vorgelegt von Diplom-Mathematikerin
Susanne Hannappel aus Berlin
Vom Institut f¨ur Mathematik
der Fakult¨at II, Mathematik und Naturwissenschaften der Technischen Universit¨at Berlin
zur Erlangung des akademischen Grades
Doktorin der Naturwissenschaften genehmigte Dissertation
Promotionsausschuß:
Vorsitzender: Prof. Dr. H. A. Jung Berichter: Prof. Dr. U. Pinkall Berichter: Prof. Dr. A. Bobenko
Tag der wissenschaftlichen Aussprache: 22. Juni 2001
Berlin 2001 D 83
Zusammenfassung
Die W-Eigenschaft einer diskreten Fl¨ache ist eine projektive Eigenschaft ¨
aquivalent zu der Existenz einer infinitesimalen Deformation, deren Einschr¨ankung auf eine beliebige Zelle der Bewegung eines starren K¨orpers entspricht. Die W-Eigenschaft eines Parameternetzes einer Fl¨ache ist eine projektive Eigenschaft ¨aquivalent zu der Existenz einer infinitesimalen Verbiegung, f¨ur die der Koeffizient der gemischten Ableitung der zweiten Fundamentalform bis zur ersten Ordnung erhalten bleibt.
Diskrete Fl¨achen mit ebenen Vierecken haben die W-Eigenschaft genau dann, wenn f¨ur jedes Viereck die Diagonalen und die Eckenspuren tan-gential an einen Kegelschnitt sind. Konjugierte Parameternetze f¨ur die die zweite Fundamentalform konform ist, haben die W-Eigenschaft genau dann, wenn die Fl¨ache eine Jonas Fl¨ache ist.
Diskrete Jonas Fl¨achen sind per Definition diskrete Fl¨achen mit ebenen Vierecken mit Diagonalen und Eckenspuren tangential an einen in zwei Geradenb¨uschel zerfallenden Kegelschnitt. Das folgende Resultat zeigt, daß es einen Zusammenhang zwischen der diskreten und der kontinuier-lichen Definition gibt.
Wenn f¨ur eine parametrisierte Fl¨ache eine Funktion g existiert, so daß die infinitesimalen Vierecke der Maschenweite ε der Taylor Entwicklung von
ˆ
f = f + ε2 g bis zur 6. Ordnung die Ebenheitsbedingung erf¨ullen und f¨ur jedes Viereck die Bedingung bis zur 2. Ordnung gilt, daß die Diag-onalen und die Eckenspuren tangential an einen in zwei Geradenb¨uschel zerfallenden Kegelschnitt sind, dann ist die Fl¨ache f eine Jonas Fl¨ache, das Parameternetz ist konjugiert und konform bez¨uglich der zweiten Fun-damentalform.
Abstract
The W-property of a discrete surface is a projective property equivalent to the existence of an infinitesimal deformation whose restriction to each cell is an infinitesimal rigid motion. The W-property of a parameter net of a surface is a projective property equivalent to the existence of an in-finitesimal bending for which the coefficient of the mixed derivative of the second fundamental form is preserved to the first order.
Discrete surfaces with plane quadrilaterals have the W-property if and only if for each quadrilateral the diagonals and the traces of the vertices are tangential to a conic. Conjugate parameter nets for which the second fundamental form is conformal have the W-property if and only if the surface is a Jonas surface.
By definition discrete Jonas surfaces are discrete surfaces with planar quadrilaterals such that the diagonals and the traces of the vertices are tangent to a degenerate conic. The following result indicates that there is a relationship between the discrete and the continuous definition.
If for a parametrized surface there is a function g such that the infinites-imal quadrilaterals of size ε of a Taylor expansion of ˆf = f + ε2 g are
planar to the 6th order and for each quadrilateral the criterion that the di-agonals and the traces of the vertices are tangential to a degenerate conic is fulfilled to the 2nd order then the surface f is a Jonas surface and the parameter net is a conjugate net such that the second fundamental form is conformal.
Acknowledgements
I would like to thank my academic supervisor Prof. Dr. Ulrich Pinkall for supporting for two and a half years my double existence as a Wis-senschaftliche Mitarbeiterin at the Fachhochschule Oldenburg and as a guest at the Sonderforschungsbereich 288. I want to thank him for giv-ing me a theme in an area related to construction engineergiv-ing and for his constant support, encouragement and many helpful hints throughout the thesis.
My gratitude also goes to the Fachhochschule Oldenburg and in particular to Prof. Dr. G. Hoffmann for supporting my weekly journeys to Berlin and for his encouragement up to now.
My colleagues Tim Hoffmann, Udo Hertrich-Jeromin and Ulrich Eitner helped this work with fruitful discussions. Katrin Leschke gave many use-ful hints which led to improvements in this dissertation. In the group of the Sfb 288 I found a lively and social athmosphere.
Thanks go out to my friends and family for their frequent e-mails and phonecalls and for cheering me up.
0 Introduction 7
1 The W-Property 11
1.1 Infinitesimal Bending . . . 11
1.2 Discrete W-surfaces in RP3 . . . 15
1.2.1 Discrete FSW - surfaces in R3 . . . 15
1.2.2 W-Graphs and Discrete W-Surfaces in RP3 . . . 19
1.3 Continuous W-Nets in RP3 . . . 26
1.3.1 Continuous SF-nets in R3 . . . 26
1.3.2 Continuous W-nets in RP3 . . . 28
1.4 Examples of Reciprocally Parallel Surfaces and of Transfor-mations of Discrete W-Surfaces . . . 30
2 Conjugate W-Nets 34 2.1 W-Property of a Quadrilateral from a Discrete Net in RP3 34 2.1.1 Non-planar Quadrilateral . . . 36
2.1.2 Planar Quadrilateral . . . 37
2.2 Conjugate W-Nets on Projective Surfaces . . . 42
2.2.1 The Wilczynski Canonical Form . . . 42
2.2.2 Conjugate W-Nets . . . 44
3 Discretization of Jonas Surfaces 49 3.1 Jonas Surfaces . . . 49
3.2 Jonas Surfaces and the W-property . . . 50
3.3 Discretization of Jonas Surfaces . . . 52
3.4 Discrete Jonas Surfaces . . . 73
4 Outlook 80 4.1 Convergence of Discrete Jonas Surfaces . . . 80
4.2 Transformations . . . 81
A Polyhedral Surfaces 84 A.1 Graphs and 2-dim Cell Complexes . . . 84
A.2 Analysis on Polyhedral Surfaces . . . 86
A.3 Dual of a Polyhedral Surface . . . 90
A.4 Discrete Surfaces and Graphs in RP3 . . . . 94
B Properties of W-surfaces and W-nets 95 B.0.1 Discrete W-Surfaces . . . 95 B.0.2 Continuous W-Nets . . . 96
Introduction
In 1926 Felix Klein wrote about Cremona’s work on reciprocal force dia-grams used in graphical statics [Kl], [Cr].
In graphical statics force diagrams were used, before computers, to deter-mine tension and thrust in planar frames made of timber [SyGr]. A frame consists of rigid weightless bars which are connected at joints where they can turn freely. A planar frame with n joints is geometrically determined if it consists of 2n − 3 bars and in this case it is rigid with respect to ex-terior forces acting in the plane of the frame. If one of the bars is left out the frame is not geometrically determined and it is also no longer rigid. If there are more than 2n−3 bars the frame is geometrically overdetermined, in this case bars can be left out without the frame losing its rigidity. For the construction of a part of a planar frame tension and thrust were com-puted by sketching a force diagram which consisted of all the polygons of forces, vertex by vertex.
Cremona in particular showed how for certain planar figures, as for in-stance for a tetrahedron in space projected to a plane, the reciprocal figure can be obtained. The tetrahedron in projective space is mapped with a null system to a second tetrahedron which is also projected to the plane. The two figures in the plane are reciprocally parallel. A null system is a map in projective space where points are mapped to a hyperplane of which they are a part.
Reciprocal force diagrams have also been applied to discrete quadrilateral nets in space. Wunderlich in 1951 considered discrete pseudospheres [Wu]. They can be used to materialize nets of tied strings which can take on an equilibrium state. One example is the fisherman’s net of Radon-Thomas. The equilibrium state exists if and only if upon application of an external force system in equilibrium the inner forces at each vertex add up to zero. Therefore, we have a polygon of forces at each vertex which gives us a second discrete quadrilateral net. The two discete quadrilateral nets have
parallel bars and are called reciprocally parallel. The second net is the force diagram of the first net but the first is also the force diagram of the second . Examples of this duality are Chebyshev nets and Voss surfaces. If the first net is a discrete K-surface the second net can be materialized by strings which are not tied together.
The question arises, which discrete quadrilateral nets or more generally which discrete surfaces in R3admit a reciprocally parallel discrete surface.
A theory for this problem has been developed by Sauer which he described in his first book ’Projektive Liniengeometrie’ in 1937 [Sa37] and which he fully developed in ’Differenzengeometrie’ in 1970 [Sa70].
Reciprocally parallel surfaces exist for discrete quadrilateral nets which al-low a discrete infinitesimal bending, we will call it an infinitesimal folding whose restriction to each cell is an infinitesimal rigid motion. Not every discrete quadrilateral net allows an infinitesimal folding of this kind and therefore not every net has a reciprocally parallel one.
The continuous analogue to infinitesimal foldings which leave the quadri-laterals rigid are infinitesimal bendings of a net on a continuous surface which for the 2nd fundamental form II = Ldx2+ 2M dxdy + N dy2 preserve
M to the 1st order.
In this thesis the question whether such continuous nets determine a class of surfaces is answered. One result is that every surface has a conjugate net with the above property. Only after introducing the additional con-straint that the conjugate nets are conformally parametrized with respect to the 2nd fundamental form a class of surfaces is determined which is the
class of Jonas surfaces.
Discrete quadrilateral nets with planar quadrilaterals are considered to be the discrete analogue of conjugate nets. For discrete planar quadrilateral nets which admit an infinitesimal folding which leaves the quadrilaterals rigid we prove a criterion stated by Sauer [Sa70] which he gave without proof. A planar quadrilateral of a quadrilateral net has the above prop-erty if and only if the two diagonals and the four traces of the vertices of the quadrilateral belong to a curve of 2nd class. A curve of 2nd class is a quadric not of points but of hyperplanes which in the case of RP2 are projective lines. The elements of a curve of 2nd class are tangents of a conic. A trace of a vertex is basically a projective line which is the intersection of the plane spanned by the two edges of the vertex belonging to the quadrilateral and the plane spanned by the two edges of the vertex not belonging to the quadrilateral.
A special case arises if the curve of 2nd class is degenerate as illustrated in
Fig. 1. Here we have two sets of straight lines each set intersecting exactly at one point.
Figure 1:
Diagonals and traces of the vertices belong to a degenerate curve of 2nd class
Discrete Jonas surfaces are defined in this thesis as discrete surfaces with planar quadrilaterals such that the diagonals and the traces of the vertices belong to a degenerate curve of 2nd class. We call the last property the discrete Jonas constraint.
There is a relationship between the discrete and the classical definition. But in order to prove this, we have to do more than to compute with points that come from the Taylor expansion of the continuous Jonas sur-face f .
That is, we have to compensate for the non-planarity of the continu-ous net of the Jonas surface. We introduce a function g and look at an infinitesimal quadrilateral that comes from the Taylor expansion of
ˆ
f (ξ, η) = f (ξ, η) + ε2 g(ξ, η) with respect to ξ and η. Only then can we
show that if the resulting net is planar to a higher order and the discrete Jonas constraint is fulfilled to a higher order, resulting in a partial differ-ential equation for g, that the surface is a Jonas surface.
In Chapter 1 of this thesis we will introduce the theory of reciprocally parallel discrete surfaces in R3. Discrete surfaces are a generalization of discrete quadrilateral nets as they allow the cells to by polygons with any number of vertices. They are defined as maps from abstract poly-hedral surfaces which are explained in the appendix. The existence of a reciprocally parallel surface is shown to be a projective property of the ho-mogeneous coordinates of the discrete surface in RP3 which will be called
as an analogue.
In Chapter 2 a local criterion for a planar quadrilateral of a discrete quadri-lateral net will be proved to be equivalent to the W-property. For the con-tinuous case a hyperbolic partial differential equation for conjugate nets will be derived which is equivalent to the W-property. It shows that on every surface there are conjugate nets which have the W-property. How-ever a conjugate net which is parametrized conformally with respect to the 2nd fundamental form has the W-property if and only if the surface is
a Jonas surface.
Finally, in Chapter 3 we will investigate the Taylor expansion of a general net on a surface along the parameter lines. If there is a function g such that the infinitesimal quadrilateral of ˆf = f +ε2g is planar to the 6thorder
and at the same time the discrete Jonas constraint is fulfilled to the 2nd
order then the surface is a Jonas surface and the net is a conjugate net which is conformally parametrized with respect to the second fundamental form.
Discrete Jonas surfaces are defined and we will compute examples as mini-mal surfaces, Lorentz minimini-mal surfaces, holomorphic maps projected onto the sphere and rotational discrete Jonas surfaces.
The W-Property
It is known that a discrete quadrilateral net in R3 allows an infinitesimal
bending which leaves the quadrilaterals rigid if and only if a reciprocally parallel net exists [Sa70]. We call the first property Euclidean and the second affine. Such nets are called FSW from the German ’fl¨achenstarr wackelig’. In this chapter the concept of FSW will be defined for more gen-eral discrete surfaces in R3 which come from abstract polyhedral surfaces with cells consisting of any number of vertices. A new projective defini-tion of discrete W-surfaces is introduced which is equivalent to the affine definition of FSW-surfaces. We will also show how the initial Euclidean model can be retrieved from the projective W-property. The W-property is important for Chapter 2 where we will investigate planar quadrilaterals. In the case of continuous nets we recall briefly the definition given by Sauer of SF-nets, from the German ’schr¨ankungsfeste Netze’, in R3. It will be
shown to be equivalent to a new projective definition of W-nets in RP3.
In Chapter 2 this definition will be used to develop a partial differential equation as a constraint for conjugate nets to have the W-property.
1.1
Infinitesimal Bending
Let an immersed surface be given, U being an open subset of R2 F : U −→ R3.
A bending of F (x, y) can be described as a process in time where all the single points of the surface are displaced as if they were rigid bodies. For the displacement of rigid bodies see [Sche] or [SyGr].
Mathematically, a bending of F is a 1-parameter family of immersed sur-faces
˜
F : [0, 1] × U −→ R3
such that ˜F (0) = F and ˜F (t) is an immersion with d˜s2 = ds2 for all t.
An infinitesimal bending is given by a vector field Y along F defined on U tangent to a bending ˜F at t = 0
Y (x, y) := ∂ ˜F (t,x,y)∂t |t=0 with dF · dY = 0. (1.1)
For the mathematical description of bending see [Sp]. Infinitesimal bend-ings are also called infinitesimal deformations.
Let Y be an infinitesimal bending of F , then ˜
F (t, x, y) := F (x, y) + t Y (x, y) (1.2)
in a neighbourhood of (x, y) is an immersion for sufficiently small t and d˜s2 = ds2+ t2 dY · dY. Since d˜s2 = ds2+ O(t2) ⇔ d ˜F · d ˜F = dF · dF + O(t2) ⇔ dY · dF = 0 ⇔ ∃ F∗ : U −→ R3 : dY = F∗× dF. (1.3)
the existence of an infinitesimal bending Y is equivalent to the existence and uniqueness of a map F∗ such that (1.3) is fulfilled. F∗ is called the infinitesimal rotation field.
The infinitesimal bending Y can then be expressed by a rotation field F∗ : U −→ R3 and a translation field Y∗
: U −→ R3
Y = Y∗+ F∗× F. (1.4)
Remark 1. Note that the last two equations together are equivalent to dY∗ = F × dF∗.
Remark 2. If Y is a vector field along F tangent to a bending through Euclidean motion, then Y is called a trivial infinitesimal bending.
If the rotation vector field F∗ is constant then the respective infinitesimal bending is trivial and conversely a bending is trivial if the infintesimal bendings are trivial at all times t.
F∗
F∗× F
O
Figure 1.1: The rotational part of the infinitesimal vector field
Remark 3. In the theory of kinematics of the displacement of rigid bodies Y is called the displacement vector or displacement velocity, F∗ is called the rotation vector or rotation velocity and Y∗ is called the translation vector or translation velocity, see Fig. 1.1.
The kinematic model is equivalent to the static model and displacement screws translate into wrenches with F∗ being the force vector and Y∗ be-ing the moment vector.
If several displacement screws or wrenches act on a rigid body in order to have an equilibrium of forces the following conditions must be fulfilled
n X j Fj∗ = 0 n X j Yj∗ = 0, (1.5)
where (F∗, Y∗) describes the screw displacement under an infinitesimal bending of each point of the surface given by F . The screw is a pure rotation if and only if F∗· Y∗ = 0 and it is a pure translation if and only
if F∗ = 0.
Remark 4. Note that (1.4) is equivalent to
Y∗ = Y + F × F∗ (1.6)
and (1.3) is equivalent to
dY∗ = F × dF∗. (1.7)
In particular, we have dY∗· dF∗ = 0. (F, Y ) and (F∗, Y∗) are symmetric
displacement of each point under an infinitesimal bending of the surface given by F , (F, Y ) describes the screw displacement under an infinitesimal bending of each point of the surface given by F∗.
For Y as well as Y∗ and F∗ the following compatibility conditions must be fulfilled Yxy = Yyx Fxy∗ = F ∗ yx Y ∗ xy = Y ∗ yx. We have Yx∗ = F × Fx∗ and Yy∗ = F × Fy∗: Yxy∗ = Yyx∗ ⇐⇒ Fx∗× Fy = Fy∗× Fx ⇐⇒ ∃ %, σ, δ : U ⊂ R2 −→ R Fx∗ = % Fy+ σ Fx −Fy∗ = δ Fx+ σ Fy. (1.8) Fxy∗ = Fyx∗ ⇔ %yFy+ %Fyy+ σyFx+ 2σFxy + δxFx+ δFxx+ σxFy = 0 (1.9) taking the cross product of the equation with the normal vector
%N + 2σM + δL = 0 (1.10)
follows as a necessary condition, with L, M, N the usual coefficients of II = Ldx2+ 2M dxdy + N dx2.
Remark 5. We can conclude directly that for asymptotic parameters a necessary condition for an infinitesimal bending to exist is σ = 0 and therefore
Fx∗ = %Fy − Fy∗ = δFx. (1.11)
Remark 6. For conjugate parameters the necessary condition is
%N = −Lδ. (1.12)
The problem of finding all infinitesimal bendings of an immersion is treated in [Sa70], [Sp], [Ei09] and [Bi]. It can be solved by establishing a partial differential equation for the vector field Y which is the infinitesimal bend-ing. This partial differential equation is a linear homogeneous equation of 2nd order and it is hyperbolic in the case of negative Gaussian curvature
and elliptic for positive Gaussian curvature. The solutions of the par-tial differenpar-tial equation, asymptotic directions giving the characteristics, might only lead to trivial infinitesimal bendings of F . In this case the in-finitesimal bendings are tangent to Euclidean motions and the immersion is called infinitesimally rigid.
1.2
Discrete W-surfaces in RP
31.2.1 Discrete FSW - surfaces in R3
Let a discrete surface in R3 be given by a simply connected polyhedral
surface F = (V, C) with a locally injective map F : V −→ R3, see
Ap-pendix p.94. A priori we only have cells and vertices in R3. The edges,
pairs of vertices, shall be realized by segments of straight lines connecting the vertices so that the cells become polygons in R3.
FSW, from the German ’fl¨achenstarr wackelig’ [Sa70], means that there exists an infinitesimal deformation, we will call it an infinitesimal folding, which leaves the cells of the discrete surface rigid, that is whose restriction to each cell is an infinitesimal rigid motion. If the discrete surface is FSW it can be used as a mathematical model for the construction of a physical model with rigid cells made for instance from metal. FSW also means that this physical model would not be rigid as a whole.
Remark 7. Sauer only considers cells which are quadrilaterals and in this case the polyhedral surface would be defined by cells with vertices in Z2.
Such a polyhedral surface together with a locally injective map F to R3
will be called a discrete quadrilateral net.
We consider now an infinitesimal folding of a discrete surface, the discrete analogue of a bending, where F = (V, C) is an abstract polyhedral surface, see p.90, with a locally injective map
F : V −→ R3.
A deformation of a discrete surface, we call it a folding, is as in the con-tinuous case a 1-parameter family of locally injective maps
˜
F : [0, 1] × V −→ R3
such that ˜F0 = F and ˜F is a locally injective map for all t such that for
the lengths l of the edges of F we have ˜l = l.
An infinitesimal folding is again given by a vector field Y : V −→ R3 along F with Y tangent to a folding ˜F at t = 0
Yv = ∂ ˜∂tF|t=0 such that the edges of F and of Y are orthogonal. (1.13)
Let Y be an infinitesimal folding of F , then ˜
has the property that in a neighbourhood of v ˜F is a locally injective map for sufficiently small t with the property that the lengths of the edges l are preserved to the 1st order: ˜l2 = l2+ O(t2).
The vector field Y can again be expressed by a rotation field F∗ and a translation field Y∗, as in (1.4). We will use it directly to define discrete FSW-surfaces.
Definition 1. Let a discrete surface be given by a polyhedral surface F = (V, C) and a locally injective map F : V −→ R3. F together with
F is called a discrete FSW-surface if on the dual polyhedral surface F∗ = (V∗, C∗) there are two maps F∗ : V∗
−→ R3 and Y∗ : V∗
−→ R3,
F∗ locally injective, such that
for all c and for all vertices v of c
Yv = Yc∗+ F ∗ c × Fv.
(1.15) Y∗ is the translation field whereas F∗ is the rotation field.
For the concept of the dual surface of F = (V, C) see the Appendix. Note that the vector field Y , the displacement vector field, in the case of FSW-surfaces is a special one. In general the displacement field also deforms the cells of the discrete surface.
We will use a local notation for FSW as outlined in the following figure:
c1 c2 Fv Fv2 2 Fvm2−1 2 Fv2 Fv3 2 Fv1 cn c3 Fvn Fvn−1 Fv3
Figure 1.2: Local notation with respect to Fv for a discrete surface
ci = [F∗(ϕ(~ci))] see p.85
= [(Fv0
i, Fv1i, ..., Fvmii , →)] Fv0
FSW means that for all cells we have ∀ i ∈ {1, .., n} ∀ j ∈ {0, .., mi} Yvj i = Y ∗ ci+ F ∗ ci× Fvij (1.16)
that is, for each cell ci all its vertices are displaced by the same screw
(Fc∗ i, Y
∗ ci).
From (1.16) we see that the edges of F and Y are orthogonal Yvj+1 i − Yv j i = F ∗ ci × (Fvij+1− Fvji). Likewise the edges of F∗ and Y∗ are orthogonal.
The definition of a discrete FSW surface in R3 is so far purely Euclidean. But we will see now that it is equivalent to an affine definition.
Definition 2. Let a discrete surface be given by a simply connected poly-hedral surface F = (V, C) with its dual F∗ = (V∗, C∗) and a locally injective map F : V −→ R3. Then a locally injective map F∗ : V∗
−→ R3
is called reciprocally parallel to F if
for all cells ci, cj which have a common edge {vi, vj} we have
(Fc∗j− Fc∗i) × (Fvj− Fvi) = 0.
(1.17)
Proposition 1. Let a discrete surface be given by a simply connected polyhedral surface F = (V, C) and its dual F∗ = (V∗, C∗) and a locally injective map F : V −→ R3:
F is FSW ⇐⇒ ∃ F∗ : V∗ −→ R3reciprocally parallel to F. (1.18) For the proof we will use a slightly different notation for reciprocally par-allel which is illustrated in the figure below.
Fv2 Fvn Fvn−1 Fc∗n Fc∗3 Fc∗2 Fc∗1 Fv1 c1 cn v c2 c3 Fv Fv3
Figure 1.3: Reciprocally parallel discrete surfaces
In this local notation, reciprocally parallel is expressed by ∀ v ∀ ci : (Fc∗i+1− F
∗
ci) × (Fvi− Fv) = 0 (1.19)
i ∈ {1, ..., n}.
Proof. ⇒: According to the definition of FSW this implies that ∀i ∈ {1, ..., n} in Fig. 1.2 we have
Yvi = Y ∗ ci+ F ∗ ci× Fvi Yvi = Y ∗ ci+1+ F ∗ ci+1× Fvi ⇒ Yc∗i+1− Yc∗i = Fvi× (F ∗ ci+1− F ∗ ci) and likewise Yv = Yc∗i+ F ∗ ci× Fv Yv = Yc∗i+1+ F ∗ ci+1× Fv ⇒ Yc∗ i+1 − Y ∗ ci = Fv× (F ∗ ci+1− F ∗ ci). Therefore, we have (Fc∗i+1− Fc∗i) × (Fvi− Fv) = 0.
⇐: Now let F∗ : V∗
−→ R3 be reciprocally parallel to F .
Let us define Y∗ and Y through
Yc∗i+1 − Yc∗i := Fv× (Fc∗i+1− F ∗ ci), (1.20) Yv := Yc∗i + F ∗ ci × Fv (1.21) i ∈ {1, ..., n}.
For Y∗ we choose an initial value Yc∗
i, the remaining values are computed recursively. (Fc∗i+1− Fc∗i) × (Fvi− Fv) = 0 ⇔ Fvi× (F ∗ ci+1− F ∗ ci) = Fv × (F ∗ ci+1 − F ∗ ci) ⇒ Yc∗i+1− Yc∗i = Fv × (Fc∗i+1 − F ∗ ci) is well defined. Yc∗i+1− Yc∗i = Fv×(Fc∗i+1− F ∗ ci) ⇔ Yc∗i+1 + Fc∗i+1 × Fv = Yc∗i+ F ∗ ci× Fv ⇒ Yv = Yc∗i+ F ∗ ci× Fv is well defined i ∈ {1, ..., n}.
1.2.2 W-Graphs and Discrete W-Surfaces in RP3
In the last section we showed that the Euclidean definition of a discrete FSW-surface in R3 is equivalent to an affine definition, namely the
exis-tence of a reciprocally parallel discrete surface.
Now we will see that the definition of a discrete FSW-surface in R3 can be expressed by a property which is invariant under projective transfor-mations. Mechanical equilibrium on a discrete surface holds if and only if the forces in every vertex of the discrete surface add up to zero. In R3
this is expressed by a homogeneous equation in the vectors of the edges. This property can be expressed in RP3 not only for discrete surfaces but also for discrete graphs. The only difference will be that for a discrete W-surface there will be a reciprocally parallel W-surface.
Definition 3. A W-graph in RP3 is given by a directed abstract graph
~
G = (V, ~E) together with a locally injective map
f : V −→ RP3 (1.22)
such that there exists a map
τ : ~E −→ R\{0} with τ (−~e) = τ (−→e ) defining a 1-form
ω : E~ −→ R6
(vi, vj) 7−→ τ (vi, vj) fvi∧ fvj, such that ∀v ∈ V ω has the W-property
n X i=1 ω(v, vi) = 0. (1.23) fvn fvn−1 fv2 fv1 fv3 fv c2 c1 c3 cn
Figure 1.4: Local notation with respect to fv for a discrete surface in RP3
Remark 8. p = x ∧ y denote the Pluecker coordinates in RP5 of the projective line determined by the points x, y in RP3:
p1 = x1y4− x4y1 p2 = x2y4− x4y2 p3 = x3y4− x4y3 p4 = x2y3− x3y2 p5 = x3y1− x1y3 p6 = x1y2− x2y1 If f = (F, 1) then fvi ∧ fvj = Fvj− Fvi Fvi× Fvj .
Remark 9. There is a 1-to-1 correspondence between the projective lines in RP3 given by fvi∧ fvj and the elements of RP
5 which fulfill
< p, p >= 0, (1.24)
with < p, q > = p1q4+ p2q5+ p3q6+ p4q1+ p5q2+ p6q3 being the Pluecker
sum of two elements of the Pluecker quadric is an element of the Pluecker quadric if and only if the two lines intersect. Therefore, in the W-property Pn
i=1τ (v, vi) fv ∧ fvi defines a projective line in RP
3.
Remark 10. In the following there will be neither a notational distinction between the elements of RP3 and its representatives in R4 nor between the elements of RP5 and its representatives in R6.
Remark 11. ω is well defined since a rescaling of the surface does not change ω: let ˜f = ρ(v)f (v) then
˜
τ (v1, v2) :=
τ (v1, v2)
ρ(v1)ρ(v2)
=⇒ ω = ω.˜
Definition 4. A polyhedral surface F = (V, C) together with a respec-tive locally injecrespec-tive map f is called a discrete W-surface in RP3 if its directed 1-skeleton ~G = (V, ~E) with f and a respective map τ is a W-graph.
We learnt that discrete surfaces in R3 are FSW if and only if they have a reciprocally parallel discrete surface in R3. Now we will show that having a reciprocally parallel discrete surface in R3 means that the surface is a
discrete W-surface in RP3. The reverse statement is also true as we will
see later.
Proposition 2. Let F = (V, C) be a simply connected polyhedral surface with an orientation and F∗ = (V∗, C∗) be its dual. If F : V −→ R3
has a reciprocally parallel discrete surface F∗ : V −→ R3, then the map
f : V −→ RP3 such that f = (F, 1) is a discrete W-surface in RP3. Proof. We work with the map J as in Definition 26 on p.92. It is well defined since we assumed F to have an orientation.
J (c1, c2)
c2
v1
c1
v2
Figure 1.5: The map J which maps edges of the polyhedral surface F∗ to edges of F
F : V −→ R3 and F∗ : V∗
−→ R3 are locally injective and reciprocally
parallel as defined in the local notation given in (1.19) ∀ v ∀ ci : (Fc∗i+1 − F ∗ ci) × (Fvi− Fv) = 0 i ∈ {1, ..., n}. Therefore, ∃ ki 6= 0 : (Fc∗i+1− F ∗ ci) = ki(Fvi − Fv).
With dF (vi, vj) = Fvj − Fvi (see Definition 17 on p.87) and
dF∗(ci, cj) = Fc∗j− F
∗
ci we define the function τ : ~E −→ R\{0} by
dF∗(ci, cj) = τ (vi, vj) dF (J (ci, cj)) = τ (vi, vj) dF (vi, vj). (1.25) Then τ (vi, vj) = τ (vj, vi) since dF∗(cj, ci) = τ (vj, vi) dF (J (cj, ci)) = τ (vj, vi) dF (vj, vi) ⇐⇒ dF∗(ci, cj) = τ (vj, vi) dF (vi, vj). (1.26) Furthermore, n X i=1 τ (v, vi) (Fvi− Fv) = n X i=1 (Fc∗i+1− Fc∗i) = 0 ⇒ n X i=1 τ (v, vi) (Fv× Fvi) = Fv× ( n X i=1 τ (v, vi) (Fvi− Fv)) = 0. (1.27)
Now we define a discrete surface in RP3 by
f : V −→ RP3 with f = (F, 1) therefore, fvi∧ fvj = Fvj − Fvi Fvi× Fvj .
The map τ defines a 1-form ω(vi, vj) = τ (vi, vj)fvi ∧ fvj which has the
W-property: n X i=1 τ (v, vi)fv∧ fvi = n X i=1 τ (v, vi) Fvi− Fv Fv× Fvi = 0. (1.28)
The last proposition showed that if a discrete surface F in R3 has a
recip-rocally parallel surface F∗, then F1 is a discrete W-surface in RP3. Now
we will see that the affine surface in R3 coming from a discrete W-surface
in RP3 has a reciprocally parallel surface.
We will now show how starting with a discrete W-surface we retrieve the Euclidean construction. We integrate a 1-form ∗ω = ω(J ) with ω the R6-valued co-closed 1-form of the discrete W-surface. For the map J see again Fig. 1.5 on p.21. Through integration of ∗ω we get the displacement screw (F∗, Y∗) from the Euclidean definition of FSW that defines the infinitesimal folding of F . We then integrate on F∗ a 1-form, which we will explain in the proposition and thus we will receive the displacement screw (F, Y ) that defines the infinitesimal folding of F∗.
Proposition 3. Let F = (V, C) be a simply connected polyhedral surface with an orientation and f : V −→ RP3 with f = ρ ˜f and ˜f = (F, 1) be a discrete W-surface
=⇒ there are two maps
g =F Y : V −→ R6 g∗ =F ∗ Y∗ : V∗ −→ R6
such that g∗ is given through integration of ∗ω := ω(J ) and g is given through integration of ω∗(−J∗) with ω∗ := F∗dF ◦J×dF ◦J and J∗◦ J = − id. The map f∗ : V∗ −→ RP3 with f∗ = (F∗, 1) is a discrete W-surface such
that dY (vi, vj) = Fc∗i× dF (vi, vj) (1.29) dY∗(ci, cj) = Fvi× dF ∗ (ci, cj) (1.30) (Fvj− Fvi) × (F ∗ cj− F ∗ ci) = 0. (1.31)
In particular this means that F, F∗ are two reciprocally parallel discrete surfaces in R3.
Proof. F has an orientation which defines a map J and a correspond-ing map J∗ such that J∗ ◦ J = −id. Since f is a discrete W-surface there is a map τ 6= 0 with τ (vi, vj) = τ (vj, vi) defining a co-closed 1-form
Furthermore, on F∗ we have a 1-form ∗ω(ci, ci+1) = ω(J (ci, ci+1)) =
ω(v, vi) (see Appendix p.92 Definition 26). For the notation refer to Fig.
1.2 on p.16. ∗ω is an R6-valued closed 1-form since f has the W-property:
d ∗ ω(D(v)) = d ∗ ω(c∗) = d ∗ ω((c1, c2, c3, ..., cn, →)) = n X i=1 ∗ω(ci, ci+1) = n X i=1 ω(J (ci, ci+1)) = n X i=1 ω(v, vi) = n X i=1 τ (v, vi)fv ∧ fvi = 0.
Therefore, there is a map g∗ : V∗ 7−→ R6 such that
dg∗ = ∗ω
(see Appendix equation(A.1) on p.89). In addition, we define F∗ and Y∗ by
F∗
Y∗
:= g∗
with F∗ : V∗ 7−→ R3 and Y∗ : V∗ 7−→ R3 and f∗ : V∗ −→ RP3, such that f∗ :=F
∗
1
. Now let us show that f∗ is a W-surface.
We define a 1-form ω∗ on F∗ with J again as in Definition 26 on p.92 and f = ρ F1 ω∗(ci, cj) : = dF (J (ci, cj)) F∗ ci × dF (J(ci, cj)) . (1.32)
We will have to rescale the surface ˜f = 1ρ f with f =˜ F1 and ˜ τ (vi, vj) := ρ(vi)ρ(vj) τ (vi, vj) such that 0 = n X i=1 τ (v, vi) fv∧ fvi = n X i=1 ˜ τ (v, vi) ˜fv ∧ ˜fvi = n X i=1 ˜ τ (v, vi) Fvi− Fv Fv × Fvi . (1.33)
We will make a technical computation here in order to prove Fc∗i − F∗ c =
˜
τ (vi, vi+1)(Fvi+1−Fvi) refering to Fig. 1.6 and using ˜f =
F
1 and dg
F∗ ci− F ∗ c Y∗ ci− Y ∗ c =dF ∗(c, c i) dY∗(c, c i) = dg∗(c, ci) = ∗ω(c, ci) = ω(J (c, ci)) = ω(vi, vi+1) = τ (vi, vi+1) fvi ∧ fvi+1 = ˜τ (vi, vi+1) ˜fvi ∧ ˜fvi+1 = ˜τ (vi, vi+1) Fvi+1− Fvi Fvi × Fvi+1 = ˜ τ (vi, vi+1) (Fvi+1 − Fvi) Fvi× ˜τ (vi, vi+1) (Fvi+1− Fvi) ⇒ Fc∗i− Fc∗ = ˜τ (vi, vi+1) (Fvi+1− Fvi) Yc∗i− Yc∗ = Fvi × (F ∗ ci − F ∗ c). (1.34)
We see that (1.30) and (1.31) hold. This implies that
ω∗(ci, cj) = dF (J (ci, cj)) F∗ ci × dF (J(ci, cj)) = (Fvj − Fvi) F∗ ci× (Fvj− Fvi) = 1 ˜ τ (vi, vj) F∗ cj − F ∗ ci F∗ ci × F ∗ cj .
ω∗ is then a map of the required form
ω∗ : ~E∗ −→ R6 (ci, cj) 7−→ 1 ˜ τ (J (ci, cj)) fc∗ i ∧ f ∗ cj (1.35)
and ω∗has the W-property with the notation as in Fig. 1.6, since ∀v∗ ∈ V∗ m X i=1 ω∗(v∗, vi∗) = m X i=1 ω∗(c, ci) m X i=1 dF (J (c, ci)) F∗ ci × dF (J(c,ci)) = m X i=1 (Fvi+1− Fvi) F∗ ci × (Fvi+1− Fvi) = ddF (c) F∗ ci× ddF (c) = 0 (1.36)
having used vm+1 = v1 and J (c, ci) = (vi, vi+1).
Therefore f∗ is a W-surface on F∗.
Finally, we define g. ω∗ can be pulled back to F again through (−J∗) with J∗ ◦ J = − id (see Appendix proposition 18 on p.93). ω∗(−J∗) is
v2 v1 v3 Fc∗3 Fc∗ m−1 Fc∗m Fc∗1 Fc∗ Fc∗2 vm
Figure 1.6: Local notation with respect to Fc∗ for the dual discrete surface
obviously a closed 1-form on F (see (1.32) on p.24). Therefore, there is a map g : V −→ R6 with ω∗(−J∗) = dg = dF
dY where dF comes from our
original W-surface f = ρ F1. Y is also of the form Y : V −→ R3 and we
have (1.29) Fvj− Fvi = 1 ˜ τ (J (ci, cj)) (Fc∗j − Fc∗i) Yvj− Yvi = F ∗ ci× 1 ˜ τ (J (ci, cj)) (Fc∗j − Fc∗i) = Fc∗i× (Fvj − Fvi).
1.3
Continuous W-Nets in RP
3 1.3.1 Continuous SF-nets in R3Parameter SF-nets, from the German ’schr¨ankungsfest’, are the continu-ous counterpart of discrete FSW quadrilateral nets. They were also in-troduced by Sauer [Sa37]. In the continuous and discrete case these nets share the property that the volume V of the tetrahedron of a quadrilateral or infinitesimal quadrilateral respectively is preserved to the 1storder with
an infinitesimal bending.
We will define continuous W-nets in RP3 and show that they are
contin-uous SF-nets and vice versa.
Definition 5. Let F : U −→ R3 be an immersion with U ⊆ R2an open set and with parameters F (x, y). Let F be infinitesimally bendable with the infinitesimal bending Y of F such that for ˜F = F + tY the two 2nd
funda-mental forms II = Ldx2 + 2M dxdy + N dy2
and fII = ˜Ldx2+ 2 ˜M dxdy + ˜N dy2 fulfill
˜
M = M + O(t2).
Definition 6. Let F : U −→ R3 be an immersion of an open set U ⊆ R2 and F∗ : U −→R3 with Fx∗ = % Fy −F∗ y = δ Fx (1.37)
%, δ : U −→ R\{0}, then F∗ is called reciprocally parallel to F. According to (1.8) on p.14 a necessary condition for an infinitesimal bend-ing Y = Y∗+ F∗× F to exist is that
∃ %, σ, δ : U ⊂ R2
−→ R
Fx∗ = % Fy+ σ Fx
−Fy∗ = δ Fx+ σ Fy.
(1.38) Note that for σ = 0 the immersion F∗ is reciprocally parallel to F .
Proposition 4. Let U ⊂ R2 be a simply connected open set
F : U −→R3 is a continuous SF-net ⇐⇒
there exists F∗ : U −→R3 reciprocally parallel to F.
(1.39)
Proof. ⇒: Let F be a continuous SF-net then there is in particular an infinitesimal bending of F and Y = Y∗+ F∗× F .
Using dY = F∗× dF : ˜ M − M = O(t2) ⇔ < ˜Fx, ˜Fy, ˜Fxy > − < Fx, Fy, Fxy > = O(t2) ⇔ < Fx+ tYx, Fy+ tYy, Fxy + tYxy > − < Fx, Fy, Fxy > = O(t2) ⇔ < Yx, Fy, Fxy > + < Fx, Yy, Fxy > + < Fx, Fy, Yxy > = 0 ⇔ < F∗× Fx, Fy, Fxy > + < Fx, F∗× Fy, Fxy > + < Fx, Fy, F∗× Fxy > + < Fx, Fy, Fy∗× Fx> = 0 ⇔ < Fx, Fy, Fy∗× Fx> = 0 ⇔ < Fx, Fy, (−δFx− σFy) × Fx> = 0 ⇔ σ = 0.
Therefore,
Fx∗ = % Fy − Fy∗ = δ Fx
⇐⇒
F∗is reciprocally parallel to F.
(1.40)
⇐: Now let F∗ be reciprocally parallel to F with
Fx∗ = % Fy − Fy∗ = δ Fx
then F∗ defines an infinitesimal bending of F through
Yx∗ := F × Fx∗ Yy∗ := F × Fy∗, (1.41)
since U is simply connected
Yxy∗ − Yyx∗ = Fy× Fx∗− Fx× Fy∗ = Fy × (% Fy) + Fx× (δ Fx) = 0. Then for Y := Y∗+ F∗ × F Yxy − Yyx= Yxy∗ − Y ∗ yx+ (F ∗× F ) xy − (F∗ × F )yx = 0.
According to (1.41) dY = F∗× dF and therefore dY · dF = 0 which means that Y is an infinitesimal bending of F .
1.3.2 Continuous W-nets in RP3
So far for immersed surfaces F : U −→ R3 we have that the property of
being an SF-net is equivalent to the existence of a reciprocally parallel net F∗. Now we want to define a projective notion of the SF-property.
Definition 7. A continuous W-net is given by an immersion on a sim-ply connected domain U ⊆ R2
f : U −→ RP3, such that there exists a map τ
τ : U −→ R2\{0} defining a co-closed 1-form
ω : U −→ R6
(x, y) 7−→ ( τ1f ∧ fx dx + τ2f ∧ fy dy )(x, y)
that is d ∗ ω = 0 where ∗ω = τ1f ∧ fx ∗ dx + τ2f ∧ fy ∗ dy = τ2f ∧ fy dx − τ1f ∧ fx dy (1.43)
using ∗dx = −dy and ∗dy = dx.
Remark 12. If f = (F, 1), then f ∧ fx denote the Pluecker coordinates
in RP5 of the tangent of f along x at p ∈ U : f ∧ fx = (F + Fx) − F F × (F + Fx) = Fx F × Fx . (1.44)
W-nets are well defined in RP3 since for a change of scaling ˜f = ρ f we
can define a function ˜τ := ρ12τ which results in ˜ω = ω.
The main result is that continuous SF-nets are W-nets in RP3 and vice
versa.
Proposition 5. Let F : U ⊆ R2 −→ R3 be an immersion, with U being a
simply connected domain then
F has a reciprocally parallel net ⇐⇒
f : U ⊆ R2 −→ RP3 with f = ρ (F, 1) is a W-net.
(1.45)
Proof. ⇐: Let f = ρ (F, 1) be a W-net and let ˜f = (F, 1), accordingly,
there is a function τ : U −→ R2\{0} such that
0 = d( τ2 f ∧ ˜˜ fy dx − τ1 f ∧ ˜˜ fx dy) = d( τ2 F ×FFy
y dx − τ1
Fy
F ×Fx dy). Therefore, d( τ2Fydx − τ1Fxdy) = 0 and since U is a simply connected
do-main there is a map F∗ : U −→ R3 such that F∗
x = τ2Fy and −Fy∗ = τ1Fx
which means that F∗ is reciprocally parallel to F .
⇒: Let F : U −→ R3 and F∗
: U −→ R3be reciprocally parallel therefore,
functions %, δ exist such that Fx∗ = %Fy and −Fy∗ = δFx. Since ddF∗ = 0
we get d(% Fydx − δ Fxdy) = 0.
Furthermore, define Yx∗ := F ×Fx∗ = %F ×Fyand Yy∗ := F ×F ∗
with f := F1 d( % Fy F × Fy dx − δ Fx F × Fx dy ) = ddF ∗ Y∗ = 0 ⇒ for ω := δ f ∧ fx dx + % f ∧ fy dy we have d ∗ ω = 0. (1.46)
1.4
Examples of Reciprocally Parallel Surfaces and
of Transformations of Discrete W-Surfaces
Discrete Examples of Reciprocally Parallel W-SurfacesReciprocally parallel discrete surfaces in the diagram on the following page are indicated by arrows. In particular, the discrete surfaces with planar vertices have more than one reciprocally parallel surface, which are only determined after a choice of two boundary polygons.
Discrete minimal surfaces are generated by the discrete analogue of the Weierstrass representation [BoPi94]. Let g : Z2 −→ C be a discrete
holo-morphic map, that is the cross-ratio of every elementary quadrilateral equals −1, then a discrete minimal surface is given by the discrete Weier-strass formula: Fn+1,m− Fn,m = 1 2Re 1 gn+1,m− gn,m (1 − gn+1,mgn,m, i(1 + gn+1,mgn,m), gn+1,m+ gn,m) Fn,m+1− Fn,m = − 1 2Re 1 gn,m+1− gn,m (1 − gn,m+1gn,m, i(1 + gn,m+1gn,m), gn,m+1+ gn,m) (1.47)
Nets on the sphere are generated by the stereograpic projection of a map
g : Z2 −→
C onto the sphere S2 given by
π−1(g) = 1
1 + |g|2(2 Reg, 2 Img, 1 − |g| 2).
stereographic projection of discrete identity map
having planar quadrilaterals -& Enneper surface on Z2 having planar vertices %
discrete Enneper surface .
having
planar quadrilaterals
stereographic projection of discrete exponential map
having planar quadrilaterals -& helicoid surface on Z2 having planar vertices % discrete catenoid . having planar quadrilaterals
Example for Transformations of Discrete W-Surfaces
The W-property as well as the discrete W-property are projectively invari-ant. A projective transformation of a surface does not yield a new W-surface in the projective sense. However the reciprocally parallel W-surface comes from a non-projective construction, using parallelity, and therefore it does not commute with a projective transformation.
In Fig. 1.7 we see a hyperboloid, coming from the parametrization of a Clifford torus in S3, which is then projectively transformed. On the right
the respective reciprocally parallel surfaces are shown, the first one being a Lorentz minimal surface.
Fm,n = cos(2π(m + n)/N )/(cos(2π(m − n)/N )) sin(2π(m + n)/N )/(cos(2π(m − n)/N )) sin(2π(m − n)/N )/(cos(2π(m − n)/N )) Fm,n = cos(2π(m + n)/N ) cos(2π(m − n)/N ) sin(2π(m + n)/N ) cos(2π(m − n)/N ) 2π(n − m)/N
Figure 1.7: Hyperboloid
Figure 1.8: The respective re-ciprocally parallel surface
Figure 1.9: Hyperboloid after projective transformation
Figure 1.10: The respective re-ciprocally parallel surface
Conjugate W-Nets
Conjugate nets play an important role in projective differential geometry as line congruences are tangential to conjugate nets.
Planar quadrilateral nets are considered to be a discretization of conjugate nets [Do] and in the discrete case we will prove a criterion for a planar quadrilateral of a quadrilateral net to have the W-property.
Jonas nets in particular are conjugate nets and as we want to discretize them later on we want to develop in the second part of this chapter a partial differential equation for a conjugate net which is equivalent to the W-property.
2.1
W-Property of a Quadrilateral from a Discrete
Net in RP
3It is useful to develop a criterion for a quadrilateral to have the W-property. The criteria split into two cases, the non-planar and the planar quadrilateral. The two criteria were both stated by Sauer [Sa70], however, the case of planar quadrilaterals without a proof.
Let F = (V, C) be a simply connected polyhedral surface with V ⊂ Z2
and a locally injective map
f : V −→ RP3 (2.1)
with f = ρ (F, 1).
The following notation will be useful for the description of a quadrilateral: the vertices are given by fp, fq, fs, ft, where fq∧ fs is the projective line
through fqand fsand fq∧fs∧fpthe plane spanned by fq, fsand fpas
illus-trated in
Fig. 2.1.
f
p1f
q1f
s2f
t1f
t2f
p2f
s1f
qf
pf
tf
sf
q2Figure 2.1: An inner quadrilateral
Definition 8. Let f be a quadrilateral net in RP3. The trace P q of
the vertex fq of the quadrilateral is the intersecting line of the plane
spanned by the two edges of the quadrilateral intersecting in fq with the
plane spanned by the two edges of the neighbouring quadrilateral inter-secting in fq. In the case that all edges of the vertex lie in a projective
plane any projective line through fq is a trace of that vertex.
fq2 fq3 ft fq1 fq
P
q fs fpFigure 2.2: The trace of a vertex
We assume that the edges of the vertex fq do not lie in a plane then
Pq = fq∧ fs∧ fp ∩ fq∧ fq1 ∧ fq2. The trace of a vertex of a quadrilateral is a projective line.
Pq is well defined, since fq ∧ fs, fq∧ fp and fq∧ fq1, fq∧ fq2 are pairwise linearly independent.
2.1.1 Non-planar Quadrilateral
Proposition 6. An inner non-planar quadrilateral of a quadrilateral net in RP3 has the W-property if and only if the traces of the vertices Pp, Pq, Ps, Pt
are linearly dependent. We also say that Pp, Pq, Ps, Pt are in hyperboloidic
position.
Proof. See [Sa70].
Proposition 7. A quadrilateral of a quadrilateral net in RP3 has the W-property if for each vertex all edges of the vertex lie in a projective plane.
Proof. We assume that the quadrilaterals are non-planar. For instance we look at vertex fp as in Fig. 2.1 on p.35 then any four of the vertices
ft, fp, fq, fp1 and fp2 are linearly dependent because they lie in a projective plane.
Therefore, there are λpq, λpt, λpp1, λpp2 such that
fp∧ (λpqfq+ λptft+ λpp1fp1 + λpp2fp2) = 0 ⇔ µp(λpqfp∧ fq+ λptfp∧ ft+ λpp1fp ∧ fp1+ λpp2fp∧ fp2) = 0
µp ∈ R \ 0.
(2.2)
For the other three vertices fq, fs and ft we get respective equations with
some real non-vanishing numbers µq, µs and µt such that we have four
equations 0 = µp(λptfp∧ ft+ λpqfp ∧ fq+ λpp1fp∧ fp1 + λpp2fp∧ fp2) 0 = µq(λqpfq∧ fp+ λqsfq∧ fs+ λqq1fq∧ fq1 + λqq2fq∧ fq2) 0 = µs(λsqfs∧ fq+ λstfs∧ ft+ λss1fs∧ fs1 + λss2fs∧ fs2) 0 = µt(λtsft∧ fs+ λtpft∧ fp+ λtt1ft∧ ft1 + λtt2ft∧ ft2). (2.3)
The last four equations are the closing conditions for the four vertices fp, fq, fs and ft as in (1.23) on p.20 if we can define a function τ on the
edges in the closing conditions such that for τ (i, j) =: τij we have τij = τji.
The following system
0 = µqλqp− µpλpq
0 = µqλqs− µsλsq
0 = µsλst− µtλts
0 = µtλtp− µpλpt
can be solved for instance by choosing λpq and µp, µq and µs and defining τpq := µqλqp = µpλpq = τqp τqs:= µqλqs= µsλsq = τsq τst := µsλst = µtλts = τts τtp:= µtλtp = µpλpt = τpt (2.5)
therefore the W-property holds.
However in particular Pp, Pq, Ps, Pt are linearly dependent, which means
they are in hyperboloidic position, since for the coefficients µ we have µpPp+ µqPq+ µsPs+ µtPt= 0
writing for instance the traces of the vertices as
Pp = λptfp∧ ft+ λpqfp∧ fq= −λpp1fp∧ fp1 − λpp2fp ∧ fp2.
2.1.2 Planar Quadrilateral
A curve of 2nd class in RP2 is the set of projective lines for which a given non-trivial quadratic form in R3 vanishes. They are tangents of a conic, since it is a quadric of projective lines in RP2. A special case arises if the
curve of 2nd class is degenerate then we have two sets of straight lines each
set intersecting exactly at one point.
Figure 2.3: The diagonals and the traces of the vertices belonging to a curve of 2nd class
Proposition 8. An inner planar quadrilateral of a quadrilateral net in RP3 has the W-property if and only if the traces of the vertices Pp, Pq, Ps, Pt
In RP2 six lines belong to a curve of 2nd class if and only if there is a
certain point called the point of Brianchon [Scha]. The exact construction of a point of Brianchon for a planar quadrilateral, its diagonals and its traces of the vertices is shown in Fig. 2.4.
˜
˜
f
˜
f
ˆ
f
l4 D1 l6 l2 point of Brianchon ft D2 l5 l3 l1 Ps fp fq fs Pp Pt PqFigure 2.4: The point of Brianchon for six lines lying on a curve of 2nd class
We choose a suitable order for the six lines and consider the three lines (l1∩ l2) ∧ (l4 ∩ l5) (l2∩ l3) ∧ (l5∩ l6) (l3∩ l4) ∧ (l6∩ l1).
These three lines should intersect in one point which is the point of Bri-anchon.
Now we choose the order in such a way that l1∩ l2 = fp l4∩ l5 = ft
l3∩ l4 = fq l6∩ l1 = fs.
We call ˆf the intersection of ft∧ fp and fs∧ fq. Also we call ˜f = l2∩ l3 =
Pp∩ Pq and f = l˜˜ 5∩ l6 = Ps∩ Pt.
The six respective lines belong to a curve of 2nd class if and only if the line ˜f ∧f contains ˆ˜˜ f which in this case is called the point of Brianchon. Equivalently, we can say that the six lines belong to a curve of 2nd class
Proof. Let the traces be given by
Pp = λptfp∧ ft+ λpqfp∧ fq
Pq = λqpfq∧ fp+ λqsfq∧ fs
Ps = λsqfs∧ fq+ λstfs∧ ft
Pt = λtsft∧ fs+ λtpft∧ fp.
Recall that the W-property at the vertices fp, fq, fs and ft of the
quadri-lateral means for instance for fp that there are coefficients τij = τji such
that τptfp ∧ ft+ τpqfp ∧ fq+ τp1fp ∧ fp1 + τp2fp ∧ fp2 = 0. Therefore, we can write Pp = fp ∧ (τptft+ τpqfq).
That means that by the right choice for λ we have
λpt = λtp λpq = λqp λqs = λsq λst = λts.
All the considered lines lie in one plane which is the plane of the quadrilat-eral, therefore we can examine the problem in RP2 by choosing a suitable basis.
We now want to prove that the diagonals of the quadrilateral and the four traces of the vertices lie on a curve of 2nd class.
We give an outline of how we will proceed. We compute the lines ft∧fpand
fs∧ fq in hyperplane coordinates in RP2 and the intersection ˆf , second,
we compute the traces of the vertices and the respective intersections ˜f ,f˜˜ and in the last step we show that ˆf , ˜f and f being collinear is equivalent˜˜ to
λpt = λtp λpq = λqp λqs = λsq λst = λts,
which is the W-property. As the problem is planar and the four vertices fulfill the requirement for a projective basis, that is every three vertices are linearly independent, we can choose a basis such that the vertices have the following coordinates:
ft= (1, 1, 1) fp = (1, 1, −1) fq = (1, −1, 1) fs= (−1, 1, 1).
So far the lines are given in point coordinates. In projective space how-ever, because of the duality of points and hyperplanes, we can also use hyperplane coordinates.
We compute the hyperplane coordinates for ft∧ fp and fs∧ fq which we
are solutions of
< ωtp, ft> = 0 , < ωtp, fp >= 0
< ωsq, fq> = 0 , < ωsq, fs>= 0
< ωsq, ˆf > = 0 , < ωtp, ˆf >= 0.
The traces of the vertices are so far given in Pluecker coordinates, that is Pp = λptfp ∧ ft + λpqfp ∧ fq. In the plane of the quadrilateral the
corresponding line is given by hyperplane coordinates Pp : ωp := λptωpt+ λpqωpq
with ωpt being the hyperplane coordinates of the edge through fp and
ft and ωpq, respectively. Therefore, we will compute the coordinates of
the remaining edges ωts = (0, 1, −1), ωpq = (0, 1, 1) and the traces of the vertices
Pp : ωp = λpt(−1, 1, 0) + λpq(0, 1, 1)
Pq : ωq = λqp(0, −1, −1) + λqs(−1, −1, 0)
Ps : ωs = λsq(1, 1, 0) + λst(0, −1, 1)
Pt: ωt = λts(0, 1, −1) + λtp(1, −1, 0).
For ˜f = Pp∩ Pq the two following equations must hold:
< ˜f , ωp > = 0 (2.6) < ˜f , ωq > = 0 (2.7) −λptf˜1+ (λpt+ λpq) ˜f2+ λpqf˜3 = 0 −λqsf˜1− (λqp + λqs) ˜f2− λqpf˜3 = 0 ˜ f2 = −λpqλqs− λptλqp ˜ f3 = 2λqsλpt+ λpqλqs+ λptλqp ˜ f1 = λpqλqs− λptλqp are a solution of (2.6). Therefore, we have: ˜ f = 2λqsλptf + λˆ pqλqsfq− λptλqpfp. (2.8)
Similarly, we compute the equation forf˜˜ ˜
˜
Now ˆf , ˜f andf are collinear if and only if det( ˆ˜˜ f , ˜f ,f ) = 0, which implies˜˜ conditions for the coefficients λ:
0 = det( ˆf , 2λqsλptf + λˆ pqλqsfq− λptλqpfp , −2λsqλtpf + λˆ tsλsqfs+ λstλtpft)
= λpqλqsλstλtp det( ˆf , fq, ft) − λqpλsqλtsλpt det( ˆf , fp, fs)
= 2 (λpqλqsλstλtp− λqpλsqλtsλpt).
Thus ˆf , ˜f andf are collinear ⇐⇒˜˜
λpqλqsλstλtp = λqpλsqλtsλpt.
If the net has the W-property we know that there are coefficients τ such that we can define coefficients λ:
λpt := τpt = τtp = λtp λpq := τpq = τqp = λqp
λsq := τsq = τqs = λqs λst := τst = τts = λts.
In this case, the two products are equal and the points ˆf , ˜f and f are˜˜ collinear, hence, the six lines belong to a curve of 2nd class.
If we know that the six lines belong to a regular curve of 2nd class then
λpqλqsλstλtp = λqpλsqλtsλpt.
For the net to have the W-property we must show that there exist ˜λ’s such that: ˜λpt = ˜λtp etc., by which we can define τpt:= ˜λpt.
We start by choosing ˜λpt and define:
˜ λpq := λpq ˜ λpt λpt Pp = ˜λptfp∧ ft+ ˜λpqfp∧ fq ˜ λqp := ˜λpq, ˜λqs := λqs ˜ λpq λqp Pq = ˜λpqfq∧ fp+ ˜λqsfq∧ fs ˜ λsq:= ˜λqs, ˜λst := λts ˜ λqs λsq Ps = ˜λqsfs∧ fq+ ˜λstfs∧ ft ˜ λts := ˜λst, ˜λtp := λpt ˜ λst λts Pt = ˜λstft∧ fs+ ˜λtpft∧ fp. Then we have ˜ λpq = ˜λqp ˜ λqs = ˜λsq ˜ λst = ˜λts
and since the two diagonals and Pp, Pq, Ps and Pt belong to curve of 2nd
class
˜
λqpλ˜sqλ˜tsλ˜pt= ˜λpqλ˜qsλ˜stλ˜tp ⇒ λ˜pt = ˜λtp.
That means we can define coefficients τ and the net has the W-property.
2.2
Conjugate W-Nets on Projective Surfaces
2.2.1 The Wilczynski Canonical Form
In projective differential geometry one usually works with asymptotic line parameters in the Wilczynski canonical form. This form gives the partial differential equations of the asymptotic lines in a special scaling of the surface such that only one of the tangent vectors appears in each equation [Bol1], [La], [Fe].
We first describe the case of negative Gaussian curvature. On an open set U ⊂ R2 fxx = β fy + 1 2 (V − βy)f fyy = γ fx+ 1 2(W − γx)f (2.10)
with the integrability conditions
βyyy− 2βyW − βWy = γxxx− 2γxV − γVx
Wx = 2γβy+ βγy
Vy = 2βγx+ γβx
define a surface f in RP3 in terms of the asymptotic lines. This special
form of the asymptotic line parameter equations can always be achieved and it is called Wilczynski canonical form, see [La], having used det(f, fx, fy, fxy) = const 6= 0 [Bol1],[Fe].
The asymptotic parameters are only unique up to a reparametrization x 7−→ u(x) y 7−→ v(y),
in order to preserve the Wilczynski canonical form we need to rescale the surface
f∗ =√u0v0f. (2.11)
The coefficients also transform β∗ = β v 0 (u0)2 V ∗ (u0)2 = V + S(u) γ∗ = γ u 0 (v0)2 W ∗(v0)2 = W + S(v), (2.12)
where S(u) is the Schwarzian derivative of u
S(u) = u 000 u0 − 3 2 ( u00 u0) 2.
Remark 13. The coefficients β, γ, V, W depend on the parametrization of asymptotic lines. The solutions of this system are projective surfaces which are projective transforms of each other. In projective differential geometry a classification of surfaces is given in [Fe].
Usually, one considers only the hyperbolic case of negative Gaussian cur-vature where the asymptotic lines are real. In the elliptic case of positive Gaussian curvature the asymptotic lines only exist as complex parameters. Let U ⊂ C be a domain
f : U −→ R3
fzz = β fz¯+
1
2 (V − βz¯)f (2.13)
with the integrability conditions
Vz¯ = 2β ¯βz+ ¯ββz
Im (βz ¯¯z ¯z− 2βz¯V − β ¯¯ Vz¯) = 0
having used det(f, fz, fz¯, fz ¯z) = const 6= 0 gives f in terms of the complex
2.2.2 Conjugate W-Nets Let U ⊂ R2 be an open set
f : U −→ RP3
having negative Gaussian curvature, such that f (x, y) are asymptotic pa-rameters and let ∂ξ∂,∂η∂ be conjugate vector fields in the tangent plane. We want to determine conditions which guarantee that ∂ξ∂,∂η∂ define vec-tor fields coming from coordinates.
∂ ∂ξ = m ∂ ∂x + v ∂ ∂y ∂ ∂η = n ∂ ∂x + w ∂ ∂y.
Since II(∂ξ∂ ,∂η∂) = 0 holds (m, n) and (−v, w) are linearly dependent and we can write ∂ ∂ξ = m( ∂ ∂x − p ∂ ∂y) ∂ ∂η = n ( ∂ ∂x + p ∂ ∂y), (2.14)
with m, n, p being non-vanishing functions of x and y.
Locally we can assume m, n and p to be non-vanishing functions and with-out loss of generality we assume that they are locally positive, therefore we define ea := m, eb := n, eh := p. ∂ ∂ξ = e a( ∂ ∂x − e h ∂ ∂y) ∂ ∂η = e b( ∂ ∂x + e h ∂ ∂y), (2.15)
commute, [∂ξ∂,∂η∂ ] = 0, which is equivalent to
bx − ehby = ax+ ehay (2.16)
0 = hx+ bx− ehby. (2.17)
We assume now that we have coordinates f (ξ, η).
Recalling the definition for W-Nets in (1.43) on p.29 the conjugate lines f (ξ, η) have the W-property if and only if
0 = f ∧ (τ1ξfξ+ τ2ηfη + τ1fξξ+ τ2fηη)
⇐⇒
For fxx, fyy we use (2.10) on fξξ = e2a {(ax− ehay + e2hγ) fx +(e2hay − ehax+ e2hhy − ehhx+ β) fy +1 2(V − βy+ e 2h (W − γx)) f −2eh fxy} fηη = e2b {(bx+ ehby+ e2hγ) fx +(e2hby+ ehbx+ e2hhy+ ehhx+ β) fy +1 2(V − βy+ e 2h(W − γ x)) f +2eh fyx}
The condition for having the W-property
0 = ρf + τ1ξfξ+ τ2ηfη + τ1fξξ+ τ2fηη can be rewritten as 0 = (ρ +1 2(τ1e 2a+ τ 2e2b)(V − βy+ e2h(W − γx)) f + {(e2aτ1x+ e2bτ2x+ eh(τ2ye2b− τ1ye2a) +τ1e2aax− τ1e2aehay + τ1e2ae2hγ +τ2e2bbx+ τ2e2behby+ τ2e2be2hγ)} fx + {(e2h(e2aτ1y+ e2bτ2y) + eh(e2bτ2x− e2aτ1x) +τ1e2ae2hay− τ1e2aehax+ τ1e2ae2hhy − τ1e2ahx− τ1e2aβ +τ2e2be2hby+ τ2e2behbx+ τ2e2be2hhy + τ2e2bhx+ τ2e2bβ)} fy + {(τ2e2b− τ1e2a)} fxy.
Here f (x, y) are asymptotic parameters therefore f, fx, fy and fxy must be
linearly independent and the condition is equivalent to the simultaneous vanishing of the four coefficients of f, fx, fy and fxy which are functions.
We start with the coefficient of fxy which results in
τ1e2a = τ2e2b.
The coefficient with which f is multiplied can be chosen freely, since we can choose ρ freely. Furthermore, we choose τ := 2τ1e2a = 2τ2e2b then the
condition of the two coefficients of fx, fy vanishing results in
τx = τ (−γe2h+ 1 2(ax− e ha y + bx+ ehby)) τy = τ (−hy − βe−2h+ 1 2eh(−ax+ e ha y + bx+ ehby)).
Recall conditions (2.16) and (2.17) for [∂ξ∂ ,∂η∂] = 0 implying that the sys-tem is equivalent to τx τ = −γe 2h + hx+ ax+ bx τy τ = −hy − βe −2h + e−h(bx− ax).
For the existence of a function τ we need τxy = τyx.
τxy = τyx ⇐⇒ 0 = (−hy− βe−2h+ e−h(bx− ax))x− (−γe2h+ hx+ ax+ bx)y ⇐⇒ hxy = 1 2((γe 2h) y − (βe−2h)x).
This results in the final theorems which characterize conjugate nets which are W-Nets.
Proposition 9. Let U ⊂ R2 be an open set and f : U −→ RP3 have
negative Gaussian curvature and be given in asymptotic parameters f (x, y) as the solution of the Wilczynski canonical form. If a conjugate net f (ξ, η) with real-valued functions a, b, h defined on U , with tangent vector fields
fξ = ea(fx− ehfy) fη = eb(fx+ ehfy)
has the W-property then hxy =
1 2((γe
2h
)y − (βe−2h)x).
Proposition 10. Let U ⊂ R2 be an open set and f : U −→ RP3 have
neg-ative Gaussian curvature and be given in asymptotic parameters f (x, y) as the solution of the Wilczynski canonical form and let a real-valued function h on U be a solution of hxy = 1 2((γe 2h )y − (βe−2h)x).
Let also a, b be real-valued functions such that bx − ehby = ax+ ehay
0 = hx+ bx− ehby,
there then exists a function τ = (τ1, τ2) from U to R2 defined by
τ1 := τ2e−2a and τ2 := τ2e−2b such that
fξ = ea(fx− ehfy) fη = eb(fx+ ehfy)
define a conjugate net which has the W-property, i.e. we have 0 = ρf + τ1ξfξ+ τ2ηfη + τ1fξξ+ τ2fηη.
Remark 14. The partial differential equations for h, , a and b can be solved without a constraint for β or γ therefore every surface has a con-jugate net which has the W-property.
Remark 15. A conjugate net f (ξ, η) with h = 0 has the W-property if and only if the surface is a Jonas Surface which means that we have βx = γy,
see Chapter 3.
For the case of positive Gaussian curvature, we call this the elliptic case, we have to work with complex asymptotic lines f (z, ¯z) as in (2.13). Every two conjugate vector fields can be written in the following way:
∂ ∂ξ := i e ˜ a (e−h2 ∂ ∂z − e h 2 ∂ ∂ ¯z) ∂ ∂η := e ˜ b(e−h 2 ∂ ∂z + e h 2 ∂ ∂ ¯z) (2.18)
with h a function which only takes values in i R and ˜a, ˜b being real valued. Regarding conjugate nets in equation (2.15) coming from real asymptotic lines we have the special form
∂ ∂ξ = m( ∂ ∂z − f ∂ ∂ ¯z) ∂ ∂η = n ( ∂ ∂z + f ∂ ∂ ¯z),
with m = m1 + i m2, n = n1 + i n2, f = f1 + i f2 and f ¯f = 1 which
we obtain from the condition that ∂ξ∂,∂η∂ must be real. Therefore, we can define eh := f with h taking values in i R. Furthermore, we define g := eh2
which implies e−h2 = ¯g. With g = g1+ i g2 we get ∂ ∂ξ, ∂ ∂η real =⇒ m2 m1 = g1 g2 = −n1 n2
=⇒ there exist non-vanishing real valued functions r1, r2 such that
m1+ i m2 = i r1(g1− i g2) = i r1e− h 2 n1+ i n2 = r2 (g1− i g2) = r2e− h 2. Defining e˜a:= r 1 and e ˜b := r2, we get (2.18).
The conjugate vector fields ∂ξ∂ ,∂η∂ come from coordinates if and only if [∂
∂ξ, ∂
∂η] = 0 which is equivalent to the system
0 = ˜az− ˜bz+ eh(˜az¯+ ˜b¯z− h¯z)
which is equivalent to
hz¯= e−h(˜az− ˜bz) + ˜az¯+ ˜b¯z. (2.19)
The analogous theorems for the elliptic case are:
Proposition 11. Let U ⊂ C be an open set and f : U −→ RP3 have
positive Gaussian curvature and be given in asymptotic parameters f (z, ¯z) as the solution of the Wilczynski canonical form. If a conjugate net f (ξ, η) with a, b real-valued and h imaginary-valued functions on U , with tangent vector fields fξ = iea˜(e− h 2f z− e h 2f ¯ z) fη = e ˜b (e−h2f z+ e h 2f ¯ z)
has the W-property then hz ¯z = 1 2(( ¯βe 2h) ¯ z− (βe−2h)z).
Proposition 12. Let U ⊂ C be an open set and f : U −→ RP3 have
positive Gaussian curvature and be given in asymptotic parameters f (z, ¯z) as the solution of the Wilczynski canonical form. Let also an imaginary-valued function h on U be a solution of
hz ¯z = 1 2(( ¯βe 2h) ¯ z− (βe−2h)z),
and let ˜a, ˜b be real functions such that
hz¯= e−h(˜az− ˜bz) + ˜az¯+ ˜b¯z,
There then exists a function τ = (τ1, τ2) from U to R2 defined by
τ1 := τ2e−2˜a and τ2 := −τ2e−2˜b such that
fξ = iea˜(e− h 2f z− e h 2f ¯ z) fη = e ˜b (e−h2f z+ e h 2f ¯ z)
define a conjugate net f (ξ, η) with the W-property, i.e. we have 0 = ρf + τ1ξfξ+ τ2ηfη + τ1fξξ+ τ2fηη.
Remark 16. As for the case of negative Gaussian curvature we conclude that every surface has a conjugate net which has the W-property.
Remark 17. A conjugate net f (ξ, η) with h = 0 on a surface with positive Gaussian curvature has the W-property if and only if the surface is a Jonas Surface which means βz = ¯βz¯, see Chapter 3.
Discretization of Jonas
Surfaces
3.1
Jonas Surfaces
In 1921 the German mathematician Jonas [Jo] wrote about a class of sur-faces in comparison with R-sursur-faces.
A Jonas net is a conjugate net with both equal point invariants and tan-gential invariants. A surface that has at least one Jonas net is a Jonas surface [Ca].
A Jonas net fulfills the Laplace equation
fξη = a fξ+ b fη+ c f. (3.1)
The following terms are called the Laplace-Darboux invariants and since the equation is called the point equation they are often referred to as the point invariants
H = c + a b − aξ K = c + a b − bη. (3.2)
H and K are invariant under rescaling of the surface and relatively invari-ant under a reparametrization ζ(ξ), υ(η) of the conjugate net [La]
H∗ = 1
ζ0υ0H K ∗
= 1
ζ0υ0K.
A Jonas net has equal point invariants which is an absolutely invariant property since the equation H = K is absolutely invariant.
There is a dual Laplace equation called the tangential equation which is dual in the projective sense that the points on a surface are dual to their tangential planars. In the same way we can compute invariants for the dual Laplace equation [La], called the tangential invariants H and K.
