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q-Blossoming: A new approach to algorithms and identities for q-Bernstein bases and q-Bézier curves

Plamen Simeonov, Vasilis Zafiris, Ron Goldman
2012 Journal of Approximation Theory  
We introduce a new variant of the blossom, the q-blossom, by altering the diagonal property of the standard blossom.  ...  new, affine invariant, recursive evaluation algorithms. Using two of these new recursive evaluation algorithms, we construct a recursive subdivision algorithm for q-Bézier curves.  ...  Thus in the first algorithm the intermediate values are affine invariant, but in the second algorithm only the final result is affine invariant.  ... 
doi:10.1016/j.jat.2011.09.006 fatcat:newbrv2f6vbtfb4755rlkqiupy

q-Blossoming for analytic functions

Ron Goldman, Plamen Simeonov
2018 Numerical Algorithms  
We construct a q-analogue of the blossom for analytic functions, the analytic q-blossom. This q-analogue also extends the notion of q-blossoming from polynomials to analytic functions.  ...  The goal of this paper is to generalize this analytic blossom to an analytic q-blossom, where once again the classical derivative is replaced by a quantum derivative, and then to apply this analytic q-blossom  ...  Since the q-Poisson basis functions are non-negative on [0, 1] and form a partition of unity, q-Poisson curves are affine invariant and lie in the convex hull of their control points.  ... 
doi:10.1007/s11075-018-0595-y fatcat:j727hd2yhzgixo524hzul4ofyy

Manifold splines

Xianfeng Gu, Ying He, Hong Qin
2006 Graphical Models  
We study the affine structure of domain manifolds in depth and prove that the existence of manifold splines is equivalent to the existence of a manifold's affine atlas.  ...  As a result, our new spline surface defined over any manifold is a piecewise polynomial surface with high parametric continuity without the need for any patching and/or trimming operations.  ...  Acknowledgments The authors thank the reviewers for their careful review and helpful suggestions. This work was partially supported by the NSF CAREER Award CCF-0448339 to X.  ... 
doi:10.1016/j.gmod.2006.03.004 fatcat:vlkc4cify5ffjf7dflyxy36cwa

Manifold splines

Xianfeng Gu, Ying He, Hong Qin
2005 Proceedings of the 2005 ACM symposium on Solid and physical modeling - SPM '05  
We study the affine structure of domain manifolds in depth and prove that the existence of manifold splines is equivalent to the existence of a manifold's affine atlas.  ...  As a result, our new spline surface defined over any manifold is a piecewise polynomial surface with high parametric continuity without the need for any patching and/or trimming operations.  ...  Acknowledgments The authors thank the reviewers for their careful review and helpful suggestions. This work was partially supported by the NSF CAREER Award CCF-0448339 to X.  ... 
doi:10.1145/1060244.1060249 dblp:conf/sma/GuHQ05 fatcat:inyekbp2gvf3tlerj6qoiv236i

Inflection points and singularities on C-curves

Qinmin Yang, Guozhao Wang
2004 Computer Aided Geometric Design  
We show that all so-called C-curves are affine images of trochoids or sine curves and use this relation to investigate the occurrence of inflection points, cusps, and loops.  ...  The results are summarized in a shape diagram of C-Bézier curves, which is useful when using C-Bézier curves for curve and surface modeling.  ...  Acknowledgements This work was partly supported by the National Natural Science  ... 
doi:10.1016/j.cagd.2003.11.002 fatcat:5uesosprezbxjfotireyioh5za

A Blossoming Development of Splines

Stephen Mann
2006 Synthesis Lectures on Computer Graphics and Animation  
The primary analysis tool used in this lecture is blossoming, which gives an elegant labeling of the control points that allows us to analyze their properties geometrically.  ...  Bézier/B-splines represent polynomials and piecewise polynomials in a geometric manner using sets of control points that define the shape of the surface.  ...  invariance: T( i P i B n i (u)) = i T(P i )B n i (u) .  ... 
doi:10.2200/s00041ed1v01200607cgr001 fatcat:5shteg26izdvdj4ndz2xb6xv2m

Gerhard Hochschild (1915/2010) A Mathematician of the XXth Century [article]

Walter Ferrer Santos
2011 arXiv   pre-print
Gerhard Hochschild's contribution to the development of mathematics in the XX century is succinctly surveyed.  ...  Indeed, if we have an affine algebraic group H acting on an affine variety with ring of polynomials R, as one expects that the invariant polynomials R H separate the geometric orbits -at least generically-to  ...  His collaboration with Mostow blossomed with the stream of papers written on Representative functions (see in [14] , the articles by A. Magid and D.  ... 
arXiv:1104.0335v1 fatcat:d4ylcde575grzm47junz6qfceq

Page 4693 of Mathematical Reviews Vol. , Issue 86j [page]

1986 Mathematical Reviews  
In this paper, the author shows that the Pythagoras blossoms may be obtained as the Julia set of a pair of randomly iterated affine maps.  ...  The result is the Pythagoras tree. The limit points of this construction are the blossoms on the tree.  ... 

Approximate Reachability Computation for Polynomial Systems [chapter]

Thao Dang
2006 Lecture Notes in Computer Science  
We propose a method to do so using the Bézier control net of the polynomial map and the blossoming technique to compute this control net. We also prove that our overall method is of order 2.  ...  In order to use this scheme to compute all the states reachable by the system starting from some initial set, we then consider the problem of computing the image of a set by a multivariate polynomial.  ...  invariants [29] ).  ... 
doi:10.1007/11730637_13 fatcat:hzvovfkdovc5xgw2ufvqqkhipy

Integrable Mappings from a Unified Perspective [article]

Tova Brown, Nicholas M. Ercolani
2019 arXiv   pre-print
for the associated probability distributions.  ...  The two models are integrable and our analysis uncovers the geometric sources of this integrability and uses this to conceptually explain the rigorous existence and structure of elegant closed form expressions  ...  Hence our map determines an algebro-geometric addition law on each of the affinely smooth level sets of the invariant.  ... 
arXiv:1901.08174v1 fatcat:cif7lktanfeorbd5aegwaiew3a

Page 5168 of Mathematical Reviews Vol. , Issue 2002G [page]

2002 Mathematical Reviews  
Milbrandt, A geo- metrically motivated affine invariant norm (267-280); A. Nawotki, Exploiting wavelet coefficients for modifying functions (281-292); M. Robinson, M. I. G. Bloor and M. J.  ...  Goldman, Blossoming and divided difference (155-184); S. Hahmann, G.-P. Bonneau and R. Taleb, Localizing the 4-split method for G' free- form surface fitting (185-198); B. Heckel, A. E. Uva, B.  ... 

Political Culture and Covalent Bonding. A Conceptual Model of Political Culture Change

Camelia Florela Voinea
2015 Social Science Research Network  
Project"s goals are defined with respect to the specificity of the Eastern European geopolitical space: the impact of the political culture legacy of the Eastern European communist regimes on the political  ...  Acknowledgements The ongoing research reported here represents a research project initiated and developed by the Author as the Founding President of the European Research Group on Political Attitudes and  ...  The approaches are based on rather numerical combinatorial techniques, using intensively empirical data to analytical purposes.  ... 
doi:10.2139/ssrn.2556161 fatcat:pbc725xmonbjrpfh6pbnzcuvgy

s-power series: an alternative to Poisson expansions for representing analytic functions

J. Sánchez-Reyes, J.M. Chacón
2005 Computer Aided Geometric Design  
Morin and Goldman [Computer Aided Geometric Design 17 (2000) 813] have recently presented a remarkable new framework, based on employing Poisson series, for describing analytic functions in CAD.  ...  We compare this Poisson formulation with s-power series, modified Newton series that can be regarded as the two-point analogue of Taylor expansions.  ...  Acknowledgement This work is supported by the Spanish Ministerio de Ciencia y Tecnología, under research grant DPI2003-02598.  ... 
doi:10.1016/j.cagd.2004.09.003 fatcat:lj4hwky3bjf6fbnrw5gyw2tmsi

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Rimvydas Krasauskas
2012 Advances in Computational Mathematics  
Several features of toric patches are considered: affine invariance, convex hull property, boundary curves, implicit degree and singular points.  ...  The method of subdivision into tensor product surfaces is introduced. Fundamentals of a multidimensional variant of this theory are also developed.  ...  Goldman for very helpful comments, and to referees for pointing out some errors in the first version of the paper.  ... 
doi:10.1023/a:1015289823859 fatcat:fxwvxeh7knc5rleqgzkr3imqie

First steps in symplectic and spectral theory of integrable systems [article]

Álvaro Pelayo, San Vũ Ngoc
2013 arXiv   pre-print
Even though quantum integrable systems date back to the early days of quantum mechanics, such as the work of Bohr, Sommerfeld and Einstein, the theory did not blossom at the time.  ...  The paper intends to lay out the first steps towards constructing a unified framework to understand the symplectic and spectral theory of finite dimensional integrable Hamiltonian systems.  ...  His numerous comments prompted us to rethink parts of the paper and have greatly improved the exposition and the clarity.  ... 
arXiv:1306.0124v1 fatcat:hmpcrwcaffg3zepjfkw6izke7u
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