Boris Odehnal

3D shape recognition and reconstruction based on line element geometry

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 Laguerre geometry of oriented planes, enables us to recognize a wide class of surfaces (spiral surfaces, cones, helical surfaces, rotational surfaces, cylinders, etc.), by fitting linear subspaces in an appropriate seven-dimensional image space. In combination with standard techniques such as PCA and RANSAC, line element geometry is employed to effectively perform the segmentation of complex objects according to surface type. Examples show applications in reverse engineering of CAD models and testing mathematical hypotheses concerning the exponential growth of sea shells

Line Geometry for 3D Shape Understanding and Reconstruction

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.

Convolution surfaces of quadratic triangular Bézier surfaces

In the present paper we prove that the polynomial quadratic triangular Bezier surfaces are LN-surfaces. We demonstrate how to reparameterize the surfaces such that the normals obtain linear coordinate functions. The close relation to quadratic Cremona transformations is elucidated. These reparameterizations can be effectively used for the computation of convolution surfaces.

On quadratic two-parameter families of spheres and their envelopes

In the present paper we investigate rational two-parameter families of spheres and their envelope surfaces in Euclidean R^3. The four dimensional cyclographic model of the set of spheres in R^3 is an appropriate framework to show that a quadratic triangular Bezier patch in R^4 corresponds to a two-parameter family of spheres with rational envelope surface. The construction shows also that the envelope has rational offsets. Further we outline how to generalize the construction to obtain a much larger class of surfaces with similar properties.

On optimal tolerancing in computer-aided design

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.

On generalized ln-surfaces in 4-space

The present paper investigates a class of two-dimensional rational surfaces φ in <i>R</i>⁴ whose tangent planes satisfy the following property: For any three-space <i>E</i> in <i>R</i>⁴ there exists a unique tangent plane <i>T</i> of φ which is parallel to <i>E</i>. The most interesting families of surfaces are constructed explicitly and geometric properties of these surfaces are derived. Quadratically parameterized surfaces in <i>R</i>⁴ occur as special cases. This construction generalizes the concept of LN-surfaces in <i>R</i>³ to two-dimensional surfaces in <i>R</i>⁴.

Three points related to the incenter and excenters of a triangle

Boris Odehnal studierte von 1994 bis 1999 an der Technischen Universitat Wien die Lehramtsfacher Mathematik und Darstellende Geometrie. Im Anschlus daran arbeitete er als Forschungsassistent am Institut fur Geometrie der Technischen Universitat Wien und gleichzeitig als Assistent an der Universitat fur Bodenkultur in Wien. Seit 2002 ist er am Institut fur Diskrete Mathematik und Geometrie der Technischen Universitat Wien tatig.

A Spatial Version of the Theorem of the Angle of Circumference

We try a generalization of the theorem of the angle of circumference to a version in three-dimensional Euclidean space and ask for pairs \(({\varepsilon },{\varphi })\) of planes passing through two (different skew) straight lines \(e\ni {\varepsilon }\) and \(f\ni {\varphi }\) such that the angle \(\alpha \) enclosed by \({\varepsilon }\) and \({\varphi }\) is constant. It turns out that the set of all such intersection lines is a quartic ruled surface \(\varPhi \) with \(e\cup f\) being its double curve. We shall study the surface \(\varPhi \) and its properties together with certain special appearances showing up for special values of some shape parameters such as the slope of e and f (with respect to a fixed plane) or the angle \(\alpha \).

Examples of Autoisoptic Curves

The locus \(k_\alpha \) of all points where two different tangents of a planar curve k meet at a constant angle \(\alpha \) is called the isoptic curve of k. We shall look for curves k that coincide with their isoptic curves \(k_\alpha \) and call them autoisoptic curves. Describing a planar curve k by its support function d allows us to derive a system of two linear ordinary delay differential equations that have to be fulfilled by d in order to make k an autoisoptic curve. Examples of autoisoptic curves different from the only known examples, namely logarithmic spirals, shall be given. We do not provide the most general autoisoptic curves, since these involve ordinary delay differential equations with time dependent delays. We only treat the case of constant delays.

Higher Dimensional Geometries. What Are They Good For

Geometries in higher dimensional spaces have many applications. We shall give a compilation of a few well-known examples here. The fact that some higher dimensional geometries can be found within some lower dimensional geometries makes them even more interesting. At hand of some familiar examples, we shall see what these concepts in geometry can do for us. In the beginning, the meaning of dimension will be clarified and an agreement is reached about what is higher dimensional. A few words will be said about the relations and interplay between models of various geometries. To the concept of model spaces a major part of this contribution will be dedicated to. A full section is dedicated to the applications of higher dimensional geometries.