9 research outputs found
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
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
Affine equivalences of surfaces of translation and minimal surfaces, and applications to symmetry detection and design
We introduce a characterization for affine equivalence of two surfaces of translation defined by either rational or meromorphic generators. In turn, this induces a similar characterization for minimal surfaces. In the rational case, our results provide algorithms for detecting affine equivalence of these surfaces, and therefore, in particular, the symmetries of a surface of translation or a minimal surface of the considered types. Additionally, we apply our results to designing surfaces of translation and minimal surfaces with symmetries, and to computing the symmetries of the higher-order Enneper surfaces.publishedVersio
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 (Alcázar et al., 2014b). To support this claim, we present an analysis of the arithmetic complexity of the algorithm and timings from an implementation in Sage.acceptedVersio
Design paramétrico a partir da digitalização 3D de geometrias da natureza com padrão de crescimento espiral
A modelagem de geometrias da natureza pode ser um processo complexo devido ás características orgânicas dos elementos. Propõe-se com essa dissertação identificar geometrias espaciais que sigam o padrão de crescimento espiral observado na natureza, utilizando as Tecnologias 3D como ferramentas para o processo de projeto. Para a execução do trabalho foram investigadas os Métodos de Biônica, Crescimento Espiral e a Sequência de Fibonacci, Engenharia Reversa e Design Paramétrico. O processo de representação dos elementos foi realizado em conformidade com a Metodologia para o Desenvolvimento de Produtos Baseados no Estudo da Biônica com o acréscimo das tecnologias de digitalização tridimensional e de processamento de nuvem de pontos, complementado pela parametrização de superfícies à base de curvas. Foram utilizados três processos para modelagem de curvas paramétricas representadas (i) pelo desenho de linhas sobre a malha digitalizada em 3D, (ii) por programação visual no software Grasshopper e (iii) por programação com scripts Python. Foi avaliada como melhor alternativa para o Design Paramétrico a utilização da programação visual otimizada com a programação por scripts, a qual apresentou melhor aproximação entre as curvas analisadas. Estudos de casos realizados com elementos da natureza (abacaxi e pinha) demonstraram a viabilização do método. Desta maneira a sistematização do conhecimento permitirá a proposição de um modelo paramétrico baseado na Biônica para fase inicial de inspiração e concepção de alternativas do projeto de produto.Modeling the geometries of nature can be a complex process due to the organic characteristics of the elements. It is proposed with this dissertation to identify spatial geometries that follow the pattern of spiral growth observed in nature, using 3D Technologies as tools for the design process. For the execution of the work were investigated the Bionics, Spiral Growth and Fibonacci Sequence, Reverse Engineering and Parametric Design. The process of representation of the elements was carried out in accordance with the Methodology for the Development of Products Based on the Study of the Bionics with the addition of the technologies of three-dimensional digitization and processing of cloud of points, complemented by the parameterization of surfaces based on curves. Three methods were used for modeling parametric curves represented by (i) the drawing of lines on the 3D scanned mesh, (ii) by visual programming in the Grasshopper software and (iii) by programming with Python scripts. It was evaluated as the best alternative for Parametric Design the use of optimized visual programming with programming by scripts, which presented better approximation between the analyzed curves. Case studies carried out with nature elements (pineapple and pine cone) demonstrated the viability of the method. In this way the systematization of the knowledge will allow the proposition of a parametric model based on the Bionics for the initial phase of inspiration and design of alternatives of the product design