Concentric tessellation maps and curvature continuous guided surfaces

2009

Computers & Graphics

We assemble triangular patches of total degree at most eight to form a curvature continuous surface. The construction illustrates how separation of local shape from representation and formal continuity yields an effective construction paradigm in partly underconstrained scenarios. The approach localizes the technical challenges and applies the spline approach, i.e. keeping the degree fixed but increasing the number of pieces, to deal with increased complexity when many patches join at a central point.

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Section: Reduction To a Localized C 2 Transition Between Two Patchesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…[6,2,15,21,19,3,10,11,13,5,7,8]. Here, we consider three-sided patches as in [17], corresponding to control nets with triangular facets.…”

Section: Introductionmentioning

confidence: 99%

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Assembling curvature continuous surfaces from triangular patches

2009

Computers & Graphics

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“…The C 2 bi-2 characteristic ring of the bi-3 subdivision presented in the previous section advertises itself as concentric tessellation map for guided subdivision [5]. Our construction and sampling of a guide surface of degree bi-5 to derive subdivision rules are sketched in Figures 8 and 9.…”

Section: Bi-4 Subdivisionmentioning

confidence: 99%

Lens-shaped surfaces and C 2 subdivision

2009

Computing

Lens-shaped surfaces (with vertices of valence 2) arise for example in automatic quad-remeshing. Applying standard Catmull-Clark subdivision rules to a vertex of valence 2, however, does not yield a C 1 surface in the limit. When correcting this flaw by adjusting the vertex rule, we discover a variant whose characteristic ring is z → z 2 . Since this conformal ring is of degree bi-2 rather than bi-3, it allows constructing a subdivision algorithm that works directly on the control net and generates C 2 limit surfaces of degree bi-4 for lens-shaped surfaces. To further improve shape, a number of re-meshing and re-construction options are discussed indicating that a careful approach pays off. Finally, we point out the analogy between characteristic configurations and the conformal maps z 4/n , cos z and e z .Keywords lens-shaped, C 2 , subdivision surface, re-meshing, polar Mathematics Subject Classification (2000)Computer aided design (modeling of curves and surfaces) · Splines · Computer graphics and computational geometry MotivationThe prescription of the original Catmull-Clark construction [1] for creating a new mesh from an arbitrary existing one applies also when a vertex has valence n = 2 (Figure 2): For each facet, a new face node is computed as the average of the facet's old vertices; for each edge, a new edge node is introduced as the average of the edge's endpoints and the two new vertices of the faces joined by the edge; and for each vertex of valence n a new vertex node is computed as (Q + 2R + (n − 3)S)/n

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Subdivision and G-Spline Hybrid Constructions for High-Quality Geometric and Analysis-Suitable Surfaces

Geometric Challenges in Isogeometric Analysis

2022