15,310 research outputs found
Determination and (re)parametrization of rational developable surfaces
The developable surface is an important surface in computer aided design, geometric modeling and industrial manufactory. It is often given in the standard parametric form, but it can also be in the implicit form which is commonly used in algebraic geometry. Not all algebraic developable surfaces have rational parametrizations. In this paper, the authors focus on the rational developable surfaces. For a given algebraic surface, the authors first determine whether it is developable by geometric inspection, and then give a rational proper parametrization in the affirmative case. For a rational parametric surface, the authors also determine the developability and give a proper reparametrization for the developable surface
An axiomatic approach for solving geometric problems symbolically
technical reportThis paper describes a new approach for solving geometric constraint problems and problems in geometry theorem proving. We developed a rewrite-rule mechanism operating on geometric predicates. Termination and completeness of the problem solving algorithm can be obtained through well foundedness and confluence of the set of rewrite rules. To guarantee these properties we adapted the Knuth-Bendix completion algorithm to the specific requirements of the geometric problem. A symbolic, geometric solution has the advantage over the usual algebraic approach that it speaks the language of geometry. Therefore, it has the potential to be used in many practical applications in interactive Computer Aided Design
Reverse engineering of CAD models via clustering and approximate implicitization
In applications like computer aided design, geometric models are often
represented numerically as polynomial splines or NURBS, even when they
originate from primitive geometry. For purposes such as redesign and
isogeometric analysis, it is of interest to extract information about the
underlying geometry through reverse engineering. In this work we develop a
novel method to determine these primitive shapes by combining clustering
analysis with approximate implicitization. The proposed method is automatic and
can recover algebraic hypersurfaces of any degree in any dimension. In exact
arithmetic, the algorithm returns exact results. All the required parameters,
such as the implicit degree of the patches and the number of clusters of the
model, are inferred using numerical approaches in order to obtain an algorithm
that requires as little manual input as possible. The effectiveness, efficiency
and robustness of the method are shown both in a theoretical analysis and in
numerical examples implemented in Python
Involutions of polynomially parametrized surfaces
We provide an algorithm for detecting the involutions leaving a surface
defined by a polynomial parametrization invariant. As a consequence, the
symmetry axes, symmetry planes and symmetry center of the surface, if any, can
be determined directly from the parametrization, without computing or making
use of the implicit representation. The algorithm is based on the fact, proven
in the paper, that any involution of the surface comes from an involution of
the parameter space (the real plane, in our case); therefore, by determining
the latter, the former can be found. The algorithm has been implemented in the
computer algebra system Maple 17. Evidence of its efficiency for moderate
degrees, examples and a complexity analysis are also given
Symmetry Detection of Rational Space Curves from their Curvature and Torsion
We present a novel, deterministic, and efficient method to detect whether a
given rational space curve is symmetric. By using well-known differential
invariants of space curves, namely the curvature and torsion, the method is
significantly faster, simpler, and more general than an earlier method
addressing a similar problem. To support this claim, we present an analysis of
the arithmetic complexity of the algorithm and timings from an implementation
in Sage.Comment: 25 page
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