Subdivision as a Sequence of Sampled Cp Surfaces

Gérot, Cédric; Barthe, Loïc; Dodgson, Neil A.; Sabin, Malcolm

doi:10.1007/3-540-26808-1_14

Cited by 3 publications

(3 citation statements)

References 11 publications

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“…Local quadratic precision (Gerot et al, 2004) with respect to the characteristic map parameterization follows from the components of the quadratic natural configurations which correspond to the eigenvalues µ c and the two µ s . We notice that there is no proof that the property here denoted as local quadratic precision and which is defined below actually implies the quadratic precision of the limit surface in the vicinity of the EV.…”

Section: Theoretical Backgroundmentioning

confidence: 99%

“…We notice that there is no proof that the property here denoted as local quadratic precision and which is defined below actually implies the quadratic precision of the limit surface in the vicinity of the EV. However as shown by Gerot et al (2004), the local quadratic precision is a necessary condition for C 2 continuity and it is a fundamental criterion that needs to be satisfied in order to improve the scheme's behavior at EVs. The eigenvectors associated with the eigenvalues µ c and the two µ s correspond to the quadratic polynomials x 2 + y 2 − Z c (with Z c > 0), 2xy and x 2 − y 2 respectively whose coefficients are geometrically interpreted as altitudes over the tangent plane defined by X and Y.…”

Section: Theoretical Backgroundmentioning

confidence: 99%

“…Even though C 1 continuity is not guaranteed theoretically, our experiments have always provided C 1 continuous schemes. The remaining degrees of freedom are used to approximately satisfy further C 2 conditions such as the well known bounded curvature condition (Doo and Sabin, 1978) or a recently investigated quadratic precision condition (Gerot et al, 2004).…”

Section: Introductionmentioning

confidence: 99%

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Subdivision scheme tuning around extraordinary vertices

Barthe

Kobbelt

2004

Computer Aided Geometric Design

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In this paper we extend the standard method to derive and optimize subdivision rules in the vicinity of extraordinary vertices (EV). Starting from a given set of rules for regular control meshes, we tune the extraordinary rules (ER) such that the necessary conditions for C 1 continuity are satisfied along with as many necessary C 2 conditions as possible. As usually done, our approach sets up the general configuration around an EV by exploiting rotational symmetry and reformulating the subdivision rules in terms of the subdivision matrix' eigencomponents. The degrees of freedom are then successively eliminated by imposing new constraints which allows us, e.g., to improve the curvature behavior around EVs. The method is flexible enough to simultaneously optimize several subdivision rules, i.e. not only the one for the EV itself but also the rules for its direct neighbors. Moreover it allows us to prescribe the stencils for the ERs and naturally blends them with the regular rules that are applied away from the EV. All the constraints are combined in an optimization scheme that searches in the space of feasible subdivision schemes for a candidate which satisfies some necessary conditions exactly and other conditions approximately. The relative weighting of the constraints allows us to tune the properties of the subdivision scheme according to application specific requirements. We demonstrate our method by tuning the ERs for the well-known Loop scheme and by deriving ERs for a √ 3-type scheme based on a 6-direction Box-spline.

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Section: Theoretical Backgroundmentioning

confidence: 99%

Section: Theoretical Backgroundmentioning

confidence: 99%