Introduction Generalized Bézier patch Concave GB patch Examples Conclusion Multi-sided Bézier surfaces over concave polygonal domains Péter Salvi, Tamás Várady Budapest University of Technology and Economics SMI 2018 Lisbon, June 6 th –8 th P. Salvi, T. Várady BME Multi-sided Bézier surfaces over concave polygonal domains
Introduction Generalized Bézier patch Concave GB patch Examples Conclusion Outline Introduction 1 Motivation Previous work Generalized Bézier (GB) patch 2 Control structure Domain & parameterization Blending functions Concave GB patch 3 Reinterpretation Building blocks Additional control Examples 4 Conclusion and future work 5 P. Salvi, T. Várady BME Multi-sided Bézier surfaces over concave polygonal domains
Introduction Generalized Bézier patch Concave GB patch Examples Conclusion Motivation Multi-sided patches Curve network based design Feature curves Automatic surface generation Hole filling E.g. vertex blends Cross-derivative constraints “Concave” configurations Representation? P. Salvi, T. Várady BME Multi-sided Bézier surfaces over concave polygonal domains
Introduction Generalized Bézier patch Concave GB patch Examples Conclusion Motivation Conventional representations used in CAD systems Trimmed tensor product surfaces Detailed interior control Continuity problems Different edge types ⇒ inherently asymmetric Division into smaller quadrilaterals (Semi-)automatic splitting curves Underdetermined entities Reduced continuity Our goals: C ∞ continuity Editing with control points (with interior control) No additional artificial curves P. Salvi, T. Várady BME Multi-sided Bézier surfaces over concave polygonal domains
Introduction Generalized Bézier patch Concave GB patch Examples Conclusion Previous work Concave surface representations Loop and DeRose (1989), Smith and Schaefer (2015) S-patches – multivariate Bézier patches Beautiful theory Difficult to use Kato (1991, 2000) Transfinite surface interpolation Supports holes No internal control Singular blends cause high curvature variation Pan et al. (2015), Stanko et al. (2016) Discrete methods minimizing fairness energies (See comparisons later) P. Salvi, T. Várady BME Multi-sided Bézier surfaces over concave polygonal domains
Introduction Generalized Bézier patch Concave GB patch Examples Conclusion ☛ ✟ Generalized Bézier (GB) patch ✡ ✠ T. Várady, P. Salvi, Gy. Karikó, A Multi-sided Bézier Patch with a Simple Control Structure . Computer Graphics Forum, Vol. 35(2), pp. 307-317, 2016. T. Várady, P. Salvi, I. Kovács, Enhancement of a multi-sided Bézier surface representation . Computer Aided Geometric Design, Vol. 55, pp. 69-83, 2017. P. Salvi, T. Várady BME Multi-sided Bézier surfaces over concave polygonal domains
Introduction Generalized Bézier patch Concave GB patch Examples Conclusion Control structure Control net derivation from the quadrilateral case Control grid → n ribbons � d � Degree: d , Layers: l = 2 Control points: C i 1/4 j , k i ∈ [ 1 .. n ] , j ∈ [ 0 .. d ] , k ∈ [ 0 .. l − 1 ] 1/2 1/2 1 1/2 1/2 1/2 1/2 1 1/2 1/2 Weighting functions: µ i j , k α α 1 β β α α 1 β β P. Salvi, T. Várady BME Multi-sided Bézier surfaces over concave polygonal domains
Introduction Generalized Bézier patch Concave GB patch Examples Conclusion Domain & parameterization Domain Regular domain in the ( u , v ) plane Side-based local parameterization functions s i and h i Based on Wachspress barycentric coordinates λ i ( u , v ) s 4 s 3 s 3 h 4 h 3 h 3 h 4 s 4 s 2 h 5 h 2 s 5 s 2 h 2 h 1 s 1 h 1 v s 1 u P. Salvi, T. Várady BME Multi-sided Bézier surfaces over concave polygonal domains
Introduction Generalized Bézier patch Concave GB patch Examples Conclusion Domain & parameterization Local parameters λ i s i = λ i − 1 + λ i h i = 1 − λ i − 1 − λ i Barycentric coordinates λ i λ i ≥ 0 [positivity] � n i = 1 λ i = 1 [partition of unity] � n i = 1 λ i ( u , v ) · P i = ( u , v ) [reproduction] λ i ( P j ) = δ ij [Lagrange property] P. Salvi, T. Várady BME Multi-sided Bézier surfaces over concave polygonal domains
Introduction Generalized Bézier patch Concave GB patch Examples Conclusion Blending functions Half-Bézier ribbons d l − 1 � � C i j , k · µ i j , k B i R i ( s i , h i ) = j , k ( s i , h i ) j = 0 k = 0 C i j , k : j -th CP on side i , layer k B i j , k ( s i , h i ) = B d j ( s i ) · B d k ( h i ) bivariate Bernstein polynomials µ i j , k rational function of h i , h i ± 1 1 on side i , 0 on the others The surface interpolates the ribbons at the boundary ( G 1 / G 2 ) P. Salvi, T. Várady BME Multi-sided Bézier surfaces over concave polygonal domains
Introduction Generalized Bézier patch Concave GB patch Examples Conclusion Blending functions Central weight & patch equation Weights do not add up to 1 Deficiency ⇒ weight of the central control point: n d l − 1 � � � µ i j , k B i B 0 ( u , v ) = 1 − j , k ( s i , h i ) i = 1 j = 0 k = 0 Patch equation: n � S ( u , v ) = R i ( s i , h i ) + C 0 B 0 ( u , v ) i = 1 P. Salvi, T. Várady BME Multi-sided Bézier surfaces over concave polygonal domains
Introduction Generalized Bézier patch Concave GB patch Examples Conclusion Algorithms Degree elevation Linear and bilinear combinations Modifies the surface interior Control net generated by reductions and elevations Default positions Merging Bézier ribbons of different degrees P. Salvi, T. Várady BME Multi-sided Bézier surfaces over concave polygonal domains
Introduction Generalized Bézier patch Concave GB patch Examples Conclusion ☛ ✟ Concave GB patch ✡ ✠ P. Salvi, T. Várady BME Multi-sided Bézier surfaces over concave polygonal domains
Introduction Generalized Bézier patch Concave GB patch Examples Conclusion Reinterpretation Problem: ribbon orientation Convex case: prev. tangent → next tangent Does not work for the concave case Interpolants should point towards the interior of the surface Control point placement? P. Salvi, T. Várady BME Multi-sided Bézier surfaces over concave polygonal domains
Introduction Generalized Bézier patch Concave GB patch Examples Conclusion Reinterpretation Independent ribbons Original constraints of the GB patch: Common d degree, same l = ⌈ d / 2 ⌉ number of layers Corresponding control points of adjacent ribbons are identical These can be lifted! ⇒ Ribbons become independent entities µ i j , k weight function still ensures the interpolating property Local d i and l i values for each ribbon Various possible configurations: 4 + 4 → 4 4 + 4 → 5 9 + 9 → 13 9 + 4 → 12 4 + 4 → 7 P. Salvi, T. Várady BME Multi-sided Bézier surfaces over concave polygonal domains
Introduction Generalized Bézier patch Concave GB patch Examples Conclusion Building blocks Ribbons d i l i − 1 � � C i j , k · µ i j , k B i R i ( s i , h i ) = j , k ( s i , h i ) j = 0 k = 0 ( d i + 1 ) × l i control points Degrees: d i (edgewise) 2 l i − 1 (cross-boundary) Degree elevation: Independently in the two parametric directions Adding a layer increases the degree by 2 P. Salvi, T. Várady BME Multi-sided Bézier surfaces over concave polygonal domains
Introduction Generalized Bézier patch Concave GB patch Examples Conclusion Building blocks Blending functions Blend of C i j , k is µ i j , k B i j , k ( s i , h i ) , where j , k ( s i , h i ) = B d i j ( s i ) · B 2 l i − 1 B i ( h i ) k α i = h 2 h 2 i − 1 + h 2 � � i − 1 / , when 2 j < d i µ i j , k = 1 , when 2 j = d β i = h 2 � h 2 i + 1 + h 2 � i + 1 / , when 2 j > d i No central control point ⇒ weight deficiency solved by normalization: n 1 � S ( u , v ) = B sum ( u , v ) · R i ( s i , h i ) i = 1 P. Salvi, T. Várady BME Multi-sided Bézier surfaces over concave polygonal domains
Introduction Generalized Bézier patch Concave GB patch Examples Conclusion Building blocks Blending function examples µ 1 0 , 0 B 3 0 ( s 1 ) B 3 µ 1 1 , 0 B 3 1 ( s 1 ) B 3 0 ( h 1 ) 0 ( h 1 ) µ 1 1 , 1 B 3 1 ( s 1 ) B 3 µ 1 0 , 1 B 3 0 ( s 1 ) B 3 1 ( h 1 ) 1 ( h 1 ) P. Salvi, T. Várady BME Multi-sided Bézier surfaces over concave polygonal domains
Introduction Generalized Bézier patch Concave GB patch Examples Conclusion Building blocks Domain generation – projection Project vertices on a best fit plane Simple Works well on: Relatively flat objects Fails for: Highly curved models Goals: Preserve angles Preserve arc lengths P. Salvi, T. Várady BME Multi-sided Bézier surfaces over concave polygonal domains
Introduction Generalized Bézier patch Concave GB patch Examples Conclusion Building blocks Domain generation – heuristic algorithm Normalize the angles p 5 p 3 Draw an open polygon Distribute the deviation p 4 Proportionally to edges p 2 p 6 Better results: p 7 p 1 p 8 =(0,0) v P. Salvi, T. Várady BME Multi-sided Bézier surfaces over concave polygonal domains
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