Interpolatory convexity-preserving subdivision schemes for curves and surfaces

Dyn, Nira; Levin, David; Liu, Dingyuan

doi:10.1016/0010-4485(92)90057-h

Cited by 54 publications

(30 citation statements)

References 7 publications

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“…• the parameters α and β defined in equation (16) are also estimated in order to check the convergence of the proposed scheme. In addition to the previous experiment, a "real world" surface is also used to study the convergence parameters in a concrete case.…”

Section: Surface Subdivision Resultsmentioning

confidence: 99%

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Non-linear subdivision using local spherical coordinates

Aspert

Ebrahimi

2003

Computer Aided Geometric Design

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In this paper, we present an original non-linear subdivision scheme suitable for univariate data, plane curves and discrete triangulated surfaces, while keeping the complexity acceptable. The proposed technique is compared to linear subdivision methods having an identical support. Numerical criteria are proposed to verify basic properties, such as convergence of the scheme and the regularity of the limit function.

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Section: Surface Subdivision Resultsmentioning

confidence: 99%

“…An method of convexity-preserving surface subdivision has been proposed in [16]. Another interpolating approach has been described in [20], which resembles by some aspects the proposed scheme.…”

Section: Review Of Subdivisionmentioning

confidence: 99%

Non-linear subdivision using local spherical coordinates

Aspert

Ebrahimi

2003

Computer Aided Geometric Design

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“…These schemes are convergent, and the limit curves interpolate the initial control points [22]. Interpolatory schemes in general are discussed in [34]. Recently this construction was extended to non-interpolatory schemes [30], by using (11) instead of (10) with w i (x) defined in (12).…”

Section: The Main Construction Methods Of Schemesmentioning

confidence: 99%

Linear subdivision schemes for the reﬁnement of geometric objects

Dyn¹

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

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Abstract. Subdivision schemes are efficient computational methods for the design, representation and approximation of surfaces of arbitrary topology in R 3 . Subdivision schemes generate curves/surfaces from discrete data by repeated refinements. This paper reviews some of the theory of linear stationary subdivision schemes and their applications in geometric modelling. The first part is concerned with "classical" schemes refining control points. The second part reviews linear subdivision schemes refining other objects, such as vectors of Hermite-type data, compact sets in R n and nets of curves in R 3 . Examples of various schemes are presented. Mathematics Subject Classification (2000). Primary 65D07, 65D17, 65D18; Secondary 41A15, 68U07.

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“…Thus the above problem can be reformulated as: Let f be a spherical convex function and let D=[x 1 , ..., x n ] be a set of discrete data sites located on 0 such that the data points f (x i ) x i , i=1, ..., n, are strictly convex, i.e., such that they are extreme points of their convex hull. Then there exists a C convex spherical function, interpolating f at D. In fact, as in Section 4.2, one can show that such interpolating functions can also approximate f. Finally, we note that some constructive methods for smooth convexity preserving interpolation can be found in [12,14].…”

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confidence: 96%