On Organizing Principles of Discrete Differential Geometry...
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On Organizing Principles of DiscreteDifferential Geometry. Geometry of Spheres
Alexander Bobenko
Technical University Berlin
LMS Durham Symposium “Methods of Integrable Systems inGeometry", August 11-21, 2006
DFG Research Unit 565 “Polyhedral Surfaces”Alexander Bobenko On organizing principles of DDG
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Discrete Differential Geometry
I Aim: Development of discrete equivalents of the geometricnotions and methods of differential geometry. The latterappears then as a limit of refinements of the discretization.
I Question: Which discretization is the best one?I (Theory): preserves fundamental properties of the smooth
theoryI (Applications): represent smooth shape by a discrete shape
with just few elements; best approximation
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This talk
I Discretization PrinciplesI SurveyI New results on discrete curvature line parametrized
surfaces in Lie, Laguerre and Möbius geometry, joint withYu.B. Suris [arXiv:math.DG/0608291]
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Transformation Group Principle
Smooth geometric objects and their discretizations belong tothe same geometry, i.e. are invariant with respect to the sametransformation group⇒ discrete Klein’s Erlangen Program
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Möbius transformations
Definition. A Möbius transformation in R3 is a composition ofreflections in spheres.Properties.
I ConformalI preserve spheresI Willmore energy W = 1
4
∫(k1 − k2)
2
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Discrete surfaces in Euclidean and Möbius geometries
Euclidean geometry Möbius geometry
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Discrete surfaces in Euclidean and Möbius geometries
Euclidean geometry Möbius geometry
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Discrete Willmore energy for simplicial surfaces
[B. ’04] Theory[B., Schröder ’05] Applications
Definition. W (v) =∑
i βi − 2π
W (S) = 12
∑v∈V W (v) =
∑e∈E β(e)− π|V |
I Möbius invariantI W (S) ≥ 0, and W (S) = 0 iff S is a spherical convex
polyhederI The discrete Willmore energy W approximates the smooth
Willmore energy 14
∫(k1 − k2)
2; special smooth limitAlexander Bobenko On organizing principles of DDG
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Application. Geometric fairing
Smooth filling. Position and tangency constraints.Alexander Bobenko On organizing principles of DDG
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Application. Geometric C1-surface restoration
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Quadrilateral Surfaces
Quadrilateral surfaces as discrete parametrized surfaces
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Surfaces and transformations
Classical theory of (specialclasses of) surfaces (constantcurvature, isothermic, etc.)
General and specialQuad-surfaces
special transformations(Bianchi, B"acklund, Darboux)
discrete → symmetric
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Basic idea
Do not distinguish discrete surfaces and their transformations.Discrete master theory.
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Example - planar quadrilaterals as discrete conjugate systems.Multidimensional Q-nets [Doliwa, Santini ’97].
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Basic idea
Do not distinguish discrete surfaces and their transformations.Discrete master theory.
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Example - planar quadrilaterals as discrete conjugate systems.Multidimensional Q-nets [Doliwa, Santini ’97].
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Integrability as Consistency
I Equation I Consistency
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Integrability as Consistency
I Equation I Consistency
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a b
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Integrability as Consistency
I Equation I Consistency
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Integrability as Consistency
I Equation I Consistency
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Alexander Bobenko On organizing principles of DDG
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Discrete integrable systems 2D
I Equation I Consistency ⇒Integrability
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x u
v yI Lax representation,
Darboux transformation[Adler, B., Suris ’03],[Nijhoff ’02]
Q(x , y , u, v) = 0, Q affine with respect to all variables, squaresymmetry
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Discrete integrable systems 2D. Classification
[Adler, B., Suris ’03]
(Q1) α(x − v)(u − y)− β(x − u)(v − y) + δ2αβ(α− β) = 0,(Q2)
α(x − v)(u − y)− β(x − u)(v − y) + αβ(α− β)(x + y + u + v)−αβ(α− β)(α2 − αβ + β2) = 0,
(Q3) sin(α)(xu + vy)− sin(β)(xv + uy)− sin(α− β)(xy + uv)+δ2 sin(α− β) sin(α) sin(β) = 0,
(Q4) sn(α)(xu + vy)− sn(β)(xv + uy)− sn(α− β)(xy + uv)+sn(α− β)sn(α)sn(β)(1 + k2xyuv) = 0,
(H1) (x − y)(u − v) + β − α = 0,(H2) (x − y)(u − v) + (β − α)(x + y + u + v) + β2 − α2 = 0,(H3) α(xu + vy)− β(xv + uy) + δ(α2 − β2) = 0
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Discrete integrable systems 3D
Q(f , . . . , f123) = 0, Q affine with respect to all variables, cubesymmetry.[Wolf, Tsarev, B.] (preliminary) Classification of 3D consistentsystems.
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Discretization Principles
I Transformation Group Principle. Smooth geometricobjects and their discretizations belong to the samegeometry, i.e. are invariant with respect to the sametransformation group(discrete Klein’s Erlangen Program)
I Consistency Principle. Discretizations of smoothparametrized geometries can be extended tomultidimensional consistent nets(Integrability)
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Discretization of classical geometries
Consistency principle can be imposed for discretization ofclassical geometries (Möbius, Laguerre, Lie,...):
I transformation groups of various geometries (Möbius,Laguerre, Lie,...) are subgroups of the projectivetransformation group preserving absolute (distinguishedquadric),
I multidimensional Q-nets (projective geometry) can berestricted to an arbitrary quadric [Doliwa ’99].
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Circular nets
Martin, de Pont, Sharrock [’86], Nutbourne [’96], B. [’96],Cieslinski, Doliwa, Santini [’97], Konopelchenko, Schief [’98],Akhmetishin, Krichever, Volvovski [’99], ...
three “coordinate nets” of adiscrete orthogonal coordinatesystem
elementary cube→ Miquel theorem
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Conical nets
Conical nets as discrete curvature line parametrizations [Liu,Pottmann, Wallner, Yang, Wang ’06]
I Definition. Neighboring quads touch a common cone ofrevolution (in particular intersect at the tip of the cone)
I Conical net ⇔ circular Gauss mapI Normal shiftI Consistency
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Curvature lines through spheres
Pencil of touching spheres
Principal directions are invariant with respect to:I Möbius transformationsI normal shift
Curvature lines belong to Lie geometry.
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Möbius, Laguerre, Lie geometries
Lie geometry.Lie sphere transformations:oriented spheres (includingpoints and planes) mapped tooriented spheres preservingthe oriented contact of spherepairs
distinguished surfaces through transformationsMöbius points points MöbiusLaguerre planes tangent planes normal shift ...Lie none contact elements Lie sphere
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Lie quadric
LN+1,2 ={ξ ∈ RN+1,2 : 〈ξ, ξ〉 = 0
}.
Basis
e1, . . . , eN , e0, e∞, eN+3, ‖ei‖ = 1, ‖eN+3‖ = −1, ‖e0‖ = ‖e∞‖ = 0.
I Oriented hypersphere with center c ∈ RN and signedradius r ∈ R:
s = c + e0 + (|c|2 − r2)e∞ + reN+3.
I Oriented hyperplane 〈v , x〉 = d with v ∈ SN−1 and d ∈ R:
p = v + 0 · e0 + 2de∞ + eN+3.
I Point x ∈ RN : x = x + e0 + |x |2e∞ + 0 · eN+3.
I Infinity ∞: ∞ = e∞.
I Contact element (x , p): span(x , p) = ` ⊂ L, line.
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Line congruence
[Doliwa, Santini, Manas ’00]
I neighboring lines intersectI focal surfaces are Q-netsI consistent
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Discrete curvature line parametrization
I Line congruences can berestricted to (Lie) quadric
I Literal discretization ofBlaschke’s Lie geometricdescription of smoothcurvature lineparametrized surfaces
Definition. Discrete curvature line parametrization is a discretecongruence of isotropic lines
` : Z2 → {isotropic lines in L}
such that neighboring lines intersect
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Curvature line net. Projective model
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Curvature line net. Euclidean model
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Circular and conical nets. Relation
[B., Suris ’06], [Pottmann ’06]I Given a conical net p there exists a two-parameter family
of circular nets x such that (x , p) is curvature lineparametrized
I Given a circular net x there exists a two-parameter familyof conical nets p such that (x , p) is curvature lineparametrized
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Circular and conical nets. Relation
[B., Suris ’06], [Pottmann ’06]I Given a conical net p there exists a two-parameter family
of circular nets x such that (x , p) is curvature lineparametrized
I Given a circular net x there exists a two-parameter familyof conical nets p such that (x , p) is curvature lineparametrized
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Circular and conical nets. Relation
[B., Suris ’06], [Pottmann ’06]I Given a conical net p there exists a two-parameter family
of circular nets x such that (x , p) is curvature lineparametrized
I Given a circular net x there exists a two-parameter familyof conical nets p such that (x , p) is curvature lineparametrized
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Ribaucour congruence
I a discrete R-congruence of spheres - Q-net in Lie quadricI Ribaucour transformation of discrete curvature line
parametrized surfaces - any two corresponding contactelements have a sphere in common
I Spheres of a Ribaucour transformation build anR-congruence
I To any elementary quadrilateral of a discreteR-congruence (whose spheres span a subspace of thesignature (2,1)) there corresponds a Dupin cyclide
I Permutability of Ribaucour transformations
Permutability of Ribaucour transformations in Lie geometry ofsmooth surfaces [Burstall, Hertrich-Jeromin ’05]
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Special surfaces. Projective isothermic nets
I T-net (trapezoidal, parallel diagonals) in RN
I consistentI (equivalent) Moutard nets [Nimmo, Schief ’97] via Moutard
equation fij + f = aij(fj + fi)I Projective characterization: five “diagonal” points lie in a
three-space ([Doliwa ’05])
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Special surfaces. Möbius isothermic nets
For discrete isothermic nets (S2 is a special case):I (Möbius) isothermic net: Five “diagonal” points lie on a
common sphereI T-net in the light cone LN+1,1
I Equivalent to the cross-ratio definition of [B., Pinkall ’93]
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Special surfaces. Laguerre isothermic nets
I Definition. Five “diagonal” planes have a commontouching sphere
I Laguerre isothermic net ⇔ isothermic Gauss map in S2
Alexander Bobenko On organizing principles of DDG