16TH INTERNATIONAL CONFERENCE ON GEOMETRY AND GRAPHICS 4–8 AUGUST, 2014, INNSBRUCK, AUSTRIA
©2014 ISGG
CONTROL METHOD OF COMBINED DEVELOPABLE SURFACE DESIGN BY AFFINE TRANSFORMATION AND LOCUS Hirotaka SUZUKI1 1 Kobe University, Japan ABSTRACT: Developable surfaces have significant advantage in manufacturing process, though designing of the surfaces have much restrictions. Author have already proposed combined developable surface design method and introduced the method to CG exercise class of graphic science education. In this method, each developable surface which composes designed shape is generated by manipulation (translation, scaling and rotation) of curved line. Though this method have much design potential, complexity level of designed examples is remaining in a certain level. Because if manipulation of curved line is extended to much complicated level, the features of designed shapes cannot be easily understood at designing phase. To solve this problem, author introduced affine transformation expression for understanding features of designed shapes and locus diagram for visualization of series of manipulation. In this paper, principle of proposed design method is explained simply at first. Then relationship among manipulation of curved line, affine transformation and locus diagram is explained. Next, examples of affine transformation expression and locus diagram expression of designed shapes by manipulation in higher level are indicated. Finally, submitted works by students are shown as examples of proposed design process. Keywords: Developable surface, Cylindrical surface, Conical surface, Directing line, Generating line. 1. INTRODUCTION Developable surfaces have significant advantage in manufacturing process, though designing of the surfaces have much restrictions. Therefore a lot of methods have been proposed to combine developable surfaces to increase freedom of shape design. The research about developable surfaces are classified into two categories, research which enable approximation of given curved surface with combination of developable surfaces and research which enable easy designing with combination of developable surfaces. As a research of easy designing, author have already proposed combined developable surface design method[1] and introduced the method to CG exercise class of graphic science education[2]. As shown in Figure 1, each developable surface which composes designed shape is generated by manipulation (translation, scaling and rotation) of curved line in this Paper #044
method. Though this method have much design potential, complexity level of designed examples is remaining in a certain level. Because if manipulation of curved line is extended to much complicated level, for example manipulation which includes combination of rotations around different axes as shown in Figure 2, the features of designed shape cannot be easily
Figure 1: An example of designed shape generated by manipulation of given curve [1].
understood at designing phase. To solve this problem, author introduced affine transformation for understanding features of designed shape and locus diagram for visualization of series of manipulation. As each manipulation in proposed design method is corresponding to affine transformation mathematically, combination of manipulations can be expressed as product of affine transformation matrix. With expression of product of matrix, features of designed shape, for examples property of continuity or symmetry of the shape, can be easily understood. And each manipulation of proposed design method can be expressed by locus diagram which can easily express content of manipulations visually and simply. In this paper, principle of proposed design method is explained simply at first. Then relationship among manipulation of curved line, affine transformation and locus is explained. Next, examples of affine transformation expression and locus diagram expression of designed shapes by manipulation in higher level are indicated. Finally, submitted works by students are shown as examples of proposed design process. Iannis Xenakis, an architect, once tried similar designing method. When he designed ‘Philips Pavilion (1958, Brussels)’, he combined hyperbolic paraboloid surfaces and conical surfaces as shown in Figure 3. In proposing method, some of directing lines are become intersection lines between combined surfaces. Different from proposing method, some of generating lines are become intersection lines between combined surfaces in method by Xenakis. Some of globes on the market are approximation of sphere making use of several cylindrical surfaces. Or, Mitani[3] proposed manufacturing method of approximated solid of revolution by folded paper (ORIGAMI). These methods can be thought as a part of proposing method. Proposing method is further developed and generalized comparing to these existing methods.
Figure 2: An example of designed shape generated by manipulation which includes combination of rotations around different axes.
Figure 3: A sketch of Philips Pavilion (1958, Brussels) drawn by Prof. Koji MIYAZAKI. Though designing of developable surfaces have much restrictions, manufacturing of developable surfaces is very easy. Therefore a lot of methods have been proposed to combine developable surfaces to increase freedom of shape design. These methods are classified to 2 categories, research which enable approximation of given curved surface with combination of developable surfaces and research which enable easy designing with combination of developable surfaces. As research for approximation, Mitani et al. [4] proposed the method of approximation with plane triangles, Shatz et al.[5] proposed the method of approximation with conical surfaces and planes, Massawri et al.[6] proposed the 2
method of approximation with tubes which are constructed by triangles and Pottman et al.[7] proposed the method of approximation with developable strips. As research for easy designing, Rose et al.[8] proposed the method of 3 dimensional shape generation from 2 dimensional perspective drawing, Kilian et al.[9] proposed method of shape generation with repetition of curved line folding and Suzuki[10] proposed method of shape generation with combination of tangent surfaces. The objective of the proposing method in this paper is offering the way of designing and manufacturing for graphic science education and design workshop of restricted time. The method belongs to the research for easy designing. Though proposed method restricts freedom of design in certain level, the method enables designing and manufacturing in very short time. In this paper, right-handed coordinate system as shown in Figure 4 is adopted for coordinate expression.
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X
X
(
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(3)
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(4)
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(5)
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(6)
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(7)
(1)
Considering easy operation for combination of transformations, Equation (1) is often expressed as homogeneous coordinate system as following Equation (2):
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In this paper, affine transformation is expressed as homogeneous coordinate system as in Equation (2). If we assume transformation for translation with moving vector (xt, yt, zt) as T(xt, yt, zt), transformation for scaling from origin point with scaling rate k (0 < k < 1) as S(k) and transformation for rotation around XYZ axis with rotating angle as Rx(), Ry(), Rz(), these transformations are expressed as following matrix:
2.1 Relations between manipulation and affine transformation In general, affine transformation is matrix transformation expressed by following Equation (1): (
Z
Figure 4: Left-handed coordinate system(left) and right-handed coordinate system(right).
2. EXPRESSION OF CURVED LINE MANIPULATION BY AFFINE TRANSFORMATION
( )
Z
(2)
3
Table 1:Implemented functions in POV-Ray and content of the functions. Functions in POV-Ray Translate() Scale(k) Rotate_X(theta) Rotate_Y(theta) Rotate_Z(theta) Scale_frompoint(k, ) Rotate_Xparallel(theta, ) Rotate_Yparallel(theta, ) Rotate_Zparallel(theta, )
Z ( 10, 0, 40 )
Content of function (xt, yt, zt) (k)
(k, xs, ys, zs)
Equation (3) Equation (4) Equation (5) Equation (6) Equation (7) Equation(8)
( 10, 0, 30 )
( 15, 0, 20 )
( 30, 0, 10 )
(theta, xr, yr, zr) Equation (9) (theta, xr, yr, zr) Equation (10)
( 10, 0, 0 )
(theta, xr, yr, zr) Equation (11)
X
( 0, 0, -10 )
Figure 6: Six control points and a spline curved line generated from the points. #declare Number_of_ControlPoints = 6; #declare Points_for_CurvedLine =array[MAX_NUMBER_POINT][2] { {0, -10}, //point1 {10, 0 }, // point 2 {30, 10}, // point 3 {15, 20}, // point 4 {10, 30}, // point 5 {10, 40}, // point 6 //point 7 and after {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0} }
Figure 5: An example shape made from 8 times rotations of given curve.
Figure 7: A definition of spline curved line for the shape shown in Figure 5.
And, if we assume transformation of scaling from non-origin point (xs, ys, zs) with scaling rate k as S’(k, xs, ys, zs) and transformation of rotation around the line parallel to XYZ axis with rotating angle as R’x(), R’y(), R’z(), these transformations are expressed by combination of matrix in Equation (3) to (7) as following:
program, repetition or conditional branch or functions can be used in POV-Ray. As shown in Table 1, we implemented all transformation matrices shown in Equation (3) to (11) as function in POV-Ray. 2.3 Description of transformation matrix in CG freeware POV-Ray Figure 5 shows an example shape made from 8 times rotations of given curve. Rotation angles in 8 rotations are the same angle /4. Given curve is spline curve defined from six control points as shown in Figure 6. In six control points, the first and the last points are defined for angle at edges of the curved line. As shown in Figure 7 and 8, definitions for spline curve (number of control points and coordinates of control points) and definition of manipulations (number of manipulations and content of ma-
(8) (9) (10) (11)
2.2 Description of transformation matrix in CG freeware POV-Ray POV-Ray is freeware application for CG modeling and rendering. All objects in a scene are located by text expression in POV-Ray. As same as description of general computer 4
#declare Number_of_Manipulations = 8; #declare A[1] = Rotate_Z(2*pi/8); #declare A[2] = Rotate_Z(2*pi/8); #declare A[3] = Rotate_Z(2*pi/8); #declare A[4] = Rotate_Z(2*pi/8); #declare A[5] = Rotate_Z(2*pi/8); #declare A[6] = Rotate_Z(2*pi/8); #declare A[7] = Rotate_Z(2*pi/8); #declare A[8] = Rotate_Z(2*pi/8);
Figure 8: A definition of manipulations for the shape shown in Figure 5. object { combination_of_developablesurfaces ( Number_of_Manipulation, 0, //Draw all surfaces if the number is 0 //Draw directed surface if the number is not 0 Number_of_ControlPoints, Points_for_CurvedLine, A //Affine transformations ) }
Figure 10: An example shape made from 12 manipulations of given curve. #declare A[1] = Scale_frompoint(3/4, ); #declare A[2] = Rotate_Zparallel(pi/2, ); #declare A[3] = Scale_frompoint(4/3, ); #declare p_offset = ; #declare A[4] = Translate(p_offset); #declare A[5] = Rotate_Zparallel(pi*4, p_offset); #declare A[6] = Rotate_Zparallel(pi*4, p_offset); #declare A[7] = Scale_frompoint(3/4, + p_offset); #declare A[8] = Rotate_Zparallel(pi/2, + p_offset); #declare A[9] = Scale_frompoint(4/3, + p_offset); #declare A[10] = Rotate_Zparallel(pi/4, p_offset); #declare A[11] = Rotate_Zparallel(pi/4, p_offset); #declare A[12] = Translate(-p_offset);
Figure 9: An example of the function which constructs the shape from given curved line and given transformations. nipulations) are declared beforehand. Then these definitions are passed to shape construction function ‘combination_of_developablesurfaces’ as shown in Figure 9. After camera settings (location, direction and angle of view) and information of other objects are described, user can obtain perspective view of defined shape. As shown in Figure 7, curved line is defined as plane curve. When given manipulations do not include rotation, space curve can be used to get developable surface. At present, only plane curve can be given in this system. Figure 10 is another example of the shape made from manipulation of given curved line. In this shape, 12 manipulations including translation, scaling and rotation are used as shown in Figure 11.
Figure 11: A definition of manipulations for the shape shown in Figure 10. ( where
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(12)
Obviously, A8 equals unit matrix. If given curved line overlap to original line after a series manipulations, the product of all matrices corresponding to manipulations must be unit matrix. In this case, the last curved surface is connected to the first without excess and deficiency and users can easily understand this feature strictly from the viewpoint of mathematics. In case of the shape shown in Figure 10, manipulations for given curved line are described in Figure 11 and the manipulations can be described in affine transformation format as following equation:
2.4 Understanding of features of shapes with affine transformation The manipulations given for constructing the shape shown in Figure 5 can be expressed as following equation (12): 5
(
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Y 100
(
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(
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( )
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(13)
where P = (5, -5, 0) . The product of 12 matrices in Equation (13) is actually unit matrix and as same as the shape shown in Figure 5, the last curved surface is connected to the first without excess and deficiency. And, A[3] and A[1], A[4] and A[12], A[5] and A[11], A[6] and A[10] are symmetry with respect to the surface y = x. And each transformation of A[2] and A[8] are also symmetry with respect to the surface y = x. We can understand that the shape constructed by transformations described in Figure 11 or Equation (13) is symmetry with respect to the surface y = x. In this way, users can understand features of constructed shape, symmetry or continuity, by considering manipulations as affine transformations.
O
100
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Figure 12: An example of locus diagram corresponding to translation T (-60, 60, 0). Y 100
O
100
X
Figure 13: An example of locus diagram corresponding to scaling S’ (3/4, 0, 80, 0).
3. EXPRESSION OF CURVED LINE MANIPULATION BY LOCUS DIAGRAM
Y 100
3.1 Locus diagram expression of translation, scaling and rotation of curved line. A point given in spatial field may move by manipulation matrices. Locus diagram expression of manipulation enables easy understanding for series of manipulations. To realize easy understanding of manipulations, we developed function of generating locus of given 2 points, main point and sub point as shown in Figure 12 to 14. Though locus diagram can be drawn with only 1 point, 2 points are needed to make sure direction of manipulated curved line. In case of rotation, given point may move along an arc of circle. However, generating lines on the cylindrical surface generated by rotation of plane curve are straight lines in this design method. Considering the difference, locus lines by rotation are drawn as straight lines
O
100
X
Figure 14: An example of locus diagram corresponding to rotation RZ (/2). with arcs which indicate the given manipulations are rotation as shown in Figure 14. Concretely, in case of translation, the segment which links main point and sub point, and the line which corresponding to manipulated segment are drawn in black. And the segment 6
which links main point and manipulated main point is drawn also as shown in Figure 12. In case of scaling, 3 black lines are drawn as same as case of translation. In addition to these 3 black lines, 2 blue lines are also drawn among center point of scaling, main point and sub point as shown in Figure 13. In case of rotation, 3 black lines are drawn as same as case of translation also. In addition to these lines, 2 red lines are also drawn among center point of rotation, main point and manipulated main point, and the arc which corresponding to manipulation of the rotation is drawn as shown in Figure 14. We can easily discriminate manipulations in Figure 12 from manipulation in Figure 14 though start point and manipulated start point are exactly same in this case. Though an original idea of locus diagram was indicated by Suzuki[1], several information is added to realize easy understanding of manipulations in proposed diagram. Mitani[11] also proposed a design method of folded paper with given profile polyline and given trajectory polyline. There are similar points between proposed locus diagram and design method by Mitani. In case of method by Mitani, there is advantageous point that 2 developable surfaces are generated by folding along given polyline, though trajectory polyline do not include scaling which is included in proposing method.
Y 50
50 X
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Figure 15: The locus diagram corresponding to series of manipulations shown in Figure 5. Y 60
O
60 X
Figure 16: The locus diagram corresponding to series of manipulations shown in Figure 10. stood only with locus diagram, however, we can understand the features intuitively and strictly with both affine transformation expressions and locus diagram expressions.
3.2 Understanding of features of shapes with locus diagram Figure 15 shows the locus diagram corresponding to series of manipulations shown in Figure 5. The shape in the figure is obviously rotational symmetry and continuous without excess and deficiency. Though, the features can be understood from affine transformation, the locus diagram shown in Figure 15 can tell the features easier. Figure 16 shows the locus diagram corresponding to series of manipulations shown in Figure 10. Though continuity and symmetry of the shape are mentioned in 2.4 from viewpoint of affine transformation, we can understand the features easier with the locus diagram. Strict features cannot be under-
4. EXAMPLES OF DESIGNED SHAPES AND EXPRESSIONS OF THE SHAPES In this chapter, several examples of designed shapes and affine transformation expressions and locus diagram expressions of corresponding shapes are indicated. At first, an example of designed shape made from series of manipulations given as Equation (14) is shown. (
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where l = 24,
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Figure 17: The locus diagram corresponding to manipulations shown in Equation (14).
Figure 19: The locus diagram corresponding to manipulations shown in Equation (15).
Figure 18: The shape generated from manipulations shown in Equation (14) 4 times.
Figure 20: The shape generated from manipulations shown in Equation (15) and Figure 19.
Figure 17 shows the locus diagram expression of given manipulations in Equation (14), and Figure 18 shows the shape generated by giving the manipulations 4 times. So far, in this paper, shapes generated from rotations of same direction are introduced. As shown in Figure 18, rotations of inverse directions can be given to original curved line. Next, Figure 19 shows the locus diagram of 12 manipulations expressed as Equation (15) (
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So far, in this paper, shapes in which the last curved surface is connected to the first without excess and deficiency are introduced. As shown in Figure 19, spiral shape in which the last curved surface is not connected to the first can be generated by combination of manipulations. The shape shown in Figure 20 is an example which is made from manipulations described as Equation (15). Original curved line for the shape shown in Figure 20 is exactly same as that for the shape shown in Figure 5. Though given manipulations for the shape in Figure 20 is similar to that for the shape in Figure 5, scaling manipulations generate such spiral
) (
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(15) 8
other. Though it is very difficult to consider continuity of the shape generated from such manipulations from perspective view of the shape, we can understand continuity strictly from affine transformation matrices. 5. IMPLEMENTATION OF THE DESIGN PROCESS IN GRAPHIC SCIENCE EDUCATION COURSE We tried to implement the design process explained at this paper in the graphic science education course. We implemented the process in the class titled ‘Environmental Equipment Planning’ which is provided for 1st grade graduate students of department of architecture, graduate school of engineering, Kobe university. In academic year 2014, 4 units (90 minutes / unit) were allocated for lampshade design exercise. Students were requested to design lampshade with the design process and manufacture the lampshade actually. In total, 18 works were submitted. Figure 22 shows the examples of submitted works by CG perspective view, locus diagram and photo of manufactured works.
Figure 21: The shape generated from manipulations shown in Equation (16) and Figure 19. shape. At last, Figure 21 shows the shape generated from series manipulations described as Equation (16). (16) So far, in this paper, shapes generated from rotations around lines which are parallel to Z axis are introduced. As shown in the figure, a series of manipulations can include rotations in which the axes of the rotations are not parallel to each
6. CONCLUSIONS AND FUTURE WORK To improve usability of design method which combine developable surfaces with given curved line and given manipulations, and to
Side
Upper
Quarter of whole manipulations
Figure 22: The examples of submitted works by CG perspective view, locus diagram and photo of manufactured works
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improve understanding of features of generated shapes, expressions of manipulations by affine transformation and by locus diagram are proposed. With expression of affine transformation, we can strictly understand the features in detail, and with locus diagram we can intuitively and roughly understand the features. We are planning to add mirroring as one of manipulation. If we adopt mirroring, generated curved surface from space curve with mirroring is cylindrical surface. As shown in Figure 7, only plane curve is allowed as given curved line at current system. We have to extend the system to allow space curve as given curved line to make good use of advantageous point by introduction of mirroring.
Strip-based Approximate Unfolding, ACM Trans., Graphics, 23(3), pp.259-263, 2004. [5] I. Shatz, A. Tal, G. Leifman, Paper craft models from meshes, The Visual Computer: International Journal of Computer Graphics archive, Vol. 22 Issue 9, pp.825-834, 2006. [6] F. Massarwi, C. Gotsman and G. Elber, Papercraft Models Using Generalized Cylinders, Proceedings of Pacific Graphics, pp.148 – 157, 2007. [7] H. Pottmann, A. Schiftner, P. Bo, H. Schmiedhofer, W. Wang, N. Baldassini and J. Wallner, Freeform surfaces from single curved panels, ACM Trans. Graphics, 27(3), Article No.76, 2008. [8] K. Rose, A. Sheffer, J. Wither, M. Cani and B. Thibert, Developable Surfaces from Arbitrary Sketched Boundaries, Eurographics Symposium on Geometry Processing, pp.163-172, 2007. [9] M. Kilian, S. Flory, Z. Chen, N. J. Mitra, A. Sheffer and H. Pottmann, Curved Folding, ACM Trans. Graphics, 27(3), Article No.75, 2008. [10] H. Suzuki, 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 CDROM, 2012. [11] J. Mitani, Column-shaped Origami Design Based on Mirror Reflections, Journal for Geometry and Graphics, Vol.16, No.2, pp.185-194, 2012.
ACKNOWLEDGMENTS Examples of works shown in chapter 5 were submitted by graduate course students of department of architecture, graduate school of engineering, Kobe university. Author would like to thank all students who participated lampshade design exercise and submit their own lampshade within a given period of time. REFERENCES [1] H. Suzuki, A generation method of developable surface by manipulation (translation, 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, in Japanese, 48(1), pp.3-10, 2014. [2] H. Suzuki, Ai Sakaki, Kensuke Yasufuku and Takashi Matsumoto, Designing of Lampshade with 3D CG Application and Manufacturing of Designed Shape in Graphic Science Education, Proceedings of The 2013 Asian Forum on Graphic Science, pp.38-43, 2013 [3] J. Mitani, A Design Method for 3D Origami Based on Rotational Sweep, Computer-Aided Design and Applications, 6(1), 69-79, 2009. [4] J. Mitani, Hiromasa Suzuki, Making Papercraft Toys from Meshes using
[12] H. Suzuki, A Study on Impact of Introduction of Lighting Equipment Design Assignment into Graphic Science Education， Proc. of 12th Intl. Conference on Geometry and Graphics, in CDROM, 2006. [13] H. Suzuki, Application of tangent surfaces in the design of lampshades, Proc. of the Fall Annul Conference of the Japan Society for Graphics Science, 133-136, in Japanese, 2010. 10
[14] H. Suzuki, ‘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, 143-144, in Japanese, 2011. [15] H. Suzuki, ‘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 CDROM, 2012. 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.
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