enhancementofamulti-sided...

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


Outline

1 IntroductionMotivationPrevious work

2 Generalized Bézier (GB) patchControl structureDomain & parameterizationBlending functions

3 EnhancementsProblemsNew algorithms

4 Examples5 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?




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




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




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




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




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]




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 )




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)




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.




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




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




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




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 )




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




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




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




Example 1

Torso




Example 1

Torso – detail




Example 2

Gamepad




Example 2

Gamepad – detail




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




Future work

Patches over concave domains




Any questions?

Thank you for your attention.



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