17TH INTERNATIONAL CONFERENCE ON GEOMETRY AND GRAPHICS 4–8 AUGUST, 2016, BEIJING, CHINA
©2016 ISGG
SHAPE ANALYSIS OF PCCC SHELL STRUCTURE BY GEOMETRICAL FEATURES –A STABLE SHELL STRUCTURE BETWEEN CYLINDER AND PCCP SHELLHirotaka SUZUKI1, Poul Henning KIRKEGAARD2 and Naoki ODAKA1 1 2 Kobe University, Japan Aarhus University, Denmark ABSTRACT: Skew Quadrilateral Elastic Folding paper folding method had been proposed to construct shell structures with non-flat texture in the field of lampshade design. With the methods, final shape is constructed by skew quadrilateral unit with a curved surface. Though the shape of the curved surface in the unit has been considered as almost cylindrical surface, exact shape of the curved surface is not known at now. Purpose of this paper is obtaining approximate solution of the curved surface by geometrical features of the curved surface. At first, authors compared the shell structure with Pseudo-Cylindrical Concave Polyhedral shell and normal cylinder, and measured length of diamond units of the structures. Then authors obtained exact 3D data of the shell structure making use of 3D scanner. Finally, authors concluded that the shape of the shell structure is located between the shape of Pseudo-Cylindrical Concave Polyhedral and normal cylinder, and that the diamond unit in the structure is including skew where the value of the Gaussian curvature is less than zero. Authors also indicated the shell structure made by stainless mesh to show a possibility that the exact shape of the diamond unit is free from property values of materials used for the shell structure. Keywords: Paper Folding, Developable Surface, Skew, Lampshade, Diamond Unit, 3D Scanner. 1. INTRODUCTION In 2014, Suzuki presented a lampshade work, ‘Legato’, at the fall annual conference of the Japan Society for Graphic Science[1] (see Figure 1). The structure of ‘Legato’ is constructed by a paper folding method and covered by non-flat diamond units. Luminous distribution on the surface of ‘Legato’ generated by transmitting light is continuous within the diamond unit as shown in the figure. Suzuki named the paper folding method as ‘Skew Quadrilateral Elastic Folding’ (hereafter SQEF) and showed actual luminance distribution measured by digital camera and proposed several variations of SQEF folding method[2] (see Figure 2). Though the shape of the curved surface in the unit has been considered as almost cylindrical surface, exact shape of the curved surface is not
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known at now. Purpose of this paper is obtaining approximate solution of the curved surface
Figure 1: A lamp shade work ‘Legato’. by geometrical features of the curved surface. Making use of length of the diamond unit and 3D shape of the diamond scanned by 3D scanner, authors obtained a clue of the actual shape
of curved diamond. 2. PRINCIPLE OF SQEF METHOD AND RELATED WORKS The shape constructed by SQEF method is based on the shape called ‘PCCP (Pseudo-Cylindrical Concave Polyhedral)’ shell, Yoshimura pattern or Diamond pattern. As shown in Figure 3, development of cylindrical PCCP shell has two sets of oblique parallel lines at even intervals for mountain fold and one set of horizontal parallel lines at even intervals for valley fold. Cylindrical shape shown in Figure 4 appears after folding the development along with the lines drawn in the development and bending the development to attach bilateral sides without gap and overlap. PCCP shell is one of the most famous shapes constructed by paper folding methods. Yoshimura pointed out similarity between shape of PCCP shell and shape made by buckling of cylindrical shell with compressive power[3]. Miura named the shape as Pseudo-Cylindrical Concave Polyhedral and surveyed several features of PCCP shell[4]. PCCP shell has been applied to structure of building roof[5] and shape of cans for beverage. Besides cylindrical shape, dome type shape had been proposed as well[6]. In case of PCCP structure, each flat diamond on the development of PCCP structure changes its shape to skew quadrilateral unit and the unit is constructed by two planer triangles. Consequently, luminance distribution in the area of the unit is discontinuous at boundary of the two triangles. As shown in Figure 1, similar shape appears after folding the development shown in Figure 3 along with only lines for mountain fold and bending it. Suzuki named the folding method as ‘Skew Quadrilateral Elastic Folding’[2] and authors name the cylindrical shape constructed by SQEF as Pseudo-Cylindrical Concave Curves (hereafter, PCCC) shell. In case of PCCC structure, each skew quadrilateral is constructed by continuous curved surface and illuminance distribution is continuous within
Figure 2: An example of development of conical SQEF (left) and the shape made from the development (right) (After bending, the top and the bottom of the conical shape are cut off horizontally).
Figure 3: A development of PCCP shell (Black horizontal lines for mountain fold and grey oblique lines for valley fold).
Figure 4: An example of PCCP shell. area of the unit quadrilateral as shown in Figure 1. From the viewpoint of shape, difference between two shapes is trivial, however, from the viewpoint of illuminance distribution, difference between two shapes is significant. 2
Developable Surface
SQEF method has relation to researches about developable surface. As shown in Figure 5, curved surfaces are classified into four classes from viewpoint of differential geometry[7]. At first, all curved surfaces are classified into ruled surface and double curved surface. Ruled surface can be described as locus of a straight line movement. Double curved surface, including sphere and torus, cannot be described such. Double curved surface is not developable and cannot be made from plane surface. Ruled surface is classified into developable surface and warped surface. Developable surface is including conical surface, cylindrical surface and tangent surface, and can be made from plane surface with bending. Warped surface is including hyperboloid of revolution, hyperbolic paraboloid and helicoid. Manufacturing of developable surfaces is very easy, though designing of the surfaces have much restrictions. Therefore, a lot of researches have been conducted to increase freedom of shape design. These researches are classified into two categories, research which enable approximation of given curved surface with combination of developable surfaces and research which enable easy designing. As researches for approximation, Mitani et al. proposed the method of approximation with plane triangles[8], Shatz et al. proposed the method of approximation with conical surfaces and planes[9], Massarwi et al. proposed the method of approximation with tubes which are constructed by triangles[10], Pottman et al. proposed the method of approximation with developable strips[11] and Mitani proposed manufacturing method of approximated solid of revolution by paper folding[12]. As researches for easy designing, Rose et al. proposed the method of three dimensional shape generation from two dimensional perspective drawing[13], Kilian et al. proposed
Ruled Surface
Conical Surface, Cylindrical Surface Tangent Surface Warped Surface e.g. Hyperboloid of Revolution, Hyperbolic Paraboloid, Helicoid
Curved Surface
Surface of Revolution Double Curved Surface
e.g. Sphere, Torus Double Curved Surface in General
Figure 5: Classification of Curved Surface from viewpoint of Differential Geometry. method of shape generation with repetition of curved line folding[14], Suzuki proposed method of shape generation with combination of tangent surfaces[15] and Suzuki extended the tangent surface method with hermite curve[16]. And Suzuki proposed method of shape generation with manipulation of curved line to generate connected developable surfaces[17], Suzuki implemented the design method on CG freeware POV-Ray making use of affine transformation and locus diagram[18] and Suzuki et al. introduced the curved line manipulation method into graphic science education for designing and manufacturing of lampshade to evaluate proposed method and developed interface[19]. The method proposing in this paper is considered as a research for easy designing and designing with proposed method is far easier comparing to existing researches though freedom of designing with the method is smaller. And lampshade designing is appropriate subject to learn relationship between shape and light in the field of graphic science education. Suzuki introduced lampshade design assignment into graphic science education course[20] and Suzuki introduced paper folding lampshade design assignment into the course[21]. The proposed paper folding method has high potential for assignment of descriptive geometry as well.
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To get a clue of the actual shape of the curved diamond unit in PCCC shell structure, authors make use of length of the diamond unit and 3D model of the diamond unit scanned by 3D scanner. The examinations with length and shape are explained in following chapters. 3. EXAMINATION WITH LENGTH OF DIAMOND UNIT 3.1 Fabrication of structures To examine actual shape of PCCC structure, PCCC shell structure and 2 similar structures, PCCP shell and cylinder were fabricated with the same material. The material was white woodfree paper (70kg for 788mm x 1091mm x 1000 sheets) cut into 528mm x 384mm. For development of PCCC and PCCP, 36 diamonds (88mm x 64mm) were drawn on the development as shown in Figure 6. From the development, PCCC shell, PCCP shell and cylinder were fabricated as shown in Figure 7.
Figure 6: Development for PCCP shell. (Black horizontal lines for mountain fold and grey oblique lines for valley fold. Without valley fold line, development for PCCC can be obtain.)
3.2 Examination with measurement of length of diamond unit and calculation of theoretical length To make sure actual shape of the diamond unit, length of the unit was measured, and theoretical length was calculated if possible. At first, height and width of diamond units in PCCC shell and PCCP shell were measured. In case of PCCC shell and PCCP shell, shapes of diamond units located at top and bottom were not stable as shown in Figure 7. Considering stability of the shapes, width of 6 diamond units located at middle level (level 3.5 in Figure 8) was measured and average length of them were adopted as width of diamond. As for height of diamond, height for 3 units (from level 2.5 to level 4.5 in Figure 8) was measured at 6 columns, and average length of them were adopted as height of diamond. Table 1 shows average and variance of measured length (width and height of diamond unit) of PCCC shell and PCCP shell. As shown in Figure 9, one level structure of PCCP shell shown in Figure 8 can be consi-
Figure 7: Fabricated PCCP shell (left), PCCC shell (center) and Cylinder (right).
Figure 8: Levels for measurement of unit diamonds. 4
dered as layered 2 regular hexagonal antiprisms. Therefore, width of a diamond unit in PCCP shell should be same as width of a diamond shell in development of PCCP shell and the theoretical length of width was obtained as 88.00mm. In a similar way, theoretical length of the height for a half level of PCCP shell can be considered by the height of right angled triangle described in Figure 10 and can be obtained by calculation. These theoretical values for a diamond in PCCP shell are also described in the Table 1. As shown in the Table 1, measured value and theoretical value are different due to measurement error and an effect of thickness of used paper. Calculation of the t-test using these values confirmed that difference between length of height of the diamond unit in PCCC shell and that in PCCP shell is statistically significant at the 1percent level. As for width of a diamond unit, significant difference was not observed. As shown in Figure 11, if cylinder is fabricated from the development for PCCC shell without any creases, the height of a diamond unit in the cylinder should be the same as height of diamond unit in the development and the actual length of the height is 64mm. Finally, following relationship among the height of PCCP, PCCC and cylinder can be obtained.
Table 1: Length (width and height) of diamond units in PCCP shell and PCCC shell.
PCCP PCCC
Width of diamond units (mm)
Height of diamond units (mm)
Average (Theoretical value)
Variance
Average (Theoretical value)
Variance
88.42 (88.00) 88.50 (-)
0.201
59.33 (59.50) 60.44 (-)
0.019
0.250
0.108
Figure 9: One level structure of PCCP shell as layered 2 regular hexagonal antiprisms.
PCCP 59.33mm < PCCC 60.44mm < Cylinder 64mm (1)
Figure 10: Relation of the length of height of a half level of PCCP shell and other length.
As the Equation (1) indicated, the shape of PCCC shell can be located between PCCP shell and cylinder. 3.3 Examination with vertical section lines at the center of diamond units To make sure the difference between the shape of a diamond unit in PCCC shell and that in PCCP shell in detail, actual shapes were compared using vertical section lines at the center of diamond units. Figure 12 shows a photograph of the vertical section of diamond unit in PCCC shell and approximated vertical section line using cubic spline segments. Theoretical section of diamond unit in PCCP shell is also
Figure 11: Fabricated cylinder with development for PCCC shell. 5
described in Figure 12. As shown in the figure, the absolute difference is quite smaller though the difference is relatively larger at the center of the top and bottom. When these shapes are used for lampshade as shown in Figure 7, difference of visual impression receiving from these lampshades may be quite larger, however, the visual difference is not based on difference of shapes but on continuity of luminance distribution in the diamond unit. 4. EXAMINATION WITH 3D MODEL OF DIAMOND UNIT SCANNED BY 3D SCANNER 4.1 Construction of 3D model of a diamond unit in PCCC shell Using the same PCCC shell described in previous chapter, 3D model of whole PCCC shell was obtained using 3D scanner as shown in Figure 13. The number of polygons for whole structure was 2,104,070. From the model, polygons included in a unit diamond located at the center of the top and bottom, and at opposite of paper pasting part were extracted considering stability of the shape. The number of polygons extracted was 33,800. The format of data file of these polygons was converted from ASCII STL format to POV-Ray format and position and direction of polygons were arranged from the position of 4 vertices of the diamond unit as shown in Figure 14. 4.2 Examination of a diamond unit in PCCC shell using 3D model of the unit As shown in Figure 12, shape of a diamond unit of PCCC shell is almost cylindrical surface and generating line for the cylindrical surface can be approximated by two straight line segments and one curved line segment. As shown in Figure 15, two line segments were decided using dihedral angle of two triangles in a unit of PCCP shell, and the intersection part of two lines was replaced by a circular arc. Figure 16 shows comparison of a surface of the diamond unit of PCCC shell with translated generating lines. As shown in the figure, the shape of the unit diamond of PCCC shell is
Figure 12: Horizontal section lines at the center of diamond units.
Figure 13: 3D model of whole PCCC shell.
Figure 14: 3D model of diamond unit. 6
quite similar to cylindrical surface described by single generating line, however, the shape of the unit is bending to back side at the parts close to the left and right ends of the unit. At the parts, the value of the Gaussian curvature is less than zero and this fact shows that the shape of the diamond unit cannot be a cylindrical surface as the shape includes skew at the points close to the left and right ends. 5. CONSTRUCTION OF PCCC SHELL USING METAL MESH Other than papers, metal mesh seats and plastic sheets were adopted as materials for fabrication of PCCC shell. Considering skew included in curved surface in a diamond unit, materials for PCCC shell should be sufficiently soft to allow skew and sufficiently stiff to hold skew within a diamond unit. Figure 17 shows an example of PCCC shell structure made by stainless mesh. Though property values of stainless mesh are quite different from that of papers, shapes of the unit diamonds made by stainless mesh and made by paper are quite similar. There may be possibility that the exact shape of diamond units in PCCC shell structure is free from property values of materials used for the shell structure.
Figure 15: Approximation of generating line for a diamond unit of PCCC shell.
Figure 16: Comparison of a surface of a diamond unit of PCCC shell and translated generating lines.
6. CONCLUSIONS To make clear the shape of curved surface of diamond unit in PCCC shell, authors compared the PCCC shell structure with PCCP shell structure and normal cylinder, and measured length of diamond units of the structures. From the examination, authors concluded that the shape of PCCC shell is almost cylindrical surface and the shape is located between PCCP shell and normal cylinder. Then authors obtained exact 3D data of the shell structure making use of 3D printer to compare the shape of the diamond unit with translated generating lines. From the examination, authors concluded that the shape of the diamond unit is including skew where the value of the Gaussian curvature is less than zero.
Figure 17: PCCC shell by stainless mesh. 7
[8] Mitani, J., Suzuki, H.. Making Papercraft Toys from Meshes using Strip-based Approximate Unfolding, ACM Trans., Graphics, 23(3), 259-263, (2004).
Authors also indicated the shell structure made by stainless mesh to show a possibility that the exact shape of the diamond unit is free from property value of materials used for the shell structure. Author would like to adopt spring model examination to approximate whole curved surface in the diamond unit in future.
[9] Shatz, I., Tal, A., Leifman, G.. Paper craft models from meshes, The Visual Computer: International Journal of Computer Graphics archive, Vol. 22 Issue 9, 2006: 825-834 .
ACKNOWLEDGMENTS This research activity was supported by research grant of Union Foundation for Ergodesign Culture.
[10] Massarwi, F ., Gotsman, C., Elber, G.. Papercraft Models Using Generalized Cylinders, Proceedings of Pacific Graphics, 2007: 148-157.
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[11] Pottmann, H., Schiftner, A., Bo P., Schmiedhofer, H., Wang, W., Baldassini, N ., Wallner, J.. Freeform surfaces from single curved panels, ACM Trans. Graphics, 27(3), 2008: Article No.76.
[2] Suzuki, H.. A PROPOSAL OF PAPER FOLDING METHOD FOR LAMPSHADE DESIGN -PRINCIPLE AND APPLICATION OF SKEW QUADRILATERAL ELASTIC FOLDING METHOD. Proceedings of The 10th Asian Forum on Graphic Science, 2015: F23.
[12] Mitani, J.. A Design Method for 3D Origami Based on Rotational Sweep, Computer-Aided Design and Applications, 6(1), 2009: 69-79. [13] Rose, K., Sheffer, A., Wither, J., Cani, M., Thibert, B.. Developable Surfaces from Arbitrary Sketched Boundaries, Eurographics Symposium on Geometry Processing, 2007: 163-172.
[3] Yoshimura, Y.. On the Mechanism of Buckling of a Circular Cylindrical Shell under Axial Compression. Reports of The Institute of Science and Technology (The University of Tokyo), Vol. 5, No.5, 1951: 179-198 (in Japanese).
[14] Kilian, M., Flory, S., Chen Z., Mitra N. J., Sheffer, A., Pottmann, H.. Curved Folding, ACM Trans. Graphics, 27(3), 2008: Article No.75.
[4] Miura, K.. Proposition of Pseudo-Cylindrical Concave Polyhedral Shells. Institute of Space and Aeronautical Science University of Tokyo Report No. 442, 1969: 141-163.
[15] Suzuki, H.. Application of tangent surfaces in the design of lampshades, Proc. of the Fall Annual Conference of the Japan Society for Graphics Science, 2010: 133-136 (in Japanese).
[5] Salvadori, M.. BUILDING: The Fight Against Gravity, Atheneum, 1979.
[16] Suzuki, H.. DESIGNING OF LIGHTING EQUIPMENT MAKING USE OF TANGENT SURFACE AND CONTROL METHOD OF THE SURFACE BY HERMITE CURVE, Proc. of 15th Intl. Conference on Geometry and Graphics, in 2012: CDROM.
[6] Miyazaki, K.. Encyclopaedia of Building Shape, Shokokusha Publishing, 2000 (in Japanese). [7] Isoda, H., Suzuki, K.. A Guide of Descriptive Geometry, University of Tokyo Press, 1986 (in Japanese).
[17] Suzuki, H.. A generation method of developable surface by manipulation (translation, 8
scaling and rotation) of curved line and a designing method of complicated shapes by combination of generated developable surfaces, Journal of Graphic Science of Japan, Vol. 48 Issue 1, 2014, 3-10 (in Japanese).
ate School of Human development and environment, Kobe University.
[18] Suzuki, H.. CONTROL METHOD OF COMBINED DEVELOPABLE SURFACE DESIGN BY AFFINE TRANSFORMATION AND LOCUS, Proc. of 16th Intl. Conference on Geometry and Graphics, 2014: in CDROM. [19] Suzuki, H.. Sakaki, A., Yasufuku, K., Matsumoto, T.. Designing of lampshade with 3D CG application and manufacturing of designed shape in graphic science education, Int. J. of Computer Applications in Technology, Vol.51, No.1, 9-14, (2015). [20] Suzuki, H.. A STUDY ON IMPACT OF INTRODUCTION OF LIGHTING EQUIPMENT DESIGN ASSIGNMENT INTO GRAPHIC SCIENCE EDUCATION, Proc. of 12th Intl. Conference on Geometry and Graphics, 2006: in CDROM. [21] Suzuki, H.. Education Course of Light and Shape Designing making use of Folded Papers, Proc. of the Fall Annul Conference of the Japan Society for Graphics Science, 2011: 143-144 (in Japanese). ABOUT THE AUTHORS 1. Hirotaka SUZUKI, Dr. Eng., is an associate professor of Department of Architecture, Graduate School of Engineering, Kobe University. His research interests are Lighting Environment Simulation, Geometrical Design and Graphic Science Education. He can be reached by e-mail:
[email protected], or through postal address: 1-1, Rokkodai-cho, Nada-ku, Kobe City, 657-8501, JAPAN. 2. Poul Henning KIRKEGAARD, PhD., is a professor of Department of Engineering, Aarhus University. 3. Naoki ODAKA, Dr. Eng., in a professor of Department of Human Development, Gradu9
