ergun akleman, ozan ozener and cem yuksel- designing symmetric high-genus sculptures

8/3/2019 Ergun Akleman, Ozan Ozener and Cem Yuksel- Designing Symmetric High-Genus Sculptures

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Designing Symmetric High-Genus Sculptures

ERGUN AKLEMAN

Visualization Sciences Program

OZA N OZENER

Architecture Department

CEM YUKSEL

Visualization Sciences Program

Texas A&M University

Abstract

In this paper, we present a procedure to construct a new family ofsymmetric high-genus sculptures. The procedure is not rigid andprovides users a creative flexibility. With this procedure, users caneasily create a wide variety of sculptures that still have a similarconceptual form.

1 Introduction

In this paper, we present a procedure to create a new sculpturalfamily with interactive topological modeling. Using this procedurea large set of sculptures that have a similar conceptual form can eas-

ily be created (see Figures 1 and 2). We have tested the procedure ina computer aided sculpting course [Akleman 2005b]. We observethat, using the procedure, students can rapidly create a wide vari-ety of shape. Although these shapes are completely different; theyare indistinguishably belong the same family. Figure 3 shows someexamples of shapes that were created by some of our students us-ing our procedure, as one of the biweekly assignments of computeraided sculpting course.

Figure 1: Photographs of a 3D print of one of the first symmetrichigh genus shapes designed by Ergun Akleman. The same concep-tual procedure is used to create shapes in Figure 3. This shape ismade from ABS plastic and printed using a Fused Deposition Ma-chine (FDM).

Corresponding Author: Program Coordinator, Visualization Sciences,

Department of Architecture, College of Architecture. Address: 418

Langford Center C, College Station, Texas 77843-3137. email: er-

[email protected]. phone: +(979) 845-6599. fax: +(979) 845-4491.

Figure 2: Three high-genus symmetric sculptures constructed byErgun Akleman for the identification of the procedure. Images ofthese motivating examples are rendered by Ozan Ozener. Each rowshows two views of the same shape. The ribbon shape at the toprow is really a genus-1 surface, but, during the construction of thissurface, genus changed several times.

2 Motivation

Development of new methods to create new aesthetic forms areextremely crucial for artistic applications. In computer graphics

and mathematics, it is very common to find methods to create aes-thetically pleasing results which has never been done before. Al-though, most researchers view their results as byproducts or proof-of-concepts, those results are usually very aesthetic in nature andcan be considered artworks.

In the authors opinion, Computer graphic research will eventu-ally be viewed as the major art movement of our age, since themethods developed by researchers are not only reproducible butalso they allow a wide variety of results to be produced. For in-stance, the fractal methods to create Julia sets are widely used bynon-mathematicians to create artworks [Akleman 2005a].


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Kashyap VirajBhimjiani Hankare

Gracie JulieArenas Gele

Elizabeth ElizabethNitch Nitch

Figure 3: Rendered examples of high genus symmetric sculpturesdesigned by students who take a computer aided sculpture course.Although, these images are created by seven different students;shapes and rendering styles are completely different, all the virtualsculptures are clearly from the same family.

In solid and shape modeling research, we do not usually have mo-tivation coming from aesthetic goals. Our motivations are drivenfrom engineering modeling and aesthetics has never been a majorconcern. However, identification of methods to create new aesthet-ically pleasing sculptural forms can also be a major direction forsolid and shape modeling research. And, in fact, one of the mostexciting aspects of sculpting has always been the development of

new methods to design and construct unusual, interesting, and aes-thetically pleasing shapes. One of the most interesting sculpturalshapes are high-genus surfaces [Takahashi et al. 1997; Welch andWitkin 1994; Ferguson et al. 1992; Srinivasan et al. 2002; Aklemanet al. 2003].

With the advance of computer graphics, many artists have also be-gun to use mathematics as a tool to create revolutionary forms ofsculptures. There are many contemporary sculptors such as GeorgeHart [Hart 2005], Helaman Ferguson [Ferguson 2005; Fergusonet al. 1992], Bathseba Grossman [Grossman 2005], Brent Collins,and Carlo Sequin [Sequin 2005] who successfully combine art and

mathematics to create unusual and high-genus sculptures. Thesemathematical sculptors, who have a very noticeable presence in to-days art scene, develop their own methods to model, prototype andfabricate an extraordinary variety of shapes. Most of these con-temporary sculptors who successfully combine art and mathematicsare, in fact, mathematicians or engineers who have developed theirown methods. Moreover, each individual sculpted piece sometimesrequires a different approach to be developed.

Our goal in this work is fundamentally different from mathematicalsculptors. Similar to computer graphics research, we want our re-sults to be reproducible. We also want to provide creative flexibilityto users for constructing a wide variety of sculptures that can stillbe recognizable as belonging the same family.

On the other hand, unlike classical computer graphic applicationswe want our resulting shapes to be physically realizable. In otherwords, we should be able to 3D print the resulting shapes. In thispaper, we present a procedure that satisfy all the goals to create newaesthetically pleasing sculptural forms.

Using the procedure it is possible to create a wide variety ofshapes that does not resemble any specific sculptural or archi-tectural form.

On the other hand, despite the variety all the shapes that are

created using the same procedure are still recognizable as be-longing the same family.

The resulting shapes can be physically realizable with a min-imal effort from user as shown in Figure 1.

3 Procedure

The procedure is based on a set of topological mesh modeling oper-ators. These operators guarantee that the resulting shape is topolog-ically 2-manifold. If the user avoid self-intersection, which is easyto achieve, the resulting models are 3D-printer-ready, i.e. they canbe printed by using a rapid prototyping machine as shown above.

Our procedure consists of six stages.

Stage 0: Initial Shape. Start with a symmetric convex poly-hedral shape [Cromwell 1997; Grunbaum and Shephard 1987;Wells 1991]. Any platonic solid such as tetrahedron, cube ordodecahedron [Stewart 1991; Williams 1972] are perfect can-didates for starting shape. In the example shown in Figure 4,the initial shape is a cube.

Stage 1: Extrusions. Apply the same extrusion operation toall faces of starting shape. Adding a twist or rotation to the ex-trusions can create an additional effect as shown in Figure 4.Moreover, the recently introduced local mesh operators canalso be used in this stage. These local operators can extrudegeneralized platonic solids and are called tetrahedral, cubical,octahedral, dodecahedral and icosahedral extrusions [Landre-

neau et al. 2005].

Stage 2: Creating Handles. Connect symmetrically relatedfaces using multi-segment curved handles [Srinivasan et al.2002]. This operations, along with cubical extrusions, pop-ulate the mesh with quadrilateral faces with 4-valent verticesas shown in Figure 4. After the operation, there will be onlya handful of extraordinary points (i.e. vertices with valenceother than 4 and faces with sides other than 4).

Stage 3: Doo-Sabin Smoothing. Apply Doo-Sabin subdivi-sion once [Doo and Sabin 1978]. This operation will create


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Stage 0: Stage 1: Stage 2:Initial Shape Extrusions Handle Creation

Stage 3: Stage 4: Stage 5:Doo-Sabin Rind Smoothing with

Subdivision Modeling Catmull-Clark

Figure 4: The Procedure.

visibly connected clusters of faces. These cluster of faces areformed as a result of extraordinary points. The faces in eachcluster defines a route that connect one extraordinary pointinto another. These clusters can easily be seen in Figure 4.

Stage 4: Rind Modeling. Rind modeling creates a rind struc-ture by creating a smaller replica of initial shape [Aklemanet al. 2003]. In rind modeling, users can punch holes inthis ring shape by clicking the faces. There exist two strate-gies: (1) Deleting all non-cluster faces by making the clustersclearly visible, (2) Deleting all cluster faces by making thenon-cluster structures visible. In the example shown in Fig-ure 4 we made clusters visible.

Stage 5: Final Smoothing. Final smoothing is useful to cre-ate a simple and smooth surface. Although, in final smoothing

any subdivision scheme can be used, we use Doo-Sabin [Dooand Sabin 1978] or Catmull-Clark [Catmull and Clark 1978]schemes for smoothing.

Using both stages 2 and 3 is not really required. It is even possibleto skip either one of them to create different results. As it can beseen in attached examples it is possible to create a wide variety ofshapes using this procedure.

4 Implementation

The operations used all six stages have already been imple-

mented and included in our existing 2-manifold mesh model-ing system, called TopMod [Akleman and Chen 1999; E. Ak-leman and Srinivasan 2003; Akleman et al. 2003; Srini-vasan et al. 2002]. Our system is implemented in C++ andOpenGL. All the examples in this paper were created usingthis system. You can download TopMod from http://www-viz.tamu.edu/faculty/ergun/research/topology/ for and use the soft-ware for non-commercial applications. TopMod provides onlyOpen-GL based interactive rendering. For high quality rendering,shapes can be exported in obj format and rendered in some 3D mod-eling and animation system. Our students use Maya and 3D StudioMax for rendering the final models.

5 Results

Figures 5, 6 and 7 show three students works that illustrate creativeflexibility that is provided by the procedure. Each figure show thesteps that is used by students and final virtual sculptures. Figure 5shows Cem Yuksel approach. Cem by adding handles after com-pleting the basic procedure created two forms that challenge with

each other. Lauren Simpson, as shown in Figure 6, added tetrahe-dral extrusion, which is stellation operation, before adding handles.This particular step allowed her to create handles with triangularcross-sections. Finally, rind modeling on handles with triangularcross-sections provided a simple and elegant connections. AudreyWells work shown in Figure 7 is based on octahedral extrusionsthat also allow her to create handles with both quadrilateral (thicker)and triangular (thinner) cross-sections.

Initial Shape: Doo-Sabin CubicalCube Subdivision Extrusions

Add Doo-Sabin Rindhandle Subdivision Modeling

Again Catmull-Clark SubdivisionAdd handle Subdivision Surface

Rendered Model with Another Model created withSubsurface Scattering the same procedure

starting from Icosahedron

Figure 5: The Procedure used by Cem Yuksel and final virtualsculptures. Both rendered using subsurface scattering to give anillusion of ABS plastic.


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Initial Shape: Cubical TetrahedralTetrahedron Extrusions Extrusions

Add Doo-Sabin Rindhandles Subdivision Modeling

Rendering with Rendering withDiffuse material Specular material

Figure 6: The Procedure used by Lauren Simpson.

6 Conclusions and Future Work

Sculpting and Architecture can provide major research directionsfor solid and shape modeling. In both, the precision, which isvery important for engineering shape design, is not a major con-cern. Both architecture and sculpture can provide us interestingproblems coming from aesthetics concerns. Identification of meth-ods to create new aesthetically pleasing sculptural and architecturalforms can eventually be a major direction for solid and shape mod-eling research.

In this work, we have introduced a new sculptural approach with amotivation coming from strong aesthetics concerns. Similar to re-sults from computer graphics research, our results are reproducibleand allow even novice users to create interesting sculptures with acreative flexibility. Because of their strong symmetry, these shapescan possibly be built using a few building blocks. We are currently

investigating physical construction of large versions (Medium sizeshapes such as ones larger than 1m3) of such complicated shapesusing low-cost materials such as concrete.

References

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Initial shape: Octahedral Add

Tetrahedron extrusions handles

Add smaller Doo-Sabin Final Smoothinghandles and Rind with Doo-Sabin

Rendered Another ModelModel created with

the same procedure

Figure 7: The Procedure used by Audrey Wells.

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ergun akleman, ozan ozener and cem yuksel- designing symmetric high-genus sculptures

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