Reversing subdivision rules: local linear conditions and observations on inner products

Bartels, Richard H.; Samavati, Faramarz

doi:10.1016/s0377-0427(00)00370-8

Cited by 60 publications

(64 citation statements)

References 29 publications

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“…5 Inferring L-systems from subdivision matrices Since subdivision curves are often defined using matrix notation [2,33,36,37], the inference of L-systems from the subdivision matrices is an important practical problem. Unfortunately, it cannot entirely be resolved by algorithmic means, because dots in the general subdivision matrices, e.g.…”

Section: -8mentioning

confidence: 99%

“…Bartels and Samavati introduced the notion of reverse subdivision, in which the number of points representing a curve or surface is gradually reduced, while the resulting approximations are kept within tolerable error bounds [2,33]. Specifically, local reverse subdivision [2] individual old points by subsequences of new points, it replaces subsequences of old points by individual new points.…”

Section: Reverse Subdivisionmentioning

confidence: 99%

“…Specifically, local reverse subdivision [2] individual old points by subsequences of new points, it replaces subsequences of old points by individual new points. Bartels and Samavati specify this process using reverse subdivision matrices.…”

Section: Reverse Subdivisionmentioning

confidence: 99%

“…Subdivision schemes other than Chaikin's can be specified in a similar way, using different subdivision matrices [2,33,36,37].…”

Section: Introductionmentioning

confidence: 99%

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L-System Description of Subdivision Curves

Prusinkiewicz

Samavati

Smith

et al. 2003

Int. J. Shape Model.

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In recent years, subdivision has emerged as a major geometric modeling technique. Algorithms for generating subdivision curves are often specified in terms of iterated matrix multiplication. Each multiplication maps a globally indexed sequence of points that represents a coarser approximation of the curve onto a longer sequence that represents a finer approximation. Unfortunately, an infinite set of matrices is needed to specify these mappings for sequences of points of arbitrary length. Thus, matrix algebra is not well attuned to the dynamic nature of subdivision. In addition, matrix notation and the use of indices obscure the local and stationary character of typical subdivision rules.We introduce parametric context-sensitive L-systems with affine geometry interpretation as an alternative technique for specifying and generating subdivision curves. This technique is illustrated using Chaikin, cubic B-spline, and Dyn-LevinGregory (4-point) subdivision schemes as examples. L-systems formalize subdivision algorithms in an intuitive, concise, index-free manner, reflecting the parallel and local character of these algorithms. Furthermore, L-system specification of subdivision algorithms directly leads to their computer implementation.

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Section: -8mentioning

confidence: 99%

Section: Reverse Subdivisionmentioning

confidence: 99%

Section: Reverse Subdivisionmentioning

confidence: 99%

“…Subdivision schemes other than Chaikin's can be specified in a similar way, using different subdivision matrices [2,33,36,37].…”

Section: Introductionmentioning

confidence: 99%

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L-System Description of Subdivision Curves

Prusinkiewicz

Samavati

Smith

et al. 2003

Int. J. Shape Model.

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“…To simplify this representation, we fit a curve onto these data points, using Reverse Chaikin Subdivision [6]. At every iteration of this algorithm, the data size halves.…”

Section: Sketch Acquisition and Filteringmentioning

confidence: 99%

Paper and Pen: A 3D Sketching System

Yıldız

Çapın

2012

Computer and Information Sciences III

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This paper proposes a method that resembles a natural pen and paper interface to create curve based 3D sketches. The system is particularly useful for representing initial 3D design ideas without much effort. Users interact with the system by the help of a pressure sensitive pen tablet. The input strokes of the users are projected onto a drawing plane, which serves as a paper that they can place anywhere in the 3D scene. The resulting 3D sketch is visualized emphasizing depth perception. Our evaluation involving several naive users suggest that the system is suitable for a broad range of users to easily express their ideas in 3D. We further analyze the system with the help of an architect to demonstrate the expressive capabilities.

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