by Jan Hakenberg, Ulrich Reif, Scott Schaefer, Joe Warren | published as viXra:1407.0163 – July 21st, 2014 |
Figure: We compute the exact area, centroid, and inertia of the 2-dimensional sets bounded by subdivision curves. The illustration shows the principle axes of the inertia tensor drawn at the centroid of the area; five different subdivision schemes are used to demonstrate the universality of our derivation.
Abstract: We derive the (d+2)-linear forms that compute the moment of degree d of the area enclosed by a subdivision curve in the plane. We circumvent the need to solve integrals involving the basis function by exploiting a recursive relation and calibration that establishes the coefficients of the form within the nullspace of a matrix.
For demonstration, we apply the technique to the dual three-point scheme, the interpolatory C^{1} four-point scheme, and the dual C^{2} four-point scheme.
Moments Defined by Subdivision Curves * | moments_def...pdf | 670 kB |
Moments Defined by Subdivision Curves (on viXra.org) | viXra:1407.0163 | link |
Moments Defined by Example Subdivision Curves | viXra:1407.0027 | link |
Subdivision and Moments Implementation (Mathematica 9) | moments_curves.zip | 55 kB |
The first author was partially supported by personal savings accumulated during his visit to the Nanyang Technological University as a visiting research scientist in 2012–2013. He'd like to thank everyone who worked to make this opportunity available to him.
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The moments derived in the article have diverse applications:
Our article is structured as follows. First, we recap the basics of curve subdivision: the basis function of a scheme, and refinement matrices. Chaikin's scheme serves as an example. Then, we derive the formula for the moment of degree d for binary, stationary subdivision schemes. We demonstrate the practicability of our formalism on several popular schemes. The computation of moment values defined by a number of simple example curves serves as a reference for alternative implementations.
The schemes that are covered by our treatment are listed here:
Scheme for curves | Subdivision weights | Remark |
---|---|---|
linear B-spline | interpolatory, results in a polygon | |
quadratic B-spline Chaikin 1965 | dual | |
cubic B-spline | ||
Three-point scheme Hormann/Sabin 2008, and Quartic B-spline | dual, ω=1/32, ω=-1/48, quadratic precision | |
C^{1} four-point scheme Dubuc 1986, Dyn/Gregory/Levin 1987 | interpolatory, default ω=1/16, smooth for 0<ω<0.192729... | |
C^{2} four-point scheme Dyn/Floater/Hormann 2005 | dual, default ω=1/128, smooth at least for 0<ω<1/48 | |
C^{2} six-point scheme Weissman 1990 | interpolatory, default ω=3/256, and ω=1/96 |
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Figure: The monomials 1, x, y, x^{2}, xy, y^{2} integrated over the domain bounded by subdivision curves give the moments.
Our article does not feature subdivision schemes with non-homogeneous rules. Two examples immediately come to mind:
Remark: Our subsequent preprint covers the area forms for the schemes that are listed here as future work.
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