3D-Dynamical geometry in building construction

TIME 2010 – Technology and its Integration into Mathematics Education. 3D-Dynamical Geometry in Building Construction. R. M. Falcón Department of Applied Mathematics I University of Seville Spain [email protected] 3D – Dynamical Geometry in Building Construction. R. M. Falcón. 2 TIME 2010 – Technology and its Integration into Mathematics Education. ABSTRACT In Architecture and Technical Architecture Degrees, students use CAD tools (Computer Aided Design) which are not capable, in general, of representing graphically curves or surfaces starting from its corresponding equations. To get it, users have to define specific macros or they have to create a table of points in order to convert a set of nodes into polylines. CAS tools used in Math classes allow this graphical representation of curves and surfaces starting from their parametric equations. However, they lack the dynamical development given by CAD tools, which plays a main role in the mentioned degrees. In this sense, the complementation of the algebraic and geometric tools included in the software of dynamic geometry, GeoGebra, is an attractive alternative to design and model, from a mathematical point of view, curves and rigid objects in the space. The use of sliders related to the Euler’s angles and the possibility of generating tools which project 3D into 2D, makes easier this kind of modeling. In the current workshop, we will show how to construct 3D-models of several architectonical constructions which have been made in the context of the subject called Mathematics for Building Construction II, corresponding to the Building Construction Engineering of the University of Seville, which has been implemented this academic year 2009-10. Keywords 3D-modeling. Building Construction modeling. Perspectives and projections. Dynamic geometry. GeoGebra. 1. I$TRODUCTIO$. Let O be the origin of coordinates of two orthogonal reference systems OXYZ and OX’Y’Z’ and let l be the straight line which is the intersection between the planes OXY and OX’Y’. The orientation of the latter system with respect to the former is univocally determined by the Euler angles: α: Angle formed by the straight lines OZ and OZ’. β: Angle formed by the straight lines OX’ and l. γ: Angle formed by the straight lines l and OX. The variation of these three angles implies the movement of the second system of reference with respect to the first one. Any rigid object whose coordinates are given with respect to the mobile system will be moved in the same way. Specifically, α determines the inclination angle of the rigid object with respect to the fixed system OXYZ and β determines the rotation angle in the plane OX’Y’. As a consequence, any tridimensional rigid object can be visualized by any orthographic projection which depends on the Euler angles and whose focus is fixed at infinite distance. In order to model tridimensional objects, Genevieve Tulloue [1] implemented in Cabri [2] the following orthogonal projection of a point P = (a,b,c): π r (a,b,c) = (r·(a·sin(β) + b·cos(β)), r·(-a·cos(β)·sin(α) + b·sin(β)·sin(α) + c·cos(α))), 3D – Dynamical Geometry in Building Construction. R. M. Falcón. 3 TIME 2010 – Technology and its Integration into Mathematics Education. where r determines the scale of the projection. A similar implementation in GeoGebra [3] is given by Rafael Losada [4], who extends in a natural way the point-by-point projection of Tulloue in order to project tridimensional curves and surfaces which can be defined as a mesh of curves [5]. However, the computation of sequences of curves in GeoGebra can be so slow that it is necessary a very powerful computer in order to rotate a simple solid like a cone or a sphere. In these cases, it is better to define polylines which are based in a high number of points. In our study, we are interested in the following two types of surfaces: A. Those surfaces defined as: S(u,v) = P + f(u)·g(v) = (p 1 + f 1 (u)·g 1 (v), p 2 + f 2 (u)·g 2 (v), p 3 + f 3 (u)·g 3 (v)). B. Ruled surfaces: S(u,v) = f(u) + v·g(u) = (f 1 (u)+v·g 1 (u), f 2 (u)+v·g 2 (v), f 3 (u)+v·g 3 (v)). 2. DEFI$I$G TOOLS I$ GEOGEBRA. In this Section, we will show how to define some basic tools which can be used in order to work with space curves and surfaces in GeoGebra. The first step in our worksheet of GeoGebra will be the definition of the sliders which correspond to the scalar r and the angles α and β. Once we have defined them, we have to follow the following steps: 2.1 Construction of the mobile reference system. 1) Fix the origin O. 2) Determine the coordinate system OX’Y’Z’, by defining three vectors with origin at O and extremes at: π r (1,0,0) = (r·sin(β), -r·cos(β)·sin(α)). π r (0,1,0) = (r·cos(β), r·sin(β)·sin(α)). π r (0,0,1) = (0, r·cos(α)). 3) Define a check box “Axes” related to the previous three vectors. Figure 1: 3d axes. 3D – Dynamical Geometry in Building Construction. R. M. Falcón. 4 TIME 2010 – Technology and its Integration into Mathematics Education. 2.2 Definition of the Orthogonal Scaled Projection. 1) Define any point in the space as an array of 3 elements. Ex: P = {1, 2, 3}. 2) Determine the projection π r of the previous point 1 : P’=(r/10(Element[P,1]sin(β)+Element[P,2]cos(β)),r/10(-Element[P,1]cos(β) sin(α) + Element[P,2]sin(β)sin(α)+Element[P,3]cos(α))). 3) Define a new tool OSP whose output is P’ and whose input is {P, r, α, β}. 2.3 Projection of a space curve C(x) = (c 1 (x),c 2 (x),c 3 (x)), with x in (x 0 , x 1 ). 1) Define the extremes x 0 and x 1 of the interval, like numbers or by using sliders. 2) Define the components of the curve as three functions in x. Ex: c 1 (x) = sin(x)cos(x); c 2 (x) = cos(x); c 3 (x) = sin(x). 3) Determine the projection π r of the curve C in (x 0 ,x 1 ): C= Curve[r/10(c 1 (t)sin(β)+c 2 (t)cos(β)), r/10(-c 1 (t)cos(β)sin(α) + c 2 (t)sin(β)sin(α) + c 3 (t)cos(α)), t, x 0 , x 1 ]. 4) Define a new tool Curve3d whose output is C and whose input is {r, α, β, c 1 (x), c 2 (x), c 3 (x), x 0 , x 1 }. Figure 2: Space curve in GeoGebra. 1 In all the projections, we will divide by 10 to obtain a better effect with the tridimensional axes. 3D – Dynamical Geometry in Building Construction. R. M. Falcón. 5 TIME 2010 – Technology and its Integration into Mathematics Education. 2.4 Projection of a surface S(u,v) = (p 1 +f 1 (u)·g 1 (v), p 2 + f 2 (u)·g 2 (v), p 3 +f 3 (u)·g 3 (v)) with u and v in (u 0 , u 1 ) and (v 0 ,v 1 ), respectively, by using m u-polylines and n vpolylines. A similar construction of the surface defined in 2.4 can be done by using polylines: 1) Define as numbers or sliders the elements p 1 , p 2 , p 3 , u 0 , u 1 , v 0 , v 1 , m and n. 2) Define the components of the surface as six functions of x. Ex: f 1 (x)=cos(x); f 2 (x)=cos(x); f 3 (x)=sin(x); g 1 (x)=cos(x); g 2 (x)=sin(x); g 3 (x)=1. 3) Determine the projections of the nodes of the m u-polylines and the n v-polylines: = = Sequence[Sequence[(r/10((p 1 +f 1 (u) g 1 (v)) sin(β) + (p 2 +f 2 (u) g 2 (v)) cos(β)), r/10(- (p 1 +f 1 (u) g 1 (v)) cos(β) sin(α) + (p 2 +f 2 (u) g 2 (v)) sin(β) sin(α) + (p 3 +f 3 (u) g 3 (v)) cos(α))), u, u 0 , u 1 , (u 1 - u 0 )/m], v, v 0 , v 1 , (v 1 - v 0 )/n]. 4) Define the projections of the m u-polylines and the n v-polylines: P u =Sequence[Sequence[Segment[Element[Element[=,i],j], Element[Element[=,i], j+1]],i,1,n],j,1,m]. P v =Sequence[Sequence[Segment[Element[Element[=,i],j], Element[Element[=,i+ 1],j]],i,1,n],j,1,m+1]. 5) Define a new tool Surface whose output is {P u , P v } and whose input is {r, α, β, m, n, p 1 , p 2 , p 3 , f 1 (x), f 2 (x), f 3 (x), u 0 , u 1 , g 1 (x), g 2 (x), g 3 (x), v 0 , v 1 }. Figure 3: Polygonal surface in GeoGebra. 3D – Dynamical Geometry in Building Construction. R. M. Falcón. 6 TIME 2010 – Technology and its Integration into Mathematics Education. 2.5 Projection of a ruled surface S(u,v) = (f 1 (u)+v·g 1 (u), f 2 (u)+v·g 2 (v), f 3 (u)+v·g 3 (v)) with u and v in (u 0 , u 1 ) and (v 0 ,v 1 ), respectively, by using m u-polylines and n vpolylines. A similar construction of the surface defined in 2.4 can be done by using polylines: 1) Define as numbers or sliders the elements p 1 , p 2 , p 3 , u 0 , u 1 , v 0 , v 1 , m and n. 2) Define the components of the surface as six functions of x. Ex: f 1 (x)=x; f 2 (x)=x; f 3 (x)=sin(x); g 1 (x)=sin(x); g 2 (x)=1; g 3 (x)=1. 3) Determine the projections of the nodes of the m u-polylines and the n v-polylines: = = Sequence[Sequence[(r/10((f 1 (u) + v g 1 (u)) sin(β) + (f 2 (u)+v g 2 (u)) cos(β)), r/10((f 1 (u)+v g 1 (u)) cos(β) sin(α) + (f 2 (u)+v g 2 (u)) sin(β) sin(α) + (f 3 (u)+v g 3 (u)) cos(α))), u, u 0 , u 1 , (u 1 - u 0 )/m], v, v 0 , v 1 , (v 1 - v 0 )/n]. 4) Define the projections of the m u-polylines and the n v-polylines: P u =Sequence[Sequence[Segment[Element[Element[=,i],j], Element[Element[=,i], j+1]],i,1,n+1],j,1,m]. P v =Sequence[Sequence[Segment[Element[Element[=,i],j], Element[Element[=,i+ 1],j]],i,1,n],j,1,m+1]. 5) Define a new tool RSurface whose output is {P u , P v } and whose input is {r, α, β, m, n, f 1 (x), f 2 (x), f 3 (x), u 0 , u 1 , g 1 (x), g 2 (x), g 3 (x), v 0 , v 1 }. Figure 4: Ruled surface in GeoGebra. 3D – Dynamical Geometry in Building Construction. R. M. Falcón. 7 TIME 2010 – Technology and its Integration into Mathematics Education. 3. SURFACES I$ GEOGEBRA. The tool Surface can be used in the definition of the most known surfaces. It is enough to use the parametric equations determined by the six functions given in Table 1. Surfaces S S u S v Cilinder f 1 (x)=cos(x); f 2 (x)=sin(x); f 3 (x)=1; g 1 (x)=1; g 2 (x)=1; g 3 (x)=x. Sphere f 1 (x)=cos(x); f 2 (x)=cos(x); f 3 (x)=sin(x); g 1 (x)=cos(x); g 2 (x)=sin(x); g 3 (x)=1. Elliptic paraboloid f 1 (x)=sqrt(x); f 2 (x)= sqrt(x); f 3 (x)=x; g 1 (x)=cos(x); g 2 (x)=sin(x); g 3 (x)=1. Hyperbolic paraboloid f 1 (x)=x; f 2 (x)= 1; f 3 (x)=x; g 1 (x)=1; g 2 (x)=x; g 3 (x)=x. Cone f 1 (x)=x; f 2 (x)= x; f 3 (x)=x; g 1 (x)=cos(x); g 2 (x)=sin(x); g 3 (x)=1. One-sheeted hyperboloid f 1 (x)=cos(x); f 2 (x)= cos(x); f 3 (x)=sin(x); g 1 (x)=cosh(x); g 2 (x)=sinh(x); g 3 (x)=1. Two-sheeted hyperboloid f 1 (x)=sinh(x); f 2 (x)= sinh(x); f 3 (x)=cosh(x); g 1 (x)=cos(x); g 2 (x)=sin(x); g 3 (x)=1. Torus Ex: f 1 (x)=2+cos(x); f 2 (x)=2+cos(x); f 3 (x)=sin(x); g 1 (x)=cos(x); g 2 (x)=sin(x); g 3 (x)=1. Astroid Ex: f 1 (x)=cos 3 (x); f 2 (x)=cos 3 (x); f 3 (x)=sin 3 (x); g 1 (x)=cos(x); g 2 (x)=sin(x); g 3 (x)=1. Table 1: Construction of surfaces by using u and v-parametric curves. 3D – Dynamical Geometry in Building Construction. R. M. Falcón. 8 TIME 2010 – Technology and its Integration into Mathematics Education. By using the tool RSurface, we can build all the types of ruled surfaces: a) Cylindrical surfaces: It is enough to impose g 1 , g 2 and g 3 to be constant. Figure 5: Cylindrical surface. b) Conical surfaces: It is necessary to impose all the generatrices to pass through a given point P=(a,b,c). Specifically, it must be: 3 1 2 1 2 3 ( ) ( ) ( ) . ( ) ( ) ( ) c f x a f x b f x g x g x g x − − − = = Figure 6: Conical surface. c) Conoids : They are obtained when all the generatrices of the ruled surface pass through a point of the directrix curve and a given point of the axis and they are parallel to the director plane. 3D – Dynamical Geometry in Building Construction. R. M. Falcón. 9 TIME 2010 – Technology and its Integration into Mathematics Education. Figure 7: Conoid. Some known ruled surfaces are determined by the functions given in Table 2. Ruled surfaces S S u S v Plane f 1 (x)=ax+b; f 2 (x)=cx+d; f 3 (x)=ex+f; g 1 (x)=g; g 2 (x)=h; g 3 (x)=i. Cylinder f 1 (x)=cos(x); f 2 (x)=sin(x); f 3 (x)=1; g 1 (x)=0; g 2 (x)=0; g 3 (x)=1. Hyperbolic paraboloid f 1 (x)=x; f 2 (x)= x; f 3 (x)=1; g 1 (x)=1; g 2 (x)=-1; g 3 (x)=4x. Cone f 1 (x)=cos(x); f 2 (x)= sin(x); f 3 (x)=1; g 1 (x)=cos(x); g 2 (x)=sin(x); g 3 (x)=1. One-sheeted hyperboloid f 1 (x)=cos(x); f 2 (x)= cos(x); f 3 (x)=sin(x); g 1 (x)=cosh(x); g 2 (x)=sinh(x); g 3 (x)=1. Table 2: Construction of ruled surfaces . 3D – Dynamical Geometry in Building Construction. R. M. Falcón. 16 TIME 2010 – Technology and its Integration into Mathematics Education. Creation of a tool which corresponds to a surface which can be used in building construction. Example: Let us define a tool which can be used to construct a dome. 1) Open the file Worksheet.ggb. 2) Create five sliders: a. Three sliders p1, p2 and p3, corresponding to the center of the dome, all of them defined in the interval [-5, 5]. b. The radius R of the dome, defined in the interval [0.1, 10]. c. The basis b of the dome, determined by an angle defined in the interval [0, π/2-0.1]. 3) Write in the input box: Surface[r,α,β,10,40,p 1 ,p 2 ,p 3 ,R cos(x), R cos(x),R sin(x), b, π/2, cos(x), sin(x), Function[1,1,1],0°,360°]. 4) In the menu Tools, create a tool whose output objects are the two lists which have been obtained after using the previous command. The input objects will be r, α, β, p 1 , p 2 , p 3 , R and b. Denote the new tool as Dome and write as tool help: Select r, α, β, p1, p2, p3 (center), R (radius), b (basis). 5) In the submenu Manage Tools, save the new tool as Dome.ggt. Figure 14: Construction of a dome. Exercise: Create a tool which can be used to construct a cone. You can use the construction which we have done in the previous Section. Save the new tool as Cone.ggt. Exercise: Create a tool which can be used to construct a cylindrical pillar. It has to depend on the center of the basis, the radius and the height of the pillar. Save the new tool as Pillar.ggt. 3D – Dynamical Geometry in Building Construction. R. M. Falcón. 17 TIME 2010 – Technology and its Integration into Mathematics Education. Construction of a building. Example: Let us build a cylindrical tower of two floors such that a dome covers its top. The cylinder of the first floor has a radius of 3 meters and a height of 4 meters and the one of the second floor has a radius of 1.5 meters and a height of 3 meters. 1) Open the file Worksheet.ggb. 2) Use the tool Pillar to create the first floor of the tower: Pillar[r,α,β,0,0,0,3,4]. 3) Use the tool Disk to create the ceiling of the first floor: Disk[r,α,β,0,0,4,3]. Figure 15: First floor of a tower. 4) Use the tool Pillar to create the second floor of the tower: Pillar[r,α,β,0,0,4,1.5,3]. 5) Use the tool Dome to create the dome of the tower: Dome[r,α,β,0,0,7,3,1.5,3]. Figure 16: Tower of two floors with a dome. 6) Save the worksheet as Building.ggb. Exercise: Draw a truncated cone at the top of the dome. You can use sliders in order to get the exact position of the cone. 3D – Dynamical Geometry in Building Construction. R. M. Falcón. 18 TIME 2010 – Technology and its Integration into Mathematics Education. Figure 17: Truncated cone at the top of the tower. 4. FI$AL REMARKS. In this workshop, it has been shown that it is possible to model architectonical surfaces in GeoGebra by using sliders related to the main elements of the corresponding construction. An advantage with respect to CAD’s is that it is possible to use the exact parametric equations of the surfaces. However, a disadvantage that we have observed is the need of a powerful computational engine in order to not slow down the work. Although the use of polylines improves the speed, we hope next versions of GeoGebra will develop this aspect. The implementation of tools related with Differential Geometry and an exhaustive study of different orthogonal projections which can be applied in Descriptive Geometry are two possible future works which can be interesting to deal with. REFERE$CES. [1] http://gtulloue.free.fr/Cabri3D/euler/euler.html. [2] http://www.cabri.com. [3] http://www.geogebra.org. [4] http://www.iespravia.com/rafa/3d_plantilla/3d.htm. [5] http://www.iespravia.com/rafa/superficies/index.htm. [6] http://personal.us.es/raufalgan/geogebra_archivos/Tools.zip