Circular meshes, cyclide splines and geometric algebrafolk.uio.no/ranestad/krasauskas.pdfCircular...

Circular meshes, cyclide splinesand geometric algebra

Rimas Krasauskas

Vilnius University, Lithuania

Algebraic geometry in the sciences, Oslo, January 13, 2011

R. Krasauskas (VU, Lithuania) Circular meshes, cyclide splines and GA Oslo, Jan 13, 2011 1 / 44

Outline

1 IntroductionCircular meshes in architectural design/engineeringCyclides in computer aided geometric design (CAGD)

2 Generalized cyclide splinesPrototype exampleCyclide splines from framed quasi-circular meshesMore examples

3 Using quaternions and geometric algebraBezier parametrizations with quaternionic weightsGeometric algebra and conformal model

4 Conclusions


Outline




4 Conclusions


Outline




4 Conclusions


Outline




4 Conclusions


Introduction

Circular meshes in architectural design/engineering


Freeform structures in architecture

Freeform geometries are becoming increasingly popular in contemporaryarchitecture:

Vilnius Guggenheim Hermitage Museum is a proposed art museum in Vilnius,Lithuania. Author: a British-Iraqi architect Zaha Hadid, 2008. The museumwas scheduled to open in 2011...


Freeform structures in architecture

Freeform geometries are becoming increasingly popular in contemporaryarchitecture:

Vilnius Guggenheim Hermitage Museum is a proposed art museum in Vilnius,Lithuania. Author: a British-Iraqi architect Zaha Hadid, 2008. The museumwas scheduled to open in 2011...


Planar quad meshes vs triangular meshes

Planar quad (PQ) meshes possess a number of important advantages overtriangular meshes:

a smaller number of edges, i.e. supporting beams following the edges –less steel and less costa lower node complexity – important for manufacturing










PQ meshes with offsetsFrequently two layers of an actual construction are needed:

DefinitionPQ mesh is a conical mesh if for all vertices the four incident face planes aretangent to a common oriented cone of revolution.

DefinitionPQ mesh is a circular mesh if for all faces the four incident vertices are on acircle.


Conical and circular meshes

TheoremA simply connected PQ mesh posesses a face-offsett mesh if and only if it isa conical mesh.

Pottmann, H., Liu, Y., Wallner, J., Bobenko, A., Wang, W.: Geometry of multi-layer freeformstructures for architecture. ACM Trans. Graphics 26(3) (2007)

TheoremA simply connected PQ mesh posesses a vertex-offsett mesh if and only if itis a circular mesh.

Pottmann, H., Wallner, J.: The focal geometry of circular and conical meshes. Adv. Comp. Math29, 249–268 (2008)


Duality between conical and circular meshes


Approximate conical meshes

This conical mesh in front was obtained by a combination of Catmull-Clarksubdivision and conical optimization from the control mesh behind.

Pottmann, Wallner, Yang, Wang (2007)


Introduction

Cyclides in computer aided geometric design (CAGD)


Discovery of cyclidesDupin cyclides are exceptional surfaces discovered by Charles Dupin in his1803 dissertation under Gaspard Monge.First publicated in his book Applications de Geometrie (1822).

Later studied by Maxwell (1868) and Cayley (1878).


Discovery of cyclidesDupin cyclides are exceptional surfaces discovered by Charles Dupin in his1803 dissertation under Gaspard Monge.First publicated in his book Applications de Geometrie (1822).

Later studied by Maxwell (1868) and Cayley (1878).


Dupin cyclides

An alternative viewpoint to this discovery:[...] the cyclide surface was first explored by Victorian geometers.

It is often called the Dupin cyclide after a French methematician whopublished some of its properties.

DefinitionDupin cyclides have several equivalent definitions:

any inversion of any standard torusa surface with curvature lines all circlesan envelope of a family of spheres touching 3 given spheresa canal surface in two different waysthe focal surface degenerate to a couple of conics


Dupin cyclides

An alternative viewpoint to this discovery:[...] the cyclide surface was first explored by Victorian geometers.

It is often called the Dupin cyclide after a French methematician whopublished some of its properties.

DefinitionDupin cyclides have several equivalent definitions:

any inversion of any standard torusa surface with curvature lines all circlesan envelope of a family of spheres touching 3 given spheresa canal surface in two different waysthe focal surface degenerate to a couple of conics


Different cases of Dupin cyclides

Different cases of Dupin cyclides are inversions of a standard torus:


Principal patches of cyclides

Interest in cyclides revived in the 1980’s. It was motivated by research inCAGD by R. Martin (1982), who considered principal patches of cyclides, i.e.bounded by circles which are curvature lines.Later studied by Chandru, Dutta, Hoffmann, Pratt and others.


Principal patches of cyclides

Interest in cyclides revived in the 1980’s. It was motivated by research inCAGD by R. Martin (1982), who considered principal patches of cyclides, i.e.bounded by circles which are curvature lines.Later studied by Chandru, Dutta, Hoffmann, Pratt and others.


Composing cyclide patches

The idea of smoothly composing principal Dupin cyclide patches:

Dutta, Martin and Pratt (1993) made a conclusion:Cyclides appear marginally insufficient with respect to the

degrees of freedom of shape control...


Composing cyclide patches

The idea of smoothly composing principal Dupin cyclide patches:

Dutta, Martin and Pratt (1993) made a conclusion:Cyclides appear marginally insufficient with respect to the

degrees of freedom of shape control...


Approximate methods

Pengbo Bo, Surface Fitting and Developable Surface Modeling, PhD thesis,HKU, 2010.

Computer Graphics Group, HKU: Wenping Wang, Peng Bo,...


Generalized cyclide splines

Problem formulation


Problem formulation

Cyclide splines based on regular circular meshes have restrictions:they cannot represent a surface of genus other than 0 or 1,a cyclide either have no umbilic points, or is a sphere.

ProblemFind a natural generalization of circular meshes and associated cyclidesplines, that

can represent any topology,can have isolated umbilic points.


Generalized cyclide splines

Prototype example


Filling a D3-symmetric hexagonal hole

Step 0



Step 1



Step 2



Step 3


Circular mesh extraction

The resulting circular quad mesh contains a planar hexagon in the center.


Framed mesh

A framed mesh is a mesh with 4-valence vertices and two families of vectors:normals {Nv} idexed by vertices v and tangents {Tv ,e} idexed by vertex–edgepairs (v ,e), v ∈ e, subject to the following conditions:

1 for every corner of a face defined by a vertex v and two incident edges e,e′ a triple of vectors (Nv ,Tv ,e,Tv ,e′) is an orthonormal frame;

2 any two such frames associated with adjacent corners of the same faceare symmetric w.r.t. to their common edge.


Spline construction

Starting from a framed quasi-circular mesh we construct a spline surface Sin two steps:

1 for every edge e its endpoints v , v ′ are connected by a uniquecircular/linear segment C(e) with inward tangents Tv ,e, Tv ′,e.

2 For every face f the associated circular/linear boundary⋃

e⊂f C(e) isfilled by n-sided surface S(f ) with the normals Nv at the corner vertices vaccording to three cases:

I n = 2, 3: S(f ) is a unique spherical patch;I n = 4: S(f ) is a unique principal cyclide patch;I n > 4: S(f ) is a special n-sided multipatch having a subdivision-like layout:

shrinking concentric rings composed of principal cyclide patches.

This limit procedure cannot be avoided in general: there are no isolatedumbilic points on cyclides.


Details on hole filling

Here: pi , ci,i+1 are corner vertices and boundary circles. Choose pairs ofpoints pL

i,i+1,pRi,i+1 ∈ ci,i+1, such that

ci,i+1, pRi−1,i , pL

i+1,i+2 are on the same sphere.Define new circles ci = pR

i−1,i ∧ pi ∧ pRi,i+1.

Any point q1 ∈ c1 defines unique q2 ∈ c2 and so on...


Details on hole filling

Here: pi , ci,i+1 are corner vertices and boundary circles. Choose pairs ofpoints pL

i,i+1,pRi,i+1 ∈ ci,i+1, such that

ci,i+1, pRi−1,i , pL

i+1,i+2 are on the same sphere.Define new circles ci = pR

i−1,i ∧ pi ∧ pRi,i+1.

Any point q1 ∈ c1 defines unique q2 ∈ c2 and so on...


Branching blend of cylinders

Blending with four non-symmetric pentagonal patches:


Using quaternions


Circular arcs in R3

Let p0 and p1 are two endpoints of a circular arc in R3, and let f be someinterior point on it. Then the arc can be rationally parametrized by thequaternionic formula (here a

b = ab−1)

C(t) =p0w0(1− t) + p1w1t

w0(1− t) + w1t∈ R3,

where ’weights’ are w0 = (f − p0)−1, w1 = (p1 − f )−1, and f = C( 1

2 ) is called aFarin point .Alternatively, weights can be defined via a tangent vector v0 at p0:

w0 = 1, w1 = (p1 − p0)−1v0.


Circular splinesA smooth circular spline going through a sequence of points p0, . . . ,pn−1 isuniquely defined by a tangent vector v0 at the point p0. Then other unittangent vectors are defined by recurrence: vi+1 = (pi+1 − pi)

−1vi(pi+1 − pi),i.e. its direction is an opposite to the reflected vector w.r.t. the edge pi+1 − pi .

Suppose the number of points is even n = 2k , and the spline curve is closed,i.e. p2k = p0.


Multiratio condition

DefinitionA multi-ratio of points p0, . . . ,p2k−1

mr(p0, . . . ,p2k−1) = (p0 − p1)(p1 − p2)−1 · · · (p2k−1 − p0)

−1

For example:k = 0: mr(p0,p1) = −1, when p0 6= p1.k = 2: mr(p0,p1,p2,p3) coincides with the classical cross-ratio, which isreal if and only if these four points are on a circle.

LemmaA closed circular spline exists for every initial tangent v0 and even number ofpoints⇔ their multi-ratio is real.


Quasi-circular meshes

DefinitionA Quasi-circular even (QCE) mesh is a mesh with even sided faces and4-valency vertices satisfying multi-ratio condition

mr(p0, . . . ,p2k−1) ∈ R

for every even loop of edges (p0,p1), (p1,p2), . . . , (p2k−1,p0)

This is more general definition than in:Liu, Y., Wang, W.: On vertex offsets of polyhedral surfaces. In: Advances in ArchitecturalGeometry. pp. 61–64. Vienna (2008)

Any QCE mesh with a fixed frame at a vertex naturally defines a framed meshwhich can be used to generate a cyclide spline (as was shown above).

Similarly a QCE mesh defines a vertex-offset and a face-offset meshes.


Spherical patches

Further applications of quaternions.

Let S be a spherical triangle with corner points p0, p1, p2 bounded by threecircular arcs, such that these three circles intersect in a point p∞.

Then this spherical triangle S can be rationally parametrized by thequaternionic formula

S(s, t) =p0w0(1− s − t) + p1w1s + p2w2t

w0(1− s − t) + w1s + w2t,

with weights: wi = (pi − p∞)−1, i = 0,1,2.


Principal cyclide patches

Let p0, p1, p2, p3 be any 4 points on a circle in R3, and let v1, v2 be twoorthonormal vectors.Then there is a unique principal Dupin cyclide patch D with corners in thesepoints, and bounded by circular arcs with tangent vectors v1, v2 at the cornerp0, which can be rationally parametrized by the quaternionic formula

D(s, t) =p0w0(1− s)(1− t) + p1w1s(1− t) + p2w2(1− s)t + p3w3st

w0(1− s)(1− t) + w1s(1− t) + w2(1− s)t + w3st,

where wi are defined using vectors qij = (pi − pj)/|pi − pj |:

w0 = 1, w1 = q10v1, w2 = q20v2,

w3 = |p2 − p1||p3 − p0|−1q31w1q20w2.


Geometric algebra

All formulas involving quaternions can be written in terms of geometricalgebra.

Geometric algebra (GA) = Clifford algebra + geometric content

Start from a vector space with a given inner product.The geometric product of two vectors a and b is defined to be associative anddistributive over addition, with additional rule: a · a = a2 ∈ R.Define symmetric inner and anti-symmetric exterior products:

a · b =12(ab + ba), a ∧ b =

12(ab − ba).


Conformal model of Euclidean space R3

Consider homogenious coordinates in R4 associated with a standard basis{ei}, i = 1, . . . ,5 (x1 = 0 is a plane at infinity). Then slightly change the basis:

e∞ = e4 + e5, e0 = (−e4 + e5)/2.

Use a stereographic projection R3 → S3 ⊂ R4 to the unit sphere

F : x 7→ x +12

x2e∞ + e0,

where x2 = x · x is a Euclidean inner product.


Conformal algebra of Euclidean space R3

The basis: {ei}, i = 1, . . . ,5 generates an algebra that is spanned by1 + 5 + 10 + 10 + 5 + 1 = 32 terms:

1, {ei}, {ei ∧ ej}, {ei ∧ ej ∧ ek}, {Iei}, I,

where I = e1e2e3e4e5 is called a pseudo-scalar, I2 = −1.Also a meet product will be useful:

(L1 ∨ L2)∗ = L∗1 ∧ L∗2, X ∗ = IX .


Geometric meaning

Points: p, p · p = 0.Pairs of points: p1 ∧ p2, pi – points.Circles: p1 ∧ p2 ∧ p3.Spheres: p1 ∧ p2 ∧ p3 ∧ p4.Intersection of two geometric objects: L1 ∨ L2.For example, intersection of a sphere s and a circle c with a common point p1.

q = s ∨ c

Then we compute a plane of symmetry of the pair Q := q∗ ∧ e∞ and extractthe second point by reflecting the known point: p2 = Qp1


Conclusions

We generalizedthe notion of a regular circular quad meshthe associated Dupin cyclide patchwork

to:the notion of framed quasi-circular mesh involving non-quad facesthe associated cyclide spline surface of arbitrary topology.

It seems all constructions have a nice description in terms of geometricalgebra.

Problems:hole filling problem - extra shape control requiredrelations Laguerre and Lie sphere geometries.


Conclusions






Conclusions






Conclusions






Conclusions






Conclusions






Conclusions






References (including figures)

1 Bobenko, A., Suris, Y.: Discrete differential geometry: Consistency asintegrability (2005), http://arxiv.org/abs/math.DG/0504358

2 Dutta, D., Martin, R.R., Pratt, M.J.: Cyclides in surface and solidmodeling. IEEE Comput. Graph. Appl. 13, 53–59 (1993)

3 Liu, Y., Wang, W.: On vertex offsets of polyhedral surfaces. In: Advancesin Architectural Geometry. pp. 61–64. Vienna (2008)

4 Martin, R., Pont, J.D., Harrock, T.J.S.: Cyclide surfaces in computer aideddesign. In: Gregory, J.A. (ed.) The Mathematics of Surfaces. pp.253–268. Clarendon Press, Oxford (1986)

5 Perwass, C., Geometric Algebra with Applications in Engineering,Springer, 2008.

6 Pottmann, H., Liu, Y., Wallner, J., Bobenko, A., Wang, W.: Geometry ofmulti-layer freeform structures for architecture. ACM Trans. Graphics26(3) (2007)

7 Pottmann, H., Wallner, J.: The focal geometry of circular and conicalmeshes. Adv. Comp. Math 29, 249–268 (2008)


Colaborators

Severinas Zube (Vilnius University)

Wenping Wang (The University of Hong Kong)

Pengbo Bo (The University of Hong Kong)

Acknowledgements

Kestas Karciauskas (VU), Heidi Dahl (VU, SINTEF).

CLUCalc interactive visualization software created by C. Perwass(http://www.clucalc.info/) was used intensively for preparation of this talk.


Questions

Thank you!


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