Rational curves and surfaces with rational offsets

Approximation algorithms for developable surfaces

Developable rational Be´zier and B-spline surfaces

An active contour model for parametric curve and surface approximation is presented. The active curve or surface adapts to the model shape to be approximated in an optimization algorithm. The quasi-Newton optimization procedure in each iteration step minimizes a quadratic function which is built up with the help of local quadratic approximants of the squared distance function of the model shape and an internal energy which has a smoothing and regularization effect. The approach completely avoids the parametrization problem. We also show how to use a similar strategy for the solution of variational problems for curves on surfaces. Examples are the geodesic path connecting two points on a surface and interpolating or approximating spline curves on surfaces. Finally we indicate how the latter topic leads to the variational design of smooth motions which interpolate or approximate given positions.

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Approximation with active B-spline curves and surfaces

Geometric design of quadratic and cubic developable Bezier patches from two boundary curves is studied in this paper. The conditions for developability are derived geometrically from the de Casteljau algorithm and expressed as a set of equations that must be fulfilled by the Bezier control points. This set of equations allows us to infer important properties of developable Bezier patches that provide useful parameters and simplify the solution process for the patch design. With one boundary curve freely specified, five more degrees of freedom are available for a second boundary curve of the same degree. Various methods are introduced that fully utilize these five degrees of freedom for the design of general quadratic and cubic developable Bezier patches in 3D space. A more restricted generalized conical model or cylindrical model provides simple solutions for higher-order developable patches.

Developable Bézier patches: properties and design

In the last years many efforts have been made to transfer geometric CAD methods to motion design: The generation of a (one-parametric) motion with given properties. An object (arm of a robot…) is moved with respect to a fixed system. Methods from freeform curve design provide powerful tools to handle such problems. Some of these methods and results shall be presented in this survey. Here one can find a (geometric) approach including solutions of interpolation problems. First attempts are studied. Then we state two principal demands on the result of such a design process: The motions should be rational and invariant with respect to changes of coordinates in fixed and moving frame, respectively. We use quaternions to describe displacements and motions and give an overview on rational spline motions. Some known results on interpolating motions are reviewed. Then a third demand is presented: The design process should be ‘repeatable’ in the following sense: If a solution is computed the algorithm should give the same solution (including parametrisation), if we start with new input data generated by our first solution. The paper includes suggestions for further research.

Rational motion design—a survey

Developable (1, n )-Be´zier surfaces

We study architecturally shaky Stewart-Gough platforms (SGP) and characterize the geometry of its anchor points. The main tool is the connection of geometric objects with mappings which preserve the structure of the problem. Here, the geometric way is the use of linear manifolds of correlations and quadratic transformations. By these methods we show that the anchor points have to be conjugate points with respect to 3-dimensional linear manifolds of correlations. This result is used to give all possible configurations of anchor points of architecturally shaky SGP.

Characterisation of Architecturally Shaky Platforms

Quantitative EDXS: Influence of geometry on a four detector system.

In this paper there are studied mozions σ / σo in the isotropic plane. Given a C2 — curve k in the moving frame σ we find the enveloped curve ko in the fixed frame σo and consider the correspondance between the isotropic curvatures A and Ao of k and ko. Than we investigate third-order properties of the pointpaths.

Otto Röschel

Papers

Rational motion design—a survey

Developable (1, n )-Be´zier surfaces

Characterisation of Architecturally Shaky Platforms

Quantitative EDXS: Influence of geometry on a four detector system.

Zur Kinematik der isotropen Ebene II