A subdivision scheme for surfaces of revolution

Morin, Géraldine; Warren, Joe; Weimer, Henrik

doi:10.1016/s0167-8396(01)00043-7

Cited by 73 publications

(43 citation statements)

References 8 publications

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“…Polar subdivision differs structurally from tensored univariate schemes with singularities, e.g. [Morin et al 2001], in that the number of neighbors of the extraordinary point does not double with each polar subdivision step but stays fixed. Quadrilaterals in a polar net are not split and the control net refines only towards the extraordinary point ( Figure 1).…”

Section: Introduction Circular Radialmentioning

confidence: 99%

Bicubic polar subdivision

Karčiauskas

Peters

2007

ACM Trans. Graph.

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We describe and analyze a subdivision scheme that generalizes bicubic spline subdivision to control nets with polar structure. Such control nets appear naturally for surfaces with the combinatorial structure of objects of revolution and at points of high valence in subdivision meshes. The resulting surfaces are C 2 except at a finite number of isolated points where the surface is C 1 and the curvature is bounded.

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Section: Introduction Circular Radialmentioning

confidence: 99%

Bicubic polar subdivision

Karčiauskas

Peters

2007

ACM Trans. Graph.

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“…This does not gives nonstationary subdivision schemes which reproduce a circle as in the sense of book [7]. A circle reproducing subdivision scheme (in [7]) is a circle approximative scheme which uses linear formulas for computation of new control points by old points. Below, we present subdivision of a circular arc which is not linear and not approximative.…”

Section: Subdivision Of a Circular Arcmentioning

confidence: 99%

A circle representation using complex and quaternion numbers

Zubė

2006

Lith Math J

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“…A main task of curve modeling in product design is to reproduce segments of a variety of basic shapes, such as conics, spirals and clothoids exactly, and to transition smoothly between them. Since the standard uniform, polynomial subdivision algorithms cannot reproduce these basic shapes, a number of non-stationary curve subdivision algorithms have recently been devised to reproduce, in particular, circles and ellipses [15,20,3,6,18,4,7,2,19]. However, the introduction of parameter-dependent subdivision means that explicit basis functions for the control points are no longer easily available, removing a reliable technique to compute curvature.…”

Section: Introductionmentioning

confidence: 99%

Curvature of Approximating Curve Subdivision Schemes

Karčiauskas

Peters

2012

Curves and Surfaces

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Abstract. The promise of modeling by subdivision is to have simple rules that avoid cumbersome stitching-together of pieces. However, already in one variable, exactly reproducing a variety of basic shapes, such as conics and spirals, leads to non-stationary rules that are no longer as simple; and combining these pieces within the same curve by one set of rules is challenging. Moreover, basis functions, that allow reading off smoothness and computing curvature, are typically not available. Mimicking subdivision of splines with non-uniform knots allows us to combine the basic shapes. And to analyze non-uniform subdivision in general, the literature proposes interpolating the sequence of subdivision control points by circles. This defines a notion of discrete curvature for interpolatory subdivision. However, we show that this discrete curvature generically yields misleading information for non-interpolatory subdivision and typically does not converge, not even for non-uniform spline subdivision. Analyzing the causes yields three general approaches for solving or at least mitigating the problem: equalizing parameterizations, sampling subsequences and a new skip-interpolating subdivision approach.

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A subdivision scheme for surfaces of revolution

Cited by 73 publications

References 8 publications

Bicubic polar subdivision

Bicubic polar subdivision

A circle representation using complex and quaternion numbers

Curvature of Approximating Curve Subdivision Schemes

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