Discrete Minimal Surfaces using P-Hex Meshes

Transcription

1 Discrete Minimal Surfaces using P-Hex Meshes Diplomarbeit zur Erlangung des akademischen Grades Diplom-Mathematikerin vorgelegt von: Jill Bucher Matrikel-Nr.: Fachbereich: Betreuer: Mathematik & Informatik Prof. Dr. Polthier Abgabe: 26. Juli 2011

2 Contents 1. Introduction 3 2. Smooth Surfaces Curvatures Curvature related surface types Developable Surfaces Minimal Surfaces Polyhedral Surfaces Triangular meshes Planar quad meshes (PQ meshes) Planar hex meshes (P-Hex meshes) Oset Surfaces Parallel meshes Oset meshes Gauss image meshes Oset types and related Gauss images Vertex osets Edge osets Face osets P-Hex Mesh Computations Triangle to Hex: Tangent and Dupin duality Tangent duality Dupin duality Getting the right shape Computation and optimization Quad to Hex: Conjugate curves as input Creating the initial hex mesh Perturbation Conformal Hex Conformal polygons Conformal surfaces

3 Contents Dual constructions Osetting P-Hex meshes Constant vertex distance Constant edge distance Constant face distance Discrete Minimal Surfaces Vanishing mean curvature Curvature Smooth case Discrete case Zero mean curvature Creating discrete hexagonal minimal surfaces Hexagonal Gauss image via conformal hexagons Hexagonal Gauss image via spacial hex centers Conclusion 47 A. Zusammenfassung 49 B. References 51 B.1. Books and Articles B.2. Images

4 Chapter 1. Introduction Mathematics has always played an important role in other sciences. In recent years, a new area of application is developing from challenges posed by modern day architecture. Over the last couple of years the architectural design process has changed. The designs are using less typical forms and structures and more free and smooth shapes, which leads to the socalled freeform surfaces. They often appear as roof structures or within interesting facade solutions. There are also layered constructions, mainly using glass and steel, that lead to a good view of the underlying structure of the building. Because of that, the actual realization of a construction is becoming a way of design itself. The need for new ways of design led to a new research area called architectural geometry, where geometry is used to help solve architectural problems. The main interests for the architectural design process are - creative design of freeform surfaces (roof or otherwise) - aesthetic design supported by the underlying structure; [PLBW07, PSW08] - oset possibilities; [PSW08] - light-weight structures - cost-ecient materials; [Mue09] Manufacturing costs increase dramatically, when trying to realize those new designs as smooth as possible. From the architectural point of view that means, in order to reduce costs, a freeform surface should be approximated by a surface with easy to built parts. Those so-called discrete surfaces should also support the aesthetics of the structure. Hexagonal structures often appear in nature, as e.g. honeycombs or the skeletons of radiolaria. Since those structures are highly harmonious and symmetric, they are especially appealing to the human eye; [WLY + 08]. 3

5 Chapter 1. Introduction Motivated by those circumstances, the following thesis will deal with discrete minimal surfaces using P-Hex meshes. The thesis will consist of two parts. In the rst part we will transform the architectural needs into geometric counterparts (chapter 2-4). The second part will then deal with less architectural applications and more with the actual mathematical realization, which can e.g. be used as a basis for new architectural software solutions (chapter 5-7). To create a discrete version of a smooth architectural freeform surface, it is necessary to rst analyze the given smooth surface. Chapter Ÿ2 will therefore focus on properties of smooth surfaces. To create a freeform surface, it would be useful to start with just a boundary curve of the desired surface. Given that boundary curve, it is possible to create a minimal surface within. Minimal surfaces are visually appealing and also minimize surface area. They are structurally stable and reduce weight and material due to their minimality. Thus, two of the architectural interests are already fullled. Minimal surfaces are therefore a good choice when modeling freeform surfaces. As a result, we will concentrate on them instead of general freeform surfaces. They will rst be introduced in Ÿ2.2.2 as curvature related smooth surfaces. As mentioned above, realizing smooth freeform surfaces is very cost intensive. Using segmentations to approximate smooth surfaces leads to the theory of polyhedral surfaces and their representation by meshes. Chapter Ÿ3 we will compare the three relevant mesh types: triangular meshes, planar quad meshes and planar hex meshes. The possibility of constructing parallel surfaces, like oset surfaces, should already be considered while developing the structure. In chapter Ÿ4 we will introduce discrete osets and will analyze the three mesh types with regard to their oset properties. After we stated superiority of P-Hex meshes for building constructions and chose minimal surfaces as representation for cost-ecient freeform surfaces, we will then go into more detail on how to generate P-Hex meshes (chapter Ÿ5) and their osets (chapter Ÿ6), as well as constructing discrete minimal surfaces using P-Hex meshes (chapter Ÿ7). We will present three dierent approaches to generate P-Hex meshes. The rst one uses triangles as starting point, whereas the second one 4

6 Chapter 1. Introduction uses quads. The last one already starts with hexagons. Knowing how to generate polyhedral surfaces based on P-Hex meshes, chapter Ÿ6 will be focusing on the creation of their osets. In chapter Ÿ7 we will demonstrate a process of developing discrete hexagonal minimal surfaces. We will start by deriving discrete analogues of the smooth curvature denitions to dene vanishing mean curvature for the discrete setting. At the end of this chapter we will describe two methods of creating discrete minimal surfaces using P-Hex meshes. 5

7 Chapter 2. Smooth Surfaces 2.1. Curvatures The following curvature denition will be a geometric interpretation. It will be used to give a basic understanding of the terms necessary for the following chapters. A more precise description can be found later in chapter Ÿ7 about discrete minimal surfaces. For a smooth surface S one can easily dene a unit normal vector at every point p S of the surface. A Figure 2.1.: Normal planes for principal curvatures; [7] plane that contains p and its normal vector, the so-called normal plane, intersects with the surfaces in a plane curve (see Fig. 2.1). The curvature of this curve depends on the chosen normal plane. For each normal plane the normal curvature is measured by the radius r of the osculating circle (tangent to p). The normal curvature κ then is κ = 1 r. For each point, the minimal and maximal curvatures κ min and κ max are called the principal curvatures for that point. Those principal curvatures form a network, the so-called network of principal curvature lines. In addition to this general curvature denition, there are two curvature values that describe the surfaces properties at each point in more detail. They are constructed from the principal curvatures as follows: the Gaussian curvature K is dened as K = κ min κ max and the mean curvature H is dened as H = κ min +κmax 2. 6

8 Chapter 2. Smooth Surfaces 2.2. Curvature related surface types Given those curvature denitions it is possible to dene surface types which are related to those curvatures. Surfaces with vanishing Gauss curvature will be described in Ÿ2.2.1 about developable surfaces, and surfaces with vanishing mean curvature, the so-called minimal surfaces, will be described in Ÿ Developable Surfaces Surfaces with a Gaussian curvature of zero everywhere are the so-called developable surfaces. A smooth developable surface φ is the envelope of a one-parameter family of planes. Each of these planes touch the surface along a straight line, a so-called ruling. There are three main types: Either the rulings are parallel (φ is a cylindrical surface), or they pass through a xed point s (φ is a cone with vertex s), or they are tangents of a space curve r (φ is a tangent surface and r is its singular curve). Figure 2.2.: Types of developable surfaces:(left to right) cylinder, cone, tangent of a space curve; [7] Developable surfaces have a special property: They can be mapped into the plane without distortion. This is of interest not only for metal based industries but also for architecture. The architect Frank O. Gehry often uses developable surfaces for his design, for instance for the Walt Disney Concert Hall in Los Angeles shown in Fig Figure 2.3.: Developable surfaces used in Frank O. Gehry's construction of the Walt Disney Concert Hall in Los Angeles; [7] 7

9 Chapter 2. Smooth Surfaces Our main focus will, however, not be on surfaces with vanishing Gauss curvature, but on such with vanishing mean curvature. We will therefore not go into more detail about developable surfaces and their creation or application Minimal Surfaces A special case of the surfaces with constant mean curvature (cmc-surfaces) are those, with mean curvature H = 0 everywhere. Those surfaces are the so-called minimal surfaces. They are named after their property of locally minimizing surface area for a given boundary curve, i.e. if one would slightly change the form of the surface, its surface area would increase. Soap lm experiments done by the Belgian physicist Joseph Plateau led to the question of a general existence of such minimal surfaces for any given boundary curve, known as the Plateau Problem. Soap lms. Soap lms can be used to visualize a minimal surface: They appear, when you insert a metal curve into a basin with soap-water, such that it is completely covered in soap-water and then carefully remove it from the basin. The metal curve will then be the boundary curve of a soap lm surface. If the curve is non-planar like in Fig. 2.4, so is the resulting minimal surface. Figure 2.4.: Minimal surfaces as soap lms; [3] Those surfaces are formed due to the dierent energies, like bending energies, that inuence the soap lm. The surface displays a state of equilibrium. If one would manually distort the surface at a point, it would return to the equilibrium. As it turns out, the soap lm surface also has minimal surface area with regard to all other surfaces possible with the given boundary curve, i.e. even slight perturbations in very small local areas of the surface will lead to a larger surface area. Hence the name minimal surface. 8

10 Chapter 2. Smooth Surfaces Even though one might get a better feel for minimal surfaces by studying the behavior of soap lms, in general, when you want to create a minimal surface build from other materials, you are left without the help of e.g. bending energies or gravitation. For a given boundary curve, a surface we choose will not magically transform itself into a minimal surface. More information about the surface properties of minimal surfaces are necessary. This relevant information is provided by the mean curvature H (see Ÿ7.1). From an architectural point of view smooth surfaces are just the beginning of the design process. A physical realization of such surfaces would not only be time consuming, but also extremely expensive. The smooth surface therefore should be transformed into a discrete surface, preferably into a polyhedral surface. 9

11 Chapter 3. Polyhedral Surfaces A discrete surface, i.e. a segmentation of a smooth surface into panels, is represented by a so-called mesh. A mesh consists basically of a list of vertices (or nodes), edges that connect two vertices and faces that are bounded by edges. The actual mesh combinatorics contains the information which vertices belong to common edges and faces [PLBW07]. For a polyhedral surface all faces of the surface are planar. To gain more insight into polyhedral surfaces, a closer look on dierent mesh types will be useful. From the well-known problem of tiling the plane, we know that if only congruent copies of regular polygons were allowed, there are only three types of tiling possible: equilateral triangles, squares and regular hexagons. Allowing ane transformations leads to other tilings with congruent copies of the three types. For a tiling of a surface, [WLY + 08] name those three types as the most promising candidates, though in general, the property of the identical shape will get lost due to the surface curvature Triangular meshes When looking for a planar mesh structure, triangulations are often the rst type that comes to mind. The reason for choosing triangulations is its natural face planarity. No additional eort (time or money) is needed to ensure that the faces are planar. Since every triangle has a circumcircle, triangular meshes are circular and therefore by denition capable of osets with constant vertex-vertex distance (see Ÿ4.4.1). Figure 3.1.: A triangulation; [1] 10

12 Chapter 3. Polyhedral Surfaces Triangulations are well studied and can not only be found in many building constructions, but also in renderings for the movie and games businesses. The physical realization of a triangulation, however, is highly inecient. There are high costs in manufacturing triangle faces, since it only poorly lls its bounding box. Another disadvantage is the fact, that for a triangulation typically 6 faces meet in one vertex or node (less on the boundary). This so-called high node complexity leads to higher costs when producing e.g. the needed underlying beam structures. A likely torsion in the nodes might also add signicant costs to the manufacturing of those nodes. Torsion means the absence of a common node axis of the principal planes of incoming beams. For architectural purposes as in glass-steel-constructions, triangulations compared to e.g. quad meshes lead to less light and more weight due to the extensive beam structure necessary. Even though triangulations might not be the best choice for a modern polyhedral surface, especially one with an intended physical realization, it might still be a useful starting point for the construction of other mesh types, like planar hex meshes (see Ÿ5.1) Planar quad meshes (PQ meshes) A planar quad mesh (or PQ mesh) is, as the name suggests, a quad mesh with planar faces. The typical node complexity of a PQ mesh is 4, since 4 faces meet in one vertex. The complexity can be smaller at the boundary. A single row of planar quad faces is called a PQ strip and is the simplest form of a PQ mesh. Those PQ strips are the discrete analogue of smooth developable surfaces (see Ÿ2.2.1). Complexer PQ meshes are a network of PQ strips. A good starting point for a PQ mesh is a network of principal curvature lines. [LPW + 06] developed an algorithm, that uses such a quad mesh based on principal curvature lines as input. They further optimize the mesh with regard to planarity of the faces, bending energies and closeness to the original smooth surface. The next Figure 3.2.: A PQ mesh; [4] step in their algorithm uses subdivision to get a ner mesh. Those two steps (opti- 11

13 Chapter 3. Polyhedral Surfaces mization and subdivision) alternate until the mesh fullls their expectations. The general PQ algorithm, however, only generates quad meshes with planar faces. For additional properties like e.g. computing conical meshes (see Ÿ4.4.3), the optimization step needs additional conditions that express the conical feature. A quad mesh can also be a good choice to create a P-Hex mesh from (see Ÿ5.2) Planar hex meshes (P-Hex meshes) The last type of mesh that is of interest here, is the one with planar hexagonal faces, the so-called P-Hex mesh. From an architectural point of view, hexagonal meshes provide a simple node complexity of 3, which leads to a less complex node structure in physical realizations. Additionally, every P-Hex mesh is a conical mesh (see Ÿ4.4.3) and therefore possesses oset meshes at constant face-face distance. This property has proven to be extremely useful when dealing with multi-layered structures. Figure 3.3.: A discrete surface where convex parts are modeled by P-Hex meshes; [9] The oset property of hex meshes can be used for a simple denition of the curvature (see Ÿ ) and the actual computation of the surface. Having a suitable curvature denition also makes it easier to look at certain curvature-related surfaces, such as the discrete minimal surfaces of Ÿ7. There are dierent ways of creating P-Hex meshes. The method stated by [WLY + 08] uses tangent and Dupin duality (see Ÿ5.1) on a triangulation of the underlying surface and creates a P-Hex mesh that is dual to the triangulation. Whereas the one of [WL08] starts with a quad mesh from the underlying network of principal curvature lines (see Ÿ5.2). Both methods need additional constraints to ensure the planarity of the faces. 12

14 Chapter 3. Polyhedral Surfaces Even though each method results in a P-Hex mesh, they propose a dierent view on hex meshes, due to their dierent starting point (like a triangulation or a quad mesh). Those approaches may not be equally useful and therefore need to be carefully chosen with regard to the intended outcome, i.e. additional properties that the desired P-Hex mesh should possess. The method chosen also depends on the structure of the underlying surface, since some methods are not able to deal with every type of surface. For the rest of this thesis the main mesh type focus will be on P-Hex meshes, due to their interesting properties useful for current architectural challenges. The other types and their properties might frequently resurface, e.g. when dealing with the creation of P-Hex meshes. Chapter Ÿ7 about discrete minimal surfaces will also use P-Hex meshes as their polyhedral representation. 13

15 Chapter 4. Oset Surfaces In general, for a polyhedral surface it is possible to dene parallel meshes. Those are meshes that possess the same combinatorics as the original mesh and whose edges are parallel to the corresponding edges of the original mesh. A more advanced form of a parallel mesh is the oset mesh: a mesh which is not only parallel to a given mesh, but also at a constant distance to the original mesh. The distance can be dened between corresponding vertices, edges or faces. This leads to three types of oset meshes: vertex osets, edge osets and face osets. We will start by describing parallel meshes in general and then dene oset meshes and a special type of oset meshes: the discrete Gauss image meshes, which are meshes covering the unit sphere Parallel meshes According to [PLBW07], for meshes M and M to be called parallel two main conditions need to be fullled: First, there must be a 1-on-1 correspondence between the vertices and the edges of each mesh, i.e. for each vertex in one Figure 4.1.: Parallel meshes; [8] mesh there is one unique vertex in the other mesh and vice versa. Also, if two vertices are connected by an edge in one mesh, they need to be connected in the other mesh. Second, those corresponding edges must be parallel. Fig. 4.1 shows a mesh M and a mesh M parallel to M. For a mesh M, all meshes that fulll the rst condition form a linear space denoted as C(M). Therefore, for M and M C(M), a linear 14

16 Chapter 4. Oset Surfaces combination of both can be dened (vertex-wise) and the resulting mesh is again an element of C(M). All meshes M C(M) that also fulll the second condition form the set P (M) of parallel meshes to M. [PLBW07] further shows, that P (M) is a linear subset of C(M). Since P (M) is a linear space, a linear blending between dierent members M, M, M P (M) creates another mesh M P (M) as illustrated in Fig Figure 4.2.: Linear blending of members of P (M); [8] 4.2. Oset meshes A special case of parallel meshes are the so-called oset meshes. [PLBW07] dene an oset mesh M P (M) as a parallel mesh with a constant distance to the original mesh M. Depending on the (discrete) distance denition, this leads to three dierent kinds of osets: vertex osets: The distance of corresponding vertices is constant. edge osets: The distance of corresponding parallel edges is constant. face osets: The distance of corresponding faces is constant. Using this denition of oset, to determine if a mesh M possesses an oset property of any kind, one needs a parallel mesh to check the distances Gauss image meshes The denition of a discrete Gauss image for meshes in [PLBW07] is as follows: For a mesh M and an oset mesh M with a distance d, S := (M M)/d denes the discrete Gauss image. The vertices s i S can be seen as the discrete normal vectors with s i = (m i m i )/d. The discrete Gauss image S of a mesh M is a special kind of oset mesh, and therefore also S P (M). For a given mesh and its discrete 15

17 Chapter 4. Oset Surfaces Gauss image, it is easy to produce oset surfaces for every distance d. The discrete Gauss image represents the normal vectors of the surface, so M + ds = M with M P (M) for all d. The Gauss image mesh can be used not only to create osets of M, but also for every other mesh M P (M). Therefore all meshes in P (M) are of the same oset type, since there is only one Gauss image mesh for all of them Oset types and related Gauss images Connections between the oset type and the behavior of the Gauss image mesh are stated by [PLBW07]. In their setting, M, M are parallel meshes with distance d and S = (M M)/d is the corresponding Gauss image mesh. The connections are the following: Vertex osets Circular meshes. For a circular mesh, every face has a circumcircle. A circular mesh and its oset have a constant vertex-vertex distance. For the construction of conformal hexagons in Ÿ5.3 circular meshes are necessary. Figure 4.3.: Dierent circular polygons: triangle, quad and hex; [2] Shape of the Gauss image. For a constant vertex distance d between a mesh M and a parallel mesh M the vectors s i = (m i m i )/d of the Gauss image S are of length 1. The vertices of S are therefore on the unit sphere. The faces of S are planar and intersect with the sphere in a circle. This circle is at the same time the circumcircle of the face. The Gauss image S is then by denition circular Edge osets Osets with constant edge distance are highly desirable for architectural purposes, because due to the constant distance it is possible to build a beam structure for the mesh, that is of constant height. This is especially 16

18 Chapter 4. Oset Surfaces useful in glass-steel-constructions, where the underlying beam structure is always visible. So in order to have not only an appealing outer surface, but also a nice beam structure underneath to support it, edge oset meshes are promising candidates. [PGB08] further examine edge osets from a Laguerre geometry point of view. Figure 4.4.: Gauss image mesh S for an edge oset mesh M and its oset M ; [8] Shape of the Gauss image. Edge osets have a Gauss image with edges that are tangent to the unit sphere (see Fig. 4.4). Those meshes whose edges are tangential to the unit sphere are the so-called Koebe meshes. Using a discrete Christoel transformation (see Ÿ5.3.3), the Koebe meshes are then transformed into discrete minimal surfaces Face osets Face oset meshes are relevant meshes to construct multi-layer structures. Conical meshes. [LPW + 06] dene a mesh is conical, if all vertices v of a mesh M are conical, i.e. all faces meeting in this vertex are tangent to a common oriented cone of revolution. The cone axis can be interpreted as the discrete surface normal at that vertex. Conical meshes, just as circular meshes before, discretize the network of principal curvature lines. Since for a vertex of a hexagonal mesh the three face planes incident to the vertex are always tangent to a common cone of revolution, all hexagonal meshes are conical meshes and therefore naturally possess the face oset property. Advantages of conical meshes with regard to an orthogonal support structure are described in [LPW + 06] for PQ meshes. The fact that neighboring axes lie in a common plane (as illustrated in Fig. 4.5(b)) can be used to create such an orthogonal support structure. 17

19 Chapter 4. Oset Surfaces (a) a vertex cone (b) neighboring cone axes Figure 4.5.: Properties of conical meshes; [4] Shape of the Gauss image. For meshes with a face oset property, the Gauss image has faces that are tangent to the unit sphere. After establishing the practical uses of P-Hex meshes for architectural purposes, we will now describe the actual mesh generating process. 18

20 Chapter 5. P-Hex Mesh Computations Over the last couple of years dierent ways of creating a P-Hex mesh have been established and used. The following overview is according to [WL08]. Stereographic projection. The stereographic projection method starts in a 2D setting. An extended Voronoi diagram, a so-called power diagram, of a set of 2D points is mapped onto an ellipsoid using stereographic projection. This leads to a polyhedral surface. If the power diagram consists of hexagonal faces, the resulting polyhedral surface will be a P-Hex mesh approximating the ellipsoid. A huge disadvantage of this method is its restriction to the ellipsoid. It is therefore not possible to approximate e.g. a hyperboloid of one sheet or a more general freeform surface instead of the ellipsoid. Projective duality. A dierent approach of computing a P-Hex mesh (proposed by [WL08]) uses a triangulation to generate the desired mesh. The basic idea is to start by computing the dual of a given surface. This duality maps a plane not passing through the origin, in the form ax + by + cz + 1 = 0 in primal space, to a point (a, b, c) T in dual space. During the next step a regular triangulation of the dual surface is computed. In the nal step, this triangulation is mapped to a P-Hex mesh approximating the original surface. There are, however, some problems with this approach, e.g. possible high metric distortion or singular points on the dual surfaces due to parabolic points on the original surface. Projective duality, in general, does not provide a one-on-one correspondence between the points of the original surface and those of the dual surface. Additionally, this method cannot be used to approximate a freeform surface. Even with the restriction to only convex freeform surfaces, the method might still produce 19

21 Chapter 5. P-Hex Mesh Computations faces with self-intersection. Gaussian sphere. This method of computing a P-Hex mesh starts with a freeform surface and uses the supporting function over the Gaussian sphere of the surface. This supporting function basically is the composition of the duality and the spherical inversion with respect to S 2. This piecewise linear supporting function leads to a P-Hex mesh approximating the original freeform surface. Since this method also uses projective duality, the same problems as for the plain projective duality method arise. Parallel meshes. This method was developed out of the oset theory for mesh surfaces. It can be used to develop P-Hex meshes for simple surfaces, like a surface path with K > 0 everywhere or K < 0 everywhere. A major restriction of this method is, however, that a P-Hex mesh is already needed to begin with, in order to create a new one parallel to the given mesh. Even then, this method may still produce faces with selfintersection. The disadvantages of these methods lead to the two basic attributes desirable in a method of computing P-Hex meshes: generality and validity. Generality in this context means, that there is no restriction as to what type of region (elliptic, hyperbolic, parabolic) can or cannot be computed by the method. The validity property basically ensures faces with no selfintersection. The following three approaches will present methods of computing a P-Hex mesh with respect to generality and validity: In 2008 [WLY + 08] proposed a way that starts with a triangulation and develops a P-Hex mesh using dierent types of duality (see Ÿ5.1). In the same year [WL08] described an additional way of getting a P-Hex mesh by starting with a quad mesh generated from the network of conjugate curvature lines of the underlying surface (see Ÿ5.2). The third way will be described in Ÿ5.3 and uses conformal meshes as stated by [Mue09] in Triangle to Hex: Tangent and Dupin duality This approach uses of two types of dualities: tangent duality and Dupin duality. The tangent duality describes the correspondence between a 20

22 Chapter 5. P-Hex Mesh Computations triangular mesh and a P-Hex mesh. The Dupin duality will be used to analyze the local properties of the tangent duality in more detail Tangent duality Let S be a smooth surface and T be a regular triangle mesh that approximates S. The vertices of T are on the surface S. For a triangle t T and its vertices v a, v b and v c (v i S), the tangent planes of S at v a, v b and v c intersect at a point u t. Finding all of those points u t and connecting the u t of adjacent triangles leads to a hexagonal Figure 5.1.: Tangent duality; [10] mesh H, that is combinatorially equivalent to T (see Fig. 5.1). Each face of H is planar and tangent to the surface S at a vertex of T. Without any additional properties, the tangent duality can lead to a P-Hex mesh with self-intersections in its faces. The idea now is to look for a triangle mesh, that leads to a tangent-dual P-Hex mesh with nice face shapes. The so-called Dupin duality will give more insight on this problem Dupin duality Figure 5.2.: Projections onto the tangent plane; [10] i = 0, 1,..., 5 are connected to v c. tangent plane Γ will be denoted as v For a given triangulation, the Dupin duality process will compute the socalled Dupin centers of the triangles and connect these centers into hexagons. The Dupin duality starts with a regular triangle mesh T, whose vertices are on the underlying smooth surface S. For a vertex v c S (also: v c T ), the tangent plane at this vertex will be denoted as Γ. Since T is a triangle mesh, six vertices v i with Their vertical projections onto the i. The triangles on the tangent plane will be denoted as t i, the triangles being v cv i v i, i = 0, 1,..., 5 +1 mod 6 (see Fig. 5.2). During the next step, the points v i are perturbed in a way, that they form a centrally symmetric hexagon, which is centered at v c = v c v c. All new triangles are congruent. A group of six triangles with the men- 21

23 Chapter 5. P-Hex Mesh Computations tioned property is called a triangle star and is denoted as T (Fig. 5.3, middle). One of the (congruent) triangles is arbitrarily chosen to be the so-called fundamental triangle t. Figure 5.3.: Dupin duality: fundamental triangle (left); triangle star T (middle); resulting dual hex (right); [10] In order to get the dual hex for a triangle star, we rst need to take a closer look at the so-called Dupin indicatrix. Dupin indicatrix. Given a smooth surface S and a point p S, then T p (S) denotes the tangent plane of S at p. Let there be a coordinate system on T p (S) aligned with the principal curvatures κ 1, κ 2 of the surface, then κ 1 x 2 + κ 2 y 2 = ±1 denes a conic called Dupin indicatrix. Depending on the type of point chosen, the shape of the Dupin indicatrix changes: For an elliptic point (with κ 1 < 0 and κ 2 < 0; change orientation if necessary) the Dupin indicatrix is the ellipse κ 1 x 2 + κ 2 y 2 = 1. For a hyperbolic point the Dupin indicatrix consists of the two hyperbolas κ 1 x 2 +κ 2 y 2 = ±1 with the same pair of asymptotic lines. For a parabolic point (with e.g. κ 1 0 and κ 2 = 0) the Dupin indicatrix is the pair of lines κ 1 x 2 = ±1. The Dupin indicatrix is however not dened at a planar point, i.e. a point with κ 1 = κ 2 = 0. Figure 5.4.: Dupin indicatrix: A P-Hex face for an elliptic point K > 0, for a parabolic point with K = 0 and for a hyperbolic point with K < 0; [10] Dupin conic, Dupin ellipse and Dupin hyperbola will be short for a cone / ellipse / hyperbola homothetic to the Dupin indicatrix. A homothetic transformation is a composition of a uniform scaling and a translation. 22

24 Chapter 5. P-Hex Mesh Computations Returning to the Dupin duality, we now look at the Dupin indicatrix C of S at the vertex v c. It is now possible to nd a unique Dupin conic C i, i.e. a homothetic copy of C, that circumscribes the fundamental triangle t, and therefore each triangle v i. The center of the Dupin conic is called the Dupin center of t i. The six Dupin centers of a triangle star are basically translational copies of each other, since the triangles are congruent and translational or reectional copies of each other. The connection of the Dupin centers leads to a centrally symmetric hexagon on the tangent plane, the so-called Dupin hexagon (see Fig. 5.3, middle and right) Getting the right shape Transforming a triangle mesh into an P-Hex mesh is a good start, but using the two methods mentioned above, there is no control over the resulting shapes of the hex faces. A type of P-Hex mesh that is of more interest is a valid P-Hex mesh. For a valid P-Hex mesh each face of the mesh is itself valid, i.e. without self-intersection. It is possible to translate this back to the triangulation used in the creation of the P-Hex mesh and call this triangulation valid, if the (tangent dual) P-Hex mesh is valid. The idea of a valid mesh can also be used with Dupin duality, where a triangle star is valid, if the corresponding Dupin hexagon is valid, i.e. is has no self-intersections. Elliptic point. For the case of an elliptic point (K > 0) at v c S, the equivalences of the following 3 properties are stated: 1. The triangle star is valid. 2. The Dupin center of the fundamental triangle t is inside t. 3. The three edge directions of t are not enclosed by any pair of conjugate directions with respect to the Dupin indicatrix C. Hyperbolic point. For a hyperbolic point (K < 0) the following equivalences can be derived: 1. The triangle star is valid. 2. The three vertices of the fundamental triangle t are on dierent branches of its circumscribing Dupin hyperbola. 3. The three edge directions of t are not all enclosed in the same range bounded by the two asymptotic directions of S at v c. 23

25 Chapter 5. P-Hex Mesh Computations The approach as described so far makes use of a valid triangulation to produce a valid P-Hex mesh that has no self-intersections. Additionally, one would like to have not only a valid P-Hex mesh, but also a valid P-Hex mesh with nice face shapes. Figure 5.5.: Regular (left) and quasi-regular hexagon (right); [10] In the case of Gaussian curvature K > 0 this leads to the following additional property: The P-Hex faces should be ane regular hexagons, i.e. regular hexagons under ane transformations (Fig. 5.5, left). For that to happen, the Dupin center of the fundamental triangle t of the original triangulation needs to be at the centroid of the fundamental triangle t in the tangent space. Those triangles are then called ideal triangles for K > 0. Similarly, for K < 0, the P-Hex faces should be ane copies of quasiregular hexagons. A quasi-regular hexagon is formed by juxtaposing the two halves of a regular hexagon (Fig. 5.5, right). To produce this kind of hexagon, the Dupin center of the fundamental triangle t is at the midpoint of a straight line between the centroid of t and any one of its three vertices. Those possible points can be seen on the right side in Fig Triangles fullling this condition are called ideal triangles for K < 0. For more control over the face shapes, having the Dupin centers at the ideal positions is necessary. Otherwise, the resulting face shapes will not be regular or quasi-regular hexagons anymore. Still, as long as the Dupin center is within the fundamental triangle t, the face will at least be without self-intersections Computation and optimization The algorithm proposed by [WLY + 08] consists of three parts: the computation of a valid triangulation, the conversion of this triangulation into a nearly P-Hex mesh and the optimization into an P-Hex mesh. Computation of a valid triangulation. In order to compute a valid triangulation it is necessary to have a way of constructing ideal triangles rst. Both algorithms for creating a valid triangulation described later use those ideal triangles. 24

26 Chapter 5. P-Hex Mesh Computations Since the term ideal triangle depends on the Gaussian curvature K, the computation of an ideal triangle diers slightly for K > 0 and for K < 0. We will look at the case K > 0 rst. Figure 5.6.: Properties of an ideal triangle; [10] Fig. 5.6 (left) shows an ideal triangle t: ABC. Since t is ideal, its Dupin center is located at the centroid O. It is easy to see, that the line through C and O also passes through the point M, which is the midpoint of the line segment AB. Therefore, the line CO is conjugate to the line AB at M. This property leads to the following construction of the point C: Given are the points A, B and the midpoint M on the line AB. For an also given Dupin indicatrix at M, there exists a unique Dupin ellipse (since K < 0) e that has AB as diameter. Let l be the line through M that is conjugate to the line AB. Now consider the parallelogram ABED, such that AD l BE and DE is tangent to e (see Fig. 5.6, left). The directions ME and MD are also conjugate. In the nal step the point C is marked on the line l, such that it fullls the condition CM = 3 AD. The resulting triangle ABC is an ideal triangle with its Dupin center at its centroid. For the case of K < 0 the construction is similar. The dierence is in dening the parallelogram ABED. This time the lines MD and ME are on the asymptotes of the Dupin hyperbola centered at M (see Fig. 5.6, right). The intersection point of the line parallel to l through A with the asymptotic line MD leads to the point D of the parallelogram. The point E is constructed the same way, using a parallel through B and the asymptotic line ME. The condition for the point C is again CM = 3 AD. There are two methods of getting a valid triangulation presented: the rst method is the progressive conjugation method. It generates a conjugate network on-the-y while the triangulation is in progress. Since this method uses ideal triangles (within discretization errors), the resulting 25

27 Chapter 5. P-Hex Mesh Computations P-Hex faces are nearly ane hexagons or quasi-regular hexagons. The second method is the pre-specied conjugation method. It adjusts some disadvantages of the former one with regard to width and orientation of the triangle layers. Here there are two conjugate direction elds already specied. Creation of a nearly P-Hex mesh. This step creates a nearly P-Hex mesh by determining and then connecting the Dupin centers of the triangles. The actual algorithm consists of 4 main parts. The rst part is a parabolic region detection. If the triangle is in a parabolic region, the Dupin center of that triangle is set manually to the centroid of the triangle. The next part determines the Dupin center of a triangle that is not in a parabolic region. For each vertex v i in v 1 v 2 v 3 the other two are projected onto the plane Γ i, which is determined by v i and its normal N i. Next the Dupin conic C i is computed on Γ i, so that it passes through v i and the projections of the other two vertices. The center u i of this Dupin conic represents the Dupin center for v i. The third part computes the Dupin center u for the triangle as the average of the three vertex Dupin centers: u = (u 1 + u 2 + u 3 )/3. The fourth and last part connects the Dupin centers u i of all triangles to a hexagonal mesh H 0 by using a simple mesh-duality technique. The resulting mesh H 0 is nearly planar. Optimization to create a P-Hex mesh. The hexagonal mesh H 0 created in the step before is a good initial mesh for the following optimization. To achieve planarity of a face, the volumes of all 4-point subsets of vertices of the face must be zero. The term vol(u i, u j, u k, u l ) denotes the volume of a tetrahedron (u i, u j, u k, u l ). The planarity constraints for each hex-face f i is dened as F (f i ) := 5 vol 2 (u i, u i +1, u i +2, u i +3 ) = 0 i =0 with indices modulo 6. Additionally, there will be some conditions regarding the distance to the input mesh and to the underlying surface. The resulting function will be minimized using the Gauss-Newton method. This optimization step is very fast, since the input mesh is already close to a P-Hex mesh. 26

28 Chapter 5. P-Hex Mesh Computations 5.2. Quad to Hex: Conjugate curves as input The method proposed by [WL08] starts with a quad mesh generated by the conjugate curve network of the surface. By a simple transformation, this quad mesh is then transformed into a hex mesh with nearly planar faces. To get a P-Hex mesh, this nearly planar hex mesh is then locally perturbed by using a nonlinear optimization process. (a) initial quad mesh (b) hex mesh by shifting quad mesh (c) resulting hex mesh Figure 5.7.: Dierent stages of the hex mesh generation; [9] Creating the initial hex mesh For a given surface S, a conjugate curve network will create a good initial mesh (see Fig. 5.7(a)). The conjugate curve network leads to a quad mesh, that is nearly planar. Shifting every other row of the quad mesh transforms it into a brick-wall layout using nearly planar hexagonal faces. The new faces still appear quadrilateral due to the position of the new vertices placed at the vertices of the neighboring rows. The result of this step can be seen in the middle of Fig. 5.7(b) Perturbation The next step turns the nearly planar hexagonal mesh into a P-Hex mesh using perturbation. For a hexagonal face to be planar, the volume of every 4-point subset needs to be zero. They also add a constraint, that monitors the distance to the original surface using squared distances of the mesh vertices and the surface. So the perturbation process is a constrained non-linear least squares problem. This method can be used for general surfaces which contain regions of positive curvature and regions of negative curvature. There is some control over the existence of self-intersections by adjusting the sampling size of the conjugate curve network. 27

29 Chapter 5. P-Hex Mesh Computations Unfortunately, this method only produces approximately planar hex meshes due to the optimization process Conformal Hex With a view on the parallel meshes in Ÿ4.1 and more specic the oset meshes mentioned in Ÿ4.2, we know that there is a connection between circular meshes and meshes with a vertex oset property. In this chapter about conformal hexagons, we will take a close look on the circular property Conformal polygons We know, that a polygon is called circular, if it possesses a circumcircle. A type of polygon closely related to the circular polygon is the quasi-circular polygon. It is a polygon, that is parallel to a circular polygon, i.e. they have the same combinatorics and their edges are parallel. To determine if a polygon is circular, the computation of the crossratio (for quads) or the more general multi-ratio (for polygons with an even number of vertices larger than 4) have proven to be very useful. Cross-Ratio. z 0,..., z 3 C is dened as The cross-ratio of a quadrilateral z i = (z 0, z 1, z 2, z 3 ) with cr(z 0, z 1, z 2, z 3 ) = (z 0 z 1 )(z 2 z 3 ) (z 1 z 2 )(z 3 z 0 ) The cross-ratio is Moebius invariant. An important connection between the properties of the quad and the cross-ratio is the fact, that a quad is circular if and only if the cross-ratio is real. Multi-Ratio. An extension of the cross-ratio is the multi-ratio for polygons with an even number of vertices. For a polygon z i with n vertices, the multi-ratio can be dened as q(z 0,..., z n 1 ) = (z 0 z 1 )(z 2 z 3 )... (z n 1 z n ) (z 1 z 2 )(z 3 z 4 )... (z n z 0 ) The multi-ratio is also Moebius invariant. If the multi-ratio is real, then the polygon is quasi-circular, and vice versa. Similar to the cross-ratio we get a quasi-circular polygon, if and only if the multi-ratio is real. 28

30 Chapter 5. P-Hex Mesh Computations In order to compute conformal hexagons, we will now narrow our view from arbitrary polygons (with even number of vertices) to hexagons. Conformal hexagon. A hexagon z i = (z 0,..., z 5 ) is a conformal hexagon, if and only if the cross-ratios of the two quads (z 0, z 1, z 2, z 3 ) and (z 0, z 5, z 4, z 3 ) fulll cr(z 0, z 1, z 2, z 3 ) = 1 2 = cr(z 0, z 5, z 4, z 3 ). So for a conformal hexagon, the cross-ratio of the quads is not only real, it is always equal to 1. A conformal hexagonal mesh is then dened as 2 mesh where each face is a conformal hexagon. Next we will look at the multi-ratio for conformal hexagons. Multi-Ratio of a conformal hexagon. For the multi-ratio of a hexagon we get: q(z 0, z 1, z 2, z 3, z 4, z 5 ) = (z 0 z 1 )(z 2 z 3 )(z 4 z 5 ) (z 1 z 2 )(z 3 z 4 )(z 5 z 0 ) = (z 0 z 1 )(z 2 z 3 ) (z 1 z 2 )(z 3 z 0 ) (z 3 z 0 )(z 4 z 5 ) (z 3 z 4 )(z 5 z 0 ) 1 = cr(z 0, z 1, z 2, z 3 ) ( 1) cr(z 0, z 5, z 4, z 3 ) = 1 2 ( 1) ( 1 2 ) 1 = 1 So, for a conformal hexagon, both quads are circular due to their real cross-ratio, and the hexagon itself is quasi-circular due to its real multiratio Conformal surfaces This section will describe conformal surfaces and their construction in more detail. Discrete conformal (hexagonal) surface. A discrete conformal (hex) surface is a mesh that consists of conformal hexagons (i.e. hexagons (z 1,..., z 5 ) with cr(z 0, z 1, z 2, z 3 ) = cr(z 0, z 5, z 4, z 3 ) = 1/2). Since the multi-ratio of a conformal hexagon is 1, each face of a conformal surface is quasi-circular and the surface itself possesses the vertex-oset property. 29

31 Chapter 5. P-Hex Mesh Computations A discrete conformal surface can be seen as a discrete analogue of a smooth conformal parametrized surface. Construction of planar conformal (hexagonal) meshes. The following construction technique can be found in [Mue09]. Let (z i ) be a conformal hexagon and let α and β be two similarities, which map α : (z 4, z 5 ) (z 2, z 1 ) and β : (z 3, z 4 ) (z 1, z 0 ). Then α and β commute, i.e. α β = β α. For (k, l) Z 2, β k α l (z i ) = α l β k (z i ) then is a conformal mesh with no gaps. Figure 5.8.: Construction of a planar conformal mesh; left: conformal hex (input); right: the resulting conformal (hex) mesh; [5] To create not only a conformal, but also a circular conformal mesh, the same construction process can be used, but this time the hexagon that starts the process needs to be circular. By doing so, every other hexagon created will be circular as well, which by denition leads to a circular mesh Dual constructions This paragraph will describe the creation of a dual hexagon and some properties of the original hex and its dual as mentioned in [Mue09]. Figure 5.9.: left: Edge coecients for dual conditions; middle and right: A conformal hexagon and its dual; [5] 30

32 Chapter 5. P-Hex Mesh Computations Dual of a conformal hexagon. Let (z i ) be a conformal hexagon and let a i := z i +1 z i be the edge vector (indices modulo n). A hexagon zi considered dual to z i if its edge-vectors fulll the following conditions: is z z = 1/(z z 0 ) = 1/a 0 z z = 2/(z z 1 ) = 2/a 1 z z = 1/(z z 2 ) = 1/a 2 z z = 1/(z z 3 ) = 1/a 3 z z = 2/(z z 4 ) = 2/a 4 z z = 1/(z z 5 ) = 1/a 5 Properties of conformal hexagons and their duals. We will now look at some properties of conformal hexagons with regard to a possible dual hexagon. For a conformal hexagon (z i ), its edge vectors a i vector b := z 0 z 3 from z 3 to z 0, the following are equivalent: and the diagonal (i) 5 i =0 a i = 0, a 0 + a 1 + a 2 + b = 0 and a 0a 2 a 1 b = 1/2, a 3a 5 a 4 b = 1/2 (ii) z 0 z 3 = 2/b and in particular: z 0 z 3 is parallel to z 0 z 3 (iii) the hexagon (z i ) has a dual (iv) the dual (zi ) is a conformal hexagon, unique up to translation (v) non-corresponding diagonals of both quads z 0, z 1, z 2, z 3 and z 0, z 5, z 4, z 3 are transformed according to z z = 3 z 0 z z 0 z 2, 2 z z = 3 z 3 z z 3 z 1 2 z z = 3 z 0 z z 0 z 4, 2 z z = 3 z 3 z z 3 z 5. 2 In particular they are parallel: z 2 z 0 z 1 z 3, z 1 z 3 z 0 z 2, z 4 z 0 z 5 z 3, z 5 z 3 z 4 z 0. (vi) Applying duality twice yields the original hexagon up to translation, so (z i ) = (z i ). Christoel dual. We will rst recall the denition of the smooth Christoffel dual, to derive a denition for the discrete case. 31

33 Chapter 5. P-Hex Mesh Computations Smooth Christoel dual. Let f be an isothermic parametrization. Then the Christoel dual f, dened by the formulas exists and is isothermic again. fx = f x f x and f 2 y = f y f y 2 The dual f is a minimal surface if and only if f is a sphere. Discrete Christoel dual property. Similar to the smooth case, a property of the discrete Christoel dual is the following: A discrete surface is a discrete hexagonal minimal surface, if it is the dual of a conformal mesh covering the unit sphere. This property already states to some extent the construction process described further in Ÿ7.2.1: During this construction process we rst create a conformal mesh covering the unit sphere and then construct its dual, which by the denition above will then be a discrete hexagonal minimal surface. After dealing with the general computation of P-Hex meshes, we will now provide information about how to gain an oset of a P-Hex mesh. 32

34 Chapter 6. Osetting P-Hex meshes Since there are three types of osets possible, we will give an idea for a creation method for each type Constant vertex distance The vertex oset property is not a natural property of a P-Hex mesh M, we therefore need it to be parallel to a P-Hex mesh M that is inscribed to the sphere S 2 (see Ÿ4.4). If such a parallel mesh M exists, then every hexagonal face of M possesses a circumcircle. The approach stated next is described in [WL08]. Let h be a face of M. Let α i, i = 0, 1,..., 5, denote the internal angles of h as shown in Fig Since h is circular, the following angle criterion applies α 0 +α 2 +α 4 = α 1 +α 3 +α 5. Then, let h M be the corresponding hex face and α i the corresponding interior angles of h. Since M is parallel to M, the edges of h are parallel to the edges of h. This leads to α i = α i, for all i. Figure 6.1.: Circular hex; [9] Therefore, we get α 0 + α 2 + α 4 = α 1 + α 3 + α 5 as condition for the angles of h. On the other hand, starting with the angle criterion for a planar hex face h leads to the fact, that h is parallel to a hex face h, with h being inscribed to a circle. As a result, the fulllment of the angle criterion for each face of the surface is a necessary and sucient condition for an open P-Hex mesh surface to possess an oset mesh with constant vertex-distance. For more complicated types of surfaces, this angle criterion is only a necessary condition. 33

35 Chapter 6. Osetting P-Hex meshes 6.2. Constant edge distance The following method of creating a hexagonal edge oset mesh (EO mesh) was proposed by [PLBW07]. It uses a Koebe polyhedron as starting point. Koebe Polyhedron. A Koebe polyhedron is a mesh with planar faces whose edges e touch the unit sphere S 2 at points t e. In addition to that, each face f intersects the sphere in a circle c f which touches the boundary edges of the face f. For every vertex s i, the vertex cone Γ i (see Fig. 6.2) touches the unit sphere in a circle c si. An edge e is tangent to the sphere at a point t e. In this point Figure 6.2.: Koebe polyhedron; [8] four dierent circles meet: the inscribed circles of the 2 adjacent faces and the circles of the vertex cones of each edge endpoint. Circles of the same type touch each other, whereas circles of dierent types intersect at a 90 degree angle (see Fig. 6.2). Getting an EO mesh. For a given surface φ, they start with a hexagonal Koebe polyhedron S; with s i S being a point of the Koebe polyhedron. To get the vertices m i of the oset mesh M, they parallel translate every face plane adjacent to s i until the planes are tangent to the surface φ. The intersection point of the three planes will be the vertex m i of the surface M. Following this construction ensures the edge oset property of the mesh M. This method, however, cannot rule out self-intersections Constant face distance As mentioned before in Ÿ4.4.3, P-Hex meshes are conical and therefore possess a natural face oset property. The following two methods are described in [WL08]. Method 1. A very easy approach to compute an oset surface for a given P-Hex mesh is to oset the incident faces of a vertex separately in the direction of their face normal by a constant distance d. The intersection point of three new face planes is the new corresponding vertex of the oset surface. Unfortunately, this method is not applicable in the 34

36 Chapter 6. Osetting P-Hex meshes case of three co-planar faces and will be numerically unstable when they are nearly co-planar. Method 2. This next method will take the interior angles of the faces at a vertex of the mesh into consideration when computing the vertex normal. Let M be a P-Hex mesh, v M a vertex and let f i with i = 0, 1, 2 be the faces incident at v. The corresponding vertex of the oset surface (at distance d) will be denoted as v d (with v d M d ). Let N i be the unit normal vectors of the faces f i. Let θ i be the internal angle of f i at v. The vertex normal of v will then be computed as follows: N v = 2 (tan β i + tan γ i )N i i =0 where β i = 1 (θ 2 i + θ i +1 θ i 1 ) and γ i = 1 (θ 2 i + θ i 1 θ i +1 ), i = 0, 1, 2 mod (3). With this formula, the vertex v d can be determined by intersecting the line p(t) = v + tn v with any of the oset planes of the faces. After we established the discrete oset capabilities of P-Hex meshes, we can now use an analogue of smooth curvature denitions from oset properties to dene discrete curvatures and therefore discrete minimal surfaces. 35

37 Chapter 7. Discrete Minimal Surfaces Smooth minimal surfaces are dened as surfaces with vanishing mean curvature. If we want to dene a similar version for the discrete case, we rst need to dene the curvatures of a discrete surface. The idea of transforming the smooth curvature denitions into discrete ones is a promising approach, but it is important to realize, that the equivalence of the smooth curvature denitions might not translate accordingly. Therefore a discrete realization of a smooth minimal surface might not be minimal, depending on the mesh and the curvature denition used for the realization. It is therefore a goal of the discrete minimal surface theory, to nd representations that fulll more than just one denition; see [PBCW07]. First we will present denitions of discrete curvatures and discrete minimal surfaces, which will not be restricted to any specic mesh or oset type. In the second part, we will concentrate on dierent creation methods for discrete minimal surfaces based on P-Hex meshes Vanishing mean curvature An important value of a surface with regard to minimal surfaces is the mean curvature. This section will broaden the basic denitions necessary for the smooth case as stated in Ÿ2.1 and then demonstrate, how to translate those into the discrete case. Two dierent approaches for the discrete curvature denitions will be described. 36

38 Chapter 7. Discrete Minimal Surfaces Curvature Smooth case The following denitions for the smooth case are translated versions of the ones in [Küh05]. Surface patch. Let U R 2 be open. A parametrized surface patch is an immersion of the kind f : U R 3, (u 1, u 2 ) f (u 1, u 2 ). The map f is called parametrization, the elements of U are called parameter and their images are called points or vertices. Tangent plane. The map f is an immersion. The vectors f f u and v are therefore linearly independent and span the so-called tangent plane T u f for u U. Its orthogonal complement is the one-dimensional normal space. First fundamental form. The rst fundamental form I of a surface patch is dened as I(X, Y ) := X, Y with X, Y T u f. Gauss map. Let S 2 be the unit sphere. Then, for a surface patch f : U R 3, the Gauss map ν : U S 2 is dened as ν(u 1, u 2 ) := f u 1 f u 1 f u 2 f u 2 ν(u 1, u 2 ) represents the unit normal vector placed at the origin of the surrounding space. Weingarten map. The map L := Dν (Df ) 1, with its point-wise denition L u := (Dν u ) (Df u ) 1 : T u f T u f is called Weingarten map. For every parameter u, this is a linear endomorphism of the tangent plane in f (u). The map L is independent of the parametrization and is self-adjoint with regard to the rst fundamental form I. Second fundamental form. Let f : U R 3 and ν : U S 2, L be the Weingarten map and X, Y tangent vectors. Then II(X, Y ) := I(LX, Y ) is called the second fundamental form. 37

39 Chapter 7. Discrete Minimal Surfaces Principal curvatures. Let X T u f be a unit normal vector. X is called principal curvature direction of f, if one of the two equivalent conditions is fullled: (i) II(X, X) has a stationary value for all X with I(X, X) = 1 (ii) X is eigenvector of L. The eigenvalue λ with (LX = λx) is called principal curvature. Gauss and mean curvature. Using the principal curvature denition we stated above, we get the following denitions: (i) The determinate K = det(l) = κ 1 κ 2 is the Gauss curvature of f. (ii) The value H = 1 tr(l) = 1 (κ κ 2 ) is the mean curvature of f. Those curvatures also appear within a formula that describes the variation of surface area when passing from a surface φ to an oset surface φ d. A point x φ is moved to x +dn(x), where n is the eld of unit normal vectors. Then, using Steiner's formula, the change in surface area can be described as area(φ d ) = (1 2dH(x) + d 2 K(x))dx, φ where K and H denote the Gauss and mean curvature, respectively Discrete case Contrary to the smooth case, nding a suitable curvature denition for a mesh is not as easy. Depending on the chosen part of the mesh (e.g. edges or faces), dierent denitions of curvature are possible. We will rst recall the denition of the Gauss image mesh and then dene the mixed area of two parallel polygon. Using those, we will derive the discrete curvatures related to faces and to edges. Gauss image mesh and discrete surface variation. The Gauss image mesh dened in Ÿ4.3 can be interpreted as the discrete analogue of the smooth Gauss map. Similar to the smooth case, for a mesh M one can dene an oset mesh M d = M + ds, where the Gauss image mesh σ(m) = S provides the normal vectors. 38

40 Chapter 7. Discrete Minimal Surfaces Using the same construction for the variation of surface area as for a smooth surface, [PLBW07] propose a similar denition for a mesh M and its oset M d : area(m d ) = F (1 2dH F + d 2 K F )area(f ), F: faces of M. Within this formula, H F denotes the discrete mean curvature and K F the discrete Gauss curvature of a face F. To describe H F and K F using face properties, we will now introduce the so-called mixed area of two parallel polygons, as described by [BPW09]. Mixed area of parallel polygons. For a polygon P = (p 0,..., p n 1 ) the oriented area can be described using Leibniz' sector formula. The area then is computed as area(p ) = 1 2 det(p i, p i +1 ) 0 i<n with indices modulo n. The so dened area(p ) is a quadratic form and its associated symmetric bilinear form ar ea(p, Q) is dened as area(λp + µq) = λ 2 area(p ) + 2λµ area(p, Q) + µ 2 area(q). Using this denition for a face F of the mesh M and its corresponding face F d M d of the oset mesh at distance d, we get: area(f d ) = area(f + dσ(f )) = area(f ) + 2d area(f, σ(f )) + d 2 area(σ(f )) The term area(f, σ(f )) is called mixed area of F and σ(f ). Face curvatures. Comparing the formula for the area of an oset face using mixed areas with the one for the variation of surface area, we get the following connection between the discrete curvatures and the mixed area of a face and its oset face: area(f d ) = (1 2dH f + d 2 K F ) area(f ) = area(f ) + 2d area(f, σ(f )) + d 2 area(σ(f )) area(f, σ(f )) area(σ(f )) 2 = (1 + 2d + d ) area(f ) area(f ) area(f ) 39

41 Chapter 7. Discrete Minimal Surfaces This leads to the following discrete curvature denitions: area(f, σ(f )) H F = area(f ) area(σ(f )) K F = area(f ) The denitions for the discrete Gauss and mean curvature are similar to the smooth case. Not only is the Gauss curvature dened as the quotient of areas of the Gauss image and the original surface, but also principal curvatures can be dened for most faces, such that H F = (κ 1,F + κ 2,F )/2 and K F = κ 1,F κ 2,F (see Ÿ ). Edge curvatures. A dierent approach is the one proposed by [BPW09], where the curvature terms are not associated with the faces of the mesh, but with the edges. Hence the name. In the edge curvature denition the edge vectors are interpreted as tangent vectors. Let m be a discrete surface with combinatorics (V,E,F). For an edge (i, j) E the edge m i m j of the mesh is parallel to its corresponding edge s i s j of the Gauss image mesh. The edge curvature κ e for an edge e = (i, j) E is then dened as: s j s i = κ ij (m j m i ). For a quadrilateral mesh, it is nonetheless possible to determine the curvatures associated with the face, i.e. the Gauss and the mean curvature. As stated by [BPW09], the face curvatures can be computed as follows: κ 01 κ 23 κ 12 κ 30 H F = κ 01 + κ 23 κ 12 κ 30 κ 01 κ 12 κ 23 κ 30 K F = ( ) κ 01 + κ 23 κ 12 κ 30 κ 12 κ 30 κ 01 κ Zero mean curvature In the smooth setting, a minimal surface can be dened as follows: Necessary condition for a smooth minimal surface. Let f : U R 3 be a surface patch, U R 2 open, U compact with boundary U. For the surface area of f to be smaller than or equal to the surface area of all normal variations of the type f ɛ : U R 3 with f ɛ U = f U 40

42 Chapter 7. Discrete Minimal Surfaces a necessary condition is a vanishing mean curvature H on U, everywhere; [Küh05]. Using the smooth case denition as a model, the denition for a discrete minimal surface would be to have zero (discrete) mean curvature everywhere. In the approach of [PLBW07], zero mean curvature everywhere is equivalent to vanishing mixed area for all faces of the mesh and their corresponding faces of the oset mesh. Figure 7.1 shows an example of two parallel hexagons with vanishing mixed area. Figure 7.1.: Parallel hexagons with vanishing mixed area; [6] The edge curvature approach would state a condition for the edges to ensure zero mean curvature. Since the edge curvature approach is used for quadrilateral meshes and our main focus is on hexagonal meshes, we will not go into more detail about edge curvatures Creating discrete hexagonal minimal surfaces We will describe two dierent ways of creating a hexagonal minimal surface. The rst one is restricted to conformal hexagonal meshes, whereas the second one uses non-convex hexagonal meshes Hexagonal Gauss image via conformal hexagons The method for creating discrete minimal surfaces using conformal hexagons was established by [Mue09]. A discrete (hexagonal) minimal surface is dened to be the dual of a conformal mesh covering the unit sphere. The process starts with a conformal parametrization of a part of the plane and uses stereographic projection to create the Gauss image mesh. The Christoel dual from this newfound Gauss image is then, by denition, a discrete hexagonal minimal surface. 41

43 Chapter 7. Discrete Minimal Surfaces The resulting discrete minimal surfaces of this method are dependent on the used conformal parametrization and do not approximate an underlying smooth surface. [Mue09] explains through dierent examples the relation between the input hex mesh and the resulting discrete hexagonal minimal surface. We will now describe the basic setting that is necessary for the following examples: All examples described here start with a circular hexagonal conformal mesh in C and use the approach described above to retrieve a discrete hexagonal minimal surface. The creation of the conformal hexagonal mesh is done as described in Ÿ After that, a point z C is chosen, such that α(z) z β(z). The mesh α m n β 2n (z) with (m, n) Z 2 is called a derived quad mesh. This quad mesh represents a discrete parametrization of the original conformal hexagonal mesh. Three dierent cases regarding the similarities can be distinguished: (i) Both similarities are translations. (ii) α is a rotation and β is a dilation with the same xed point. (iii) Both, α and β are similarities with the same xed point, but dierent from a pure translation, rotation or dilation. The derived quad mesh (with (m, n) Z 2 ) can be of the following forms depending on the similarity case: (i) m + in ; (ii) e a(m+in) ; (iii) e (a+ib)(m+in). Each case number relates to the appropriate case of similarities. Having those three cases, the meshes discretize the following mappings: z z, z e az and z e (a+ib)z. Next we will give an example for each type. Case 1: z z. In this case the conformal hexagonal mesh consists of regular hexagons. The resulting discrete minimal surface via Christoel duality of the Gauss image is the discrete Enneper's surface. Fig. 7.2 shows the regular hexagonal mesh on the left. The middle image represents the Gauss image mesh created by stereographic projection. And the right image displays the resulting discrete hexagonal minimal surface. 42

44 Chapter 7. Discrete Minimal Surfaces Figure 7.2.: Case 1: the initial mesh (left), the corresponding Gauss image (middle) and the resulting discrete Enneper's surface (right); [5] Case 2: z e az. In case 2 the hexagonal mesh in the beginning is not created from a regular hexagon anymore, but a symmetric hexagon that is Moebius equivalent to a regular hexagon. Applying the similarities of this case to the hexagon leads to a circular conformal mesh with rotational symmetry. This time, the resulting discrete minimal surface is a discrete catenoid. Fig. 7.3 shows the dierent stages of this case. Figure 7.3.: Case 2: the initial mesh (left), the corresponding Gauss image (middle) and the resulting discrete catenoid (right); [5] Case 3: z e (a+ib)z. For the last case, the starting hexagon can be of arbitrary shape, which is not regular but Moebius equivalent to a regular hexagon. Applying the appropriate similarities can lead to dierent discrete minimal surfaces, depending on the choice of a. For a R, it leads again to a catenoid. For a ir this process generates a helicoid. For the helicoid shown in Fig. 7.5 the choice is a = i. Figure 7.4.: Case 3: the initial mesh (left) and the corresponding Gauss image (right); [5] 43

45 Chapter 7. Discrete Minimal Surfaces Figure 7.5.: Case 3: the resulting discrete helicoid; [5] Hexagonal Gauss image via spacial hex centers The following method to create a discrete hexagonal minimal surface stated by [MW00] also uses Christoel duality. Christoel duality. The Christoel duality for this context is described in [MW00], and states that the faces of a polyhedral surface and its parallel Gauss image have vanishing mixed area. They state, that for two parallel quadrilaterals to have vanishing mixed area, the non-corresponding diagonals must also be parallel. They later use this proposition also for hexagons, by splitting a hexagon into two quads and applying the proposition for each quad separately. The method. Starting with an isothermic curvature line parametrization of a smooth minimal surface (for Fig. 7.6 and Fig. 7.7 this would be a smooth Enneper's surface), the parameter domain will be tiled with nonconvex hexagons. Mapping those hexagons onto the surface using the given parametrization leads to a hex mesh with non-planar faces inscribed to the surface. The centers of the planar hexagons are determined and then mapped using the parametrization, which leads to points that can be considered as the centers of the spacial hexagons. During the next step, we construct the Gauss image using those spacial hex centers. For each one, the tangent plane to the smooth surface is be parallel translated to touch the unit sphere. The vertices of the discrete Gauss image are represented by the intersection points of the dierent tangent planes. This procedure leads to a face oset property of the resulting mesh. To gain a vertex or edge oset property, this process changes, since then the vertices or edges, respectively, need to be inscribed (for a vertex oset) or tangent (for an edge oset) to the sphere. 44

46 Chapter 7. Discrete Minimal Surfaces By using optimization during the nal step, the non-planar hex mesh is transformed into the Christoel dual of the discrete Gauss image. The optimization ensures for one, that corresponding faces of the Gauss image and the new mesh have vanishing mixed areas. It also ensures parallelity of corresponding edges. The optimization process itself is done by minimizing a quadratic function. (a) Gauss image Σ 1 (b) Discrete Enneper's surface Φ 1 Figure 7.6.: A Gauss image and the corresponding discrete hexagonal Enneper's surface; [6] Mesh representation. The resulting meshes of this process can be seen in the following gures (Fig. 7.6 and Fig. 7.7). Each of them shows the constructed Gauss image and the corresponding discrete Enneper's surface. The one shown in Fig. 7.6 has no self-intersections. Using the same method as before, but with a dierent mesh size for the Gauss image, we get a dierent discrete Enneper's surface. This surface might have self-intersections, like the one in Fig (a) Gauss image Σ 2 (b) Discrete Enneper's surface Φ 2 Figure 7.7.: Another Gauss image and its corresponding discrete hexagonal Enneper's surface. Created the same way as Fig. 7.6, but with a dierent mesh size for the Gauss image; [6] The presented methods in Ÿ7.2.1 and Ÿ7.2.2 of creating a discrete hexagonal minimal surface are both legitimate with regard to the stated topic of this thesis, as they both create discrete hexagonal minimal surfaces. 45