Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work
Enhancement of a multi-sidedBézier surface representation
Tamás Várady, Péter Salvi, István Kovács
Budapest University of Technology and Economics
CAGD 55, pp. 69–83, 2017.
GMP 2018
T. Várady, P. Salvi, I. Kovács BME
Enhancement of a multi-sided Bézier surface representation
Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work
Outline
1 IntroductionMotivationPrevious work
2 Generalized Bézier (GB) patchControl structureDomain & parameterizationBlending functions
3 EnhancementsProblemsNew algorithms
4 Examples5 Conclusion and future work
T. Várady, P. Salvi, I. Kovács BME
Enhancement of a multi-sided Bézier surface representation
Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work
Motivation
Applications of multi-sided patches
Curve network based designFeature curvesAutomatic surface generation
Hole fillingE.g. vertex blendsCross-derivative constraints
3D point cloud approximationGiven boundary loopsSmoothly connected patches
Representation?
T. Várady, P. Salvi, I. Kovács BME
Enhancement of a multi-sided Bézier surface representation
Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work
Motivation
Conventional representations
Trimmed/split tensor product surfacesDetailed control in the interiorCAD-compatibleBut: continuity problems
Recursive subdivisionArbitrary topologyEasy to design withBut: hard to interpolateboundary cross-derivatives
Transfinite patchesInterpolates any number of sidesDepends only on the boundaryBut: little interior control
T. Várady, P. Salvi, I. Kovács BME
Enhancement of a multi-sided Bézier surface representation
Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work
Previous work
Multi-sided surfaces with control networks
Loop and DeRose (1989)S-patches – beautiful theory, difficult to use
Warren (1992)Based on Bézier triangles, max. 6 sides
Zheng and Ball (1997)High-degree expressions, max. 6 sides
Krasauskas (2002)Toric patches – lattice-based, symmetry concerns
Várady et al. (2016)Generalized Bézier patchesRegular polygonal domainSymmetric control structure
T. Várady, P. Salvi, I. Kovács BME
Enhancement of a multi-sided Bézier surface representation
Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work
Control structure
Control net derivation from the quadrilateral case
Control grid →n ribbonsDegree: dLayers:l =
⌊d+12
⌋Control points:
Cd,ij,k
i = 1 . . . nj = 0 . . . dk = 0 . . . l−1
Weights: µij ,k
1/4
11/21/2 1/2 1/2
11/21/2 1/2 1/2
1/2 1/2
11/21/2 1 1/2 1/2
11/21/2 1 1/2 1/2
1/4
α α 1 β β
α α 1 β β α α 1 1 β β
ββ11αα
1/21/2
T. Várady, P. Salvi, I. Kovács BME
Enhancement of a multi-sided Bézier surface representation
Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work
Domain & parameterization
Domain
Regular domain in the (u, v) planeSide-based local parameterization functions si and hi
Based on Wachspress barycentric coordinates λi (u, v)
u
v
s4
h4
h2
s2
s3h3
s1
h1
s1
h1
h2
s2
s3
h3
s4 h4
s5
h5
T. Várady, P. Salvi, I. Kovács BME
Enhancement of a multi-sided Bézier surface representation
Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work
Domain & parameterization
Local parameters
si = λiλi−1+λi
hi = 1− λi−1 − λi
Barycentric coordinates λi
λi ≥ 0[positivity]∑n
i=1 λi = 1[partition of unity]∑n
i=1 λi (u, v) · Pi = (u, v)[reproduction]λi (Pj) = δij[Lagrange property]
T. Várady, P. Salvi, I. Kovács BME
Enhancement of a multi-sided Bézier surface representation
Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work
Blending functions
Bernstein functions with rational weights
Cd ,ij ,k : j-th control point on side i , layer k
Multiplied by µij ,kB
dj ,k(si , hi ) = µi
j ,kBdj (si )B
dk (hi )
µij ,k is a rational function for 2× 2 CPs in each cornerαi = hi−1/(hi−1 + hi ), βi = hi+1/(hi+1 + hi )
T. Várady, P. Salvi, I. Kovács BME
Enhancement of a multi-sided Bézier surface representation
Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work
Blending functions
Central weight & patch equation
Weights do not add up to 1Deficiency ⇒ weight of the central point:
Bd0 (u, v) = 1−
n∑i=1
d∑j=0
l−1∑k=0
µij ,kB
dj ,k(si , hi )
Patch equation:
S(u, v) =n∑
i=1
d∑j=0
l−1∑k=0
Cd ,ij ,k µ
ij ,kB
dj ,k(si , hi ) + Cd
0 Bd0 (u, v)
T. Várady, P. Salvi, I. Kovács BME
Enhancement of a multi-sided Bézier surface representation
Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work
Interpolation property
DefinitionA Bézier ribbon is a Bézierpatch given by the first twolayers (rows) of control pointson a given side.
TheoremThe Generalized Bézier patch,on its boundary, interpolatesthe position and first cross-derivative of the Bézierribbons of its respective sides.
T. Várady, P. Salvi, I. Kovács BME
Enhancement of a multi-sided Bézier surface representation
Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work
Overview
Fixed issuesWeight deficiency
Increases with n and d ⇒Influence of the central control point growsStrongly oscillates between even and odd degrees
Support for G 2 continuity between patches
New/updated algorithms
Degree elevation & reductionFullness controlApproximation of point clouds
T. Várady, P. Salvi, I. Kovács BME
Enhancement of a multi-sided Bézier surface representation
Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work
Problems
Weight deficiency
No central control pointfor odd-degree patchesFor even-degree patches:Cd
l ,l ·∑n
i=1 µil ,lB
dl ,l (si , hi )
Weight deficiency isdistributed amongst theinnermost blend functions
T. Várady, P. Salvi, I. Kovács BME
Enhancement of a multi-sided Bézier surface representation
Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work
Problems
New parameterization
Better isoline distributionLower weight deficiencyConstraints:
hi = 0.5 tangentialto si−1 and si+1Middle pointon a circular arc
Mi
Mi+1
Vi-2 Vi+1
C
Vi-1 Vi
Mi-1
Vi-2 Vi+1
C
Vi-1 Vi
Mi+1Mi-1
Mi
T. Várady, P. Salvi, I. Kovács BME
Enhancement of a multi-sided Bézier surface representation
Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work
Problems
G 2 continuity
Use squared terms in the rational weights:
αi = h2i−1/(h
2i−1 + h2
i ) βi = h2i+1/(h
2i+1 + h2
i )
T. Várady, P. Salvi, I. Kovács BME
Enhancement of a multi-sided Bézier surface representation
Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work
New algorithms
Degree elevation & reduction
Essentially the same as in the original paperLinear and bilinear combinationsModifies the surface (slightly)The control net is generated by reductions and elevations
Default positions for internal control points
T. Várady, P. Salvi, I. Kovács BME
Enhancement of a multi-sided Bézier surface representation
Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work
New algorithms
Fullness control
Multi-resolution editing techniqueEdit a control point of a lower-degree patch
E.g. quartic central point
Its influence is propagated by degree elevation
T. Várady, P. Salvi, I. Kovács BME
Enhancement of a multi-sided Bézier surface representation
Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work
New algorithms
Approximation
Least-squares fit of pointsInitial surface
Generated by theboundary constraints
Initial parameterizationProjection
Iteration:Fit with smoothingDegree elevationRe-parameterization
SmoothingReduce oscillationof the control points
T. Várady, P. Salvi, I. Kovács BME
Enhancement of a multi-sided Bézier surface representation
Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work
Example 1
Torso
T. Várady, P. Salvi, I. Kovács BME
Enhancement of a multi-sided Bézier surface representation
Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work
Example 1
Torso – detail
T. Várady, P. Salvi, I. Kovács BME
Enhancement of a multi-sided Bézier surface representation
Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work
Example 2
Gamepad
T. Várady, P. Salvi, I. Kovács BME
Enhancement of a multi-sided Bézier surface representation
Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work
Example 2
Gamepad – detail
T. Várady, P. Salvi, I. Kovács BME
Enhancement of a multi-sided Bézier surface representation
Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work
Conclusion
Summary
Generalized Bézier patchesSide-based interpretationAll control points generated by the boundariesvia degree elevationInterior control
EnhancementsFollow the quadrilateral patch more closelyCentral control point / weight deficiency fixedBetter parameterizationCurvature continuityApproximation algorithm
T. Várady, P. Salvi, I. Kovács BME
Enhancement of a multi-sided Bézier surface representation
Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work
Future work
Patches over concave domains
T. Várady, P. Salvi, I. Kovács BME
Enhancement of a multi-sided Bézier surface representation
Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work
Any questions?
Thank you for your attention.
T. Várady, P. Salvi, I. Kovács BME
Enhancement of a multi-sided Bézier surface representation