where they are needed.

Industrial Geometry aims at unifying existing and developing new methods and algorithms for a var... more Industrial Geometry aims at unifying existing and developing new methods and algorithms for a variety of application areas with a strong geometric component. These include CAD, CAM, Geometric Modelling, Robotics, Computer Vision and Image Processing, Computer Graphics and Scientific Visualization. In this paper, Industrial Geometry is illustrated via the fruitful interplay of the areas indicated above in the context of novel solutions of CAD related, geometric optimization problems involving distance functions: approximation with general B-spline curves and surfaces or with subdivision surfaces, approximation with special surfaces for applications in architecture or manufacturing, approximate conversion from implicit to parametric (NURBS) representation, and registration problems for industrial inspection and 3D model generation from measurement data. Moreover, we describe a 'feature sensitive' metric on surfaces, whose definition relies on the concept of an image manifold, introduced into Computer Vision and Image Processing by Kimmel, Malladi and Sochen. This metric is sensitive to features such as smoothed edges, which are characterized by a significant deviation of the two principal curvatures. We illustrate its applications at hand of feature sensitive curve design on surfaces and local neighborhood definition and region growing as an aid in the segmentation process for reverse engineering of geometric objects.

The present paper studies Plücker coordinates for line elements in Euclidean three-space. The wel... more The present paper studies Plücker coordinates for line elements in Euclidean three-space. The well known relation between line geometry and kinematics is generalized to equiform motions and the geometry of line elements. We consider bundles and linear complexes of line elements and survey their properties.

Proceedings Geometric Modeling and Processing 2000. Theory and Applications, 2000

A geometric approach to the computation of precise or well approximated tolerance zones for CAD c... more A geometric approach to the computation of precise or well approximated tolerance zones for CAD constructions is given. We continue a previous study of linear constructions and freeform curve and surface schemes under the assumption of convex tolerance regions for points. The computation of the boundaries of the tolerance zones for curves / surfaces is discussed. We also study congruence transformations in the presence of errors and families of circles arising in metric constructions under the assumption of tolerances in the input. The classical cyclographic mapping as well as ideas from convexity and classical differential geometry appear as central geometric tools.

Developable Surfaces

Mathematics and Visualization, 2009

Developable surfaces are surfaces in Euclidean space which ‘can be made of a piece of paper’, i.e... more Developable surfaces are surfaces in Euclidean space which ‘can be made of a piece of paper’, i.e., are isometric to part of the Euclidean plane, at least locally. If we do not assume sufficient smoothness, the class of such surfaces is too large to be useful — if includes all possible aways of arranging crumpled paper in space. For C 2 surfaces, however, developability is characterized by vanishing Gaussian curvature, and by being made of pieces of torsal ruled surfaces. We will here use ‘developable’ and ‘torsal ruled’ as synonyms, because we are most interested in the ruled surface which carries a developable surface patch. We first have a look at the Euclidean differential geometry of developable surfaces, and then study developables as envelopes of their tangent planes. This view-point identifies the curves in dual projective space with the torsal ruled surfaces. We describe some fields of applications where the concept of developable surface arises naturally and knowledge of the theory leads to geometric insights. These include developables of constant slope, the cyclographic mapping, medial axis computations, geometrical optics, rational curves with rational offsets, geometric tolerancing, and the use of developable surfaces in industrial processes.