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SHAPE ANALYSIS OF PCCC SHELL STRUCTURE BY GEOMETRICAL FEATURES –A STABLE SHELL STRUCTURE BETWEEN CYLINDER AND PCCP SHELL

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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.
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17TH INTERNATIONAL CONFERENCE ON GEOMETRY AND GRAPHICS ©2016 ISGG
4–8 AUGUST, 2016, BEIJING, CHINA
Paper #00
SHAPE ANALYSIS OF PCCC SHELL STRUCTURE BY
GEOMETRICAL FEATURES
A STABLE SHELL STRUCTURE BETWEEN CYLINDER AND PCCP
SHELL-
Hirotaka SUZUKI1, Poul Henning KIRKEGAARD2 and Naoki ODAKA1
1Kobe University, Japan 2Aarhus University, Denmark
ABSTRACT: Skew Quadrilateral Elastic Folding paper folding method had been proposed to con-
struct 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 di-
amond 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 Fig-
ure 1). The structure of Legatois constructed
by a paper folding method and covered by
non-flat diamond units. Luminous distribution
on the surface of Legatogenerated by trans-
mitting 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 dig-
ital 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
known at now. Purpose of this paper is obtain-
ing approximate solution of the curved surface
by geometrical features of the curved surface.
Making use of length of the diamond unit and
3D shape of the diamond scanned by 3D scan-
ner, authors obtained a clue of the actual shape
Figure 1: A lamp shade work ‘Legato’.
2
of curved diamond.
2. PRINCIPLE OF SQEF METHOD AND
RELATED WORKS
The shape constructed by SQEF method is
based on the shape called PCCP (Pseu-
do-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 in-
tervals for valley fold. Cylindrical shape shown
in Figure 4 appears after folding the develop-
ment along with the lines drawn in the devel-
opment 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. Yoshi-
mura pointed out similarity between shape of
PCCP shell and shape made by buckling of cy-
lindrical shell with compressive power[3]. Mi-
ura named the shape as Pseudo-Cylindrical
Concave Polyhedral and surveyed several fea-
tures 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 develop-
ment of PCCP structure changes its shape to
skew quadrilateral unit and the unit is con-
structed 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
area of the unit quadrilateral as shown in Figure
1. From the viewpoint of shape, difference be-
tween two shapes is trivial, however, from the
viewpoint of illuminance distribution, differ-
ence between two shapes is significant.
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.
3
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 geome-
try[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 in-
cluding conical surface, cylindrical surface and
tangent surface, and can be made from plane
surface with bending. Warped surface is in-
cluding 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 re-
searches have been conducted to increase free-
dom 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 pers-
pective drawing[13], Kilian et al. proposed
method of shape generation with repetition of
curved line folding[14], Suzuki proposed me-
thod 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 genera-
tion with manipulation of curved line to gener-
ate connected developable surfaces[17], Suzuki
implemented the design method on CG free-
ware POV-Ray making use of affine transfor-
mation and locus diagram[18] and Suzuki et al.
introduced the curved line manipulation me-
thod into graphic science education for design-
ing and manufacturing of lampshade to eva-
luate proposed method and developed inter-
face[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 free-
dom 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 assign-
ment 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 poten-
tial for assignment of descriptive geometry as
well.
Curved
Surface
Ruled
Surface
Double
Curved
Surface
Developable Surface
Warped Surface
Surface of Revolution
Double Curved Surface in General
Conical Surface, Cylindrical Surface
Tangent Surface
e.g. Hyperboloid of Revolution,
Hyperbolic Paraboloid, Helicoid
e.g. Sphere, Torus
Figure 5: Classification of Curved Surface
from viewpoint of Differential Geometry.
4
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 develop-
ment as shown in Figure 6. From the develop-
ment, PCCC shell, PCCP shell and cylinder
were fabricated as shown in Figure 7.
3.2 Examination with measurement of length
of diamond unit and calculation of theoreti-
cal 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 di-
amond units located at top and bottom were not
stable as shown in Figure 7. Considering stabil-
ity of the shapes, width of 6 diamond units lo-
cated 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 col-
umns, 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 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.)
Figure 7: Fabricated PCCP shell (left),
PCCC shell (center) and Cylinder (right).
Figure 8: Levels for measurement of unit
diamonds.
5
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 tri-
angle described in Figure 10 and can be ob-
tained by calculation. These theoretical values
for a diamond in PCCP shell are also described
in the Table 1. As shown in the Table 1, meas-
ured 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 differ-
ence 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 differ-
ence 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 devel-
opment and the actual length of the height is
64mm. Finally, following relationship among
the height of PCCP, PCCC and cylinder can be
obtained.
PCCP 59.33mm < PCCC 60.44mm < Cylinder 64mm (1)
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 com-
pared using vertical section lines at the center
of diamond units. Figure 12 shows a photo-
graph 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
Table 1: Length (width and height) of di-
amond units in PCCP shell and PCCC shell.
Width of diamond units
(mm)
Height of diamond units
(mm)
Average
(Theoretical
value)
Variance
Average
(Theoretical
value)
Variance
88.42
(88.00)
0.201
59.33
(59.50)
0.019
88.50
( - )
0.250
60.44
( - )
0.108
Figure 9: One level structure of PCCP shell
as layered 2 regular hexagonal antiprisms.
Figure 10: Relation of the length of height
of a half level of PCCP shell and other
length.
Figure 11: Fabricated cylinder with
development for PCCC shell.
6
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 distribu-
tion 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 pre-
vious 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, po-
lygons 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 Fig-
ure 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.
7
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 dif-
ferent from that of papers, shapes of the unit
diamonds made by stainless mesh and made by
paper are quite similar. There may be possibili-
ty that the exact shape of diamond units in
PCCC shell structure is free from property val-
ues of materials used for the shell structure.
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 sur-
face and the shape is located between PCCP
shell and normal cylinder. Then authors ob-
tained 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 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.
Figure 17: PCCC shell by stainless mesh.
8
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 sur-
face in the diamond unit in future.
ACKNOWLEDGMENTS
This research activity was supported by re-
search grant of Union Foundation for Ergode-
sign Culture.
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ABOUT THE AUTHORS
1. Hirotaka SUZUKI, Dr. Eng., is an asso-
ciate professor of Department of Architecture,
Graduate School of Engineering, Kobe Univer-
sity. His research interests are Lighting Envi-
ronment Simulation, Geometrical Design and
Graphic Science Education. He can be reached
by e-mail: hirotakasuzuki@people.kobe-u.ac.jp,
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, Aar-
hus University.
3. Naoki ODAKA, Dr. Eng., in a professor
of Department of Human Development, Gradu-
ate School of Human development and envi-
ronment, Kobe University.
... As for the shape of PCCC shell structure, Suzuki et al. concluded that exact shape of diamond unit in the shell is foldable but not developable by qualitative analysis making use of 3D scanner (22) . And Okamura et al. and Odaka et al. conducted psychological evaluation test for PCCC lampshade respectively (23) (24) . ...
Conference Paper
Full-text available
In the field of lampshade design, paper folding method has been highly utilized as the non-flat texture by paper folding method can generate beautiful luminance distribution. And a method combining non-flat pattern and geometrical transmittance pattern has been already proposed to overlap luminance distribution caused by non-flat texture and that by transmittance pattern. In this paper, dissection is utilized to produce two plane tessellations. Dissection is a method to produce new shape or shapes from original shape by cutting original shape into plural pieces. Dissection can be expanded to plane tessellation to obtain two different tessellations. From two plane tessellations, development of a paper folding lampshade was obtained. And finally, the lampshade was manufactured by folding and bending of the development. In this paper, application of proposed method is explained. And interpretation of existing layered plane tessellations, which authors had been already proposed, in terms of dissection is explained as well.
... After folding the development shown in Figure 2 along with only lines for mountain fold and bending it, the shape shown in Figure 4 appears. Suzuki named the folding method as 'Skew Quadrilat- eral Elastic Folding' [6] and Suzuki et al. named the cylindrical shape constructed by SQEF as Pseudo-Cylindrical Concave Curves (hereafter, PCCC) shell [7]. In case of PCCC structure, each skew quadrilateral is constructed by con- tinuous curved surface and illuminance distri- bution is continuous within area of the unit quadrilateral as shown in Figure 4. From the viewpoint of shape, difference between two shapes is trivial, however, from the viewpoint of illuminance distribution, difference between two shapes is significant. ...
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Full-text available
Lampshade was developed as wind shield for light source when fire was used for light source. Though contemporary light sources, like LED lamp, fluorescent lamp and incandescent light, are working with electricity and wind shield is not required for the light sources, almost all light sources are still covered by lampshade to arrange or control luminous flux from light source. And combination of lampshades and light sources may give distinctive impression to surrounding people. Among many types of lampshade, authors paid attention to shades of surrounding type made by diffuse transmitting material without printed texture since the type is following basic role of lampshade, decreasing luminance level of light source and softening flow of light. Using white paper without printed texture, authors constructed several lampshades by paper folding technique and surrounded two types of light source with the shades for subjects experiment. And authors performed semantic differential for Danish students to evaluate how forms, bump texture and color temperature are affected on impression of the students.
Chapter
Lampshades were developed as wind shield for light source when fire was used for light source. Though contemporary light sources, like LED lamp, fluorescent lamp and incandescent light, are working with electricity and wind shield is not required for light source, almost all light sources are still covered by lampshade to arrange or control luminous flux from light source. And combination of lampshades and light sources may give distinctive impression to surrounding people. Until now, paper folding method ‘SQEF’ (Skew Quadrilateral Elastic Folding) was proposed to construct shapes with non-flat texture in the field of lampshade design. With SQEF methods, final shape is constructed by skew quadrilateral unit with a curved surface. As each skew quadrilateral unit is composed by continuously curved surface, luminance distribution within each unit is continuous as well. The shapes of section appeared in proposed lampshades are convex and curved surface in each unit is concave. The purpose of the research described in this paper is expansion of the horizontal sections of lampshades to nonconvex shape to generate convex unit. The buffer part was inserted between concave parts to combine them. An example of the lampshade including concave parts was manufactured. And finally, luminance distribution on the lampshade was examined.
Chapter
In the field of lampshade design, paper folding method has been highly utilized as the non-flat texture by paper folding method can generate beautiful luminance distribution. Especially, if the non-flat texture is composed of curved surface units, luminance distribution in the units is continuous and visually soft. And the method combining non-flat pattern and geometrical transmittance pattern has been already proposed to overlap luminance distribution caused by non-flat texture and that by transmittance pattern. In this paper, plane tessellations are introduced to expand diversity of lampshade design. Two methods were proposed with single tessellation and multiple tessellations. With single tessellation method, a tessellation was used for both paper folding and geometrical transmittance pattern. With multiple tessellation method, multiple tessellations were used for paper folding and geometrical transmittance pattern respectively. To verify the effect of the combination, lampshades with non-flat texture and geometrical transmittance pattern were manufactured and brightness distributions of them were measured.
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In graphic science education, understanding of knowledge about shape and light is quite significant. The knowledge includes classification of shapes, function of shapes, behavior of light and anticipation of brightness distribution. CAD and 3DCG application enabled designing of complicate shapes and rendering of brightness distribution on the shapes. However, students are difficult to learn function of the shapes and actual brightness distribution in graphic science education with CAD and 3DCG education. To improve such situation, we introduced model manufacturing of lampshade into graphic science education. In the class, students should submit both CG drawing of designed lampshade and photograph of manufactured paper model of them. To reduce restriction of lampshade designing and save the time for manufacturing, we added function of making frame of lampshade and development of curved shade surface to existing 3DCG application. Students designed lampshade shades which were based on approximation of revolution. As 3DCG application, we adopted POV-Ray, freeware ray-tracing application considering higher extensibility and higher quality of rendering. Added functions were implicated as new function of POV-Ray in include file. Ruled surface generated from plane curve and curve of its rotation is cylindrical surface, and cylindrical surface is developable surface. At first, students should decide curved line on graph paper by hand and decide coordination of control points of the curved line. Second, students should input both the coordination data of the points and number of frame. Then 3DCG application generated CG perspective view of the lampshade, drawing of lampshade frame and drawing of development of curved lampshade surface from these data. Finally, students should manufacture paper model of the lampshade with printed drawings. In this paper, we explained objective of the class, schedule of the class, detailed content of the lampshade designing exercise, submitted CG drawings and paper models and result of class evaluation by students.
Conference Paper
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