Geodesic and Panel Domes Arising from 2D Diagrams

Article Symmetry in Regular Polyhedra Seen as 2D Möbius Transformations: Geodesic and Panel Domes Arising from 2D Diagrams

Jose A. Diaz-Severiano 1, Valentin Gomez-Jauregui 2,*, Cristina Manchado 2 and Cesar Otero 2

1 School of Civil Engineering, Universidad de Cantabria, Santander 39005, Spain; [email protected] 2 R&D Group EgiCAD, School of Civil Engineering, Universidad de Cantabria, Santander 39005, Spain; [email protected] (C.M.); [email protected] (C.O.) * Correspondence: [email protected]; Tel.: +34-942-206-757

Received: 2 July 2018; Accepted: 18 August 2018; Published: 21 August 2018

Abstract: This paper shows a methodology for reducing the complex design process of space structures to an adequate selection of points lying on a plane. This procedure can be directly implemented in a bi-dimensional plane when we substitute (i) Euclidean geometry by bi-dimensional projection of the elliptic geometry and (ii) rotations/symmetries on the sphere by Möbius transformations on the plane. These graphs can be obtained by sites, specific points obtained by homological transformations in the inversive plane, following the analogous procedure defined previously in the three-dimensional space. From the sites, it is possible to obtain different partitions of the plane, namely, power diagrams, Voronoi diagrams, or Delaunay triangulations. The first would generate geo-tangent structures on the sphere; the second, panel structures; and the third, lattice structures.

Keywords: symmetry; polyhedra; space structures; stereographic projection; Möbius transformations

1. Introduction Some architectural domes, virus, fullerenes, soccer balls, and crystals share the same characteristic, they are structures that approximate the sphere. Many systems exist to replicate the shape of a sphere or any other quadric surface, most of them based on the juxtaposition of three-dimensional (3D) solids, two-dimensional (2D) panels, or one-dimensional (1D) bars + zero-dimensional (0D) nodes. The last case corresponds to the so-called space frames, a structural system assembled from linear elements arranged in such a way that the forces are transferred in a three-dimensional manner (Figure1). The aim of this paper is to show a procedure to design space structures that approximate the sphere (and other quadric surfaces), by means of the application of computational geometry methods [1]. It is possible to reduce the complex design process of 3D space structures to an adequate selection of points lying on a 2D plane. This can be achieved by substituting (i) Euclidean geometry by bi-dimensional projection of the elliptic geometry and (ii) rotations/symmetries on the sphere by Möbius transformations on the plane [2].

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Figure 1. Geodesic spatial frame approximating a quadric surface: American Society for Metals Figure 1. Geodesic spatial frame approximating a quadric surface: American Society for Metals International Headquarters and Geodesic Dome, at the Materials Park campus in Russell Township, International Headquarters and Geodesic Dome, at the Materials Park campus in Russell Township, Ohio, United States (Source: https://theredlist.com/wiki-2-19-879-605-285216-view-buckminster- Ohio, United States (Source: https://theredlist.com/wiki-2-19-879-605-285216-view-buckminster- fuller-richard-profile--1.html). fuller-richard-proﬁle--1.html).

Literature Review Literature Review Wester [3] classified space frames in three categories, (i) lattices, frames composed of one- Wester [3] classiﬁed space frames in three categories, (i) lattices, frames composed of dimensional bars interconnected at zero-dimensional nodes; (ii) plates, bi-dimensional elements that one-dimensional bars interconnected at zero-dimensional nodes; (ii) plates, bi-dimensional elements conform the faces of a polyhedron; and (iii) solids, composed of three-dimensional elements. The that conform the faces of a polyhedron; and (iii) solids, composed of three-dimensional elements. plate type structures derive from lattice type geometries by applying the principle of structural and The plate type structures derive from lattice type geometries by applying the principle of structural geometric duality, based on the concept of a point’s polarity regarding a sphere [3]. and geometric duality, based on the concept of a point’s polarity regarding a sphere [3]. One of the most paradigmatic examples of the lattice structure is the geodesic dome, originally One of the most paradigmatic examples of the lattice structure is the geodesic dome, originally applied by Bauersfeld in 1922, and then famously, thanks to Buckminster Fuller, in the early 1950s applied by Bauersfeld in 1922, and then famously, thanks to Buckminster Fuller, in the early 1950s [4]. [4]. The geodesic dome is based on the triangulated subdivision of the faces of regular polyhedra The geodesic dome is based on the triangulated subdivision of the faces of regular polyhedra (Platonic (Platonic solids) or semiregular polyhedra (Archimedean solids), in which the edges are subdivided solids) or semiregular polyhedra (Archimedean solids), in which the edges are subdivided into an into an equal number of parts (frequency). These polyhedra are highly symmetrical, belonging to equal number of parts (frequency). These polyhedra are highly symmetrical, belonging to different different symmetry groups, depending on being vertex-transitive, edge-transitive, and face- symmetry groups, depending on being vertex-transitive, edge-transitive, and face-transitive. Although transitive. Although the geodesic dome was originally conceived for spatial frames, its structure has the geodesic dome was originally conceived for spatial frames, its structure has also been applied to also been applied to other fields like optics (planetarium projections) [5], cartography (Dymaxion other ﬁelds like optics (planetarium projections) [5], cartography (Dymaxion Map, equal-area map Map, equal-area map projections) [6,7], biology (virus’ capsid design) [8], or chemistry projections) [6,7], biology (virus’ capsid design) [8], or chemistry (Buckminsterfullerenes molecules of (Buckminsterfullerenes molecules of carbon) [9]. carbon) [9]. In the late 1960s, Clinton worked on the systematization of the procedures for the design of In the late 1960s, Clinton worked on the systematization of the procedures for the design of geodesic tessellations of the sphere [10], obtaining eight different methods that could be applied to geodesic tessellations of the sphere [10], obtaining eight different methods that could be applied to two different topologies (Class I and Class II). During the next decade, H. M. Wenninger [11] two different topologies (Class I and Class II). During the next decade, H. M. Wenninger [11] proposed proposed a Class III inspired by the geometric characterization of virus capsules, which show certain a Class III inspired by the geometric characterization of virus capsules, which show certain skewed skewed topologies (Figure 2). topologies (Figure2). Yacoe and Davies proposed the geotangent structures in 1986 [12]. These panel structures can Yacoe and Davies proposed the geotangent structures in 1986 [12]. These panel structures can have two types of faces, (i) regular polygons on the equator and on the north pole of the sphere, and have two types of faces, (i) regular polygons on the equator and on the north pole of the sphere, and (ii) at least half of the other faces being non-equilateral pentagons or hexagons. The sphere that is (ii) at least half of the other faces being non-equilateral pentagons or hexagons. The sphere that is approximated by this kind of polyhedron is tangent to each edge in only one point, its section with approximated by this kind of polyhedron is tangent to each edge in only one point, its section with the the face’s plane being a circle inscribed in that face. face’s plane being a circle inscribed in that face.

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Figure 2. Different classes for geodesic structures depending on the orientation of the triangular Figure 2. Different classes for geodesic structures depending on the orientation of the triangular tessellation with respect to the basic triangle (parameters b and c) as explained in the literature [11]. tessellation with respect to the basic triangle (parameters b and c) as explained in the literature [11]. Because of the symmetry characteristics of the regular polyhedra used for this purpose, it is necessary to calculate just one single face (or a subdivision of the face, e.g., Schwartz triangle) in order Because of the symmetry characteristics of the regular polyhedra used for this purpose, it is to obtain the total definition of the global structure. necessary to calculateA specific justapplication one single of this face method (or ais subdivision the design of ofdriven the face,geometry e.g., in Schwartz architecture, triangle) using in order to obtain theparametric total definition modeling and of genetic the global algorithms structure. [13]. A specific application of this method is the design of driven geometry in architecture, using 2. Methodology parametric modeling and genetic algorithms [13]. 2.1. Geometry Concepts 2. Methodology The Poincaré model is adopted to support the approach developed in this work, consisting of a plane model of elliptic geometry obtained by means of a stereographic projection of the points 2.1. Geometrybelonging Concepts to a sphere (Pi) onto a plane (P’i), and vice versa. The equation for the sphere is x2 + y2 + (z 2 2 The Poincar− 0.5) = é(0.5)model, andis the adopted equation tofor support the plane the is z approach= 1 (upper developedtangent plane). in The this center work, of consistingthe of stereographic projection is the origin of coordinates (this projection is an inversion using the unit a plane modelsphere ofcentered elliptic at the geometry origin). In order obtained to obtain by a meanspoint-to-point of a stereographiccorrespondence between projection the sphere of the points 2 2 belongingand to athe sphere plane, the (Pi) working onto aEuclidean plane (P’i), plane and is extended, vice versa. introducing The equationa point at infinity for the (inversive sphere is x + y + (z − 0.5)plane)2 = (0.5) [14].2 , and the equation for the plane is z = 1 (upper tangent plane). The center of the stereographicAs projection a reminder, is in thethe bi-dimensional origin of coordinates projection of (thisthe elliptic projection geometry, is measurement an inversion of angles using the unit is conventional (the same as in Euclidean geometry), and lines (great circles of the sphere) are sphere centeredrepresented at the by origin).circles intersecting In order the to obtainstereographic a point-to-point projection of the correspondence equatorial great circle between in the the sphere and the plane,antipodal the (diametrically working Euclidean opposite) points. plane Henceforth, is extended, for the introducingsake of brevity, these a point lines atwill infinity be called (inversive plane) [14].as elliptic lines. As a reminder,The procedure in the bi-dimensionalstarts by selecting projectiona set of points of, S’, the inelliptic the plane geometry, (called sites), measurement from which the of angles is Delaunay triangulation (or Voronoi diagram) is calculated. The triangulation is stereographically conventionalprojected (the sameonto asthe insphere, Euclidean so that geometry), it creates an and inscribed lines (great (or circumscribed) circles of the polyhedron sphere) are that represented by circlesapproximates intersecting this the surface stereographic [15]. projection of the equatorial great circle in the antipodal (diametricallyThe opposite) projection points. is strictly Henceforth,stereographic for for triang theulations, sake of where brevity, each site, these P’i, linesof the willplane be is called as elliptic lines.transformed to a point, Pi, on the sphere. These points must be linked with respect to the connectivity of the original triangulation in z = 1, giving birth to an inscribed polyhedron. However, this cannot The procedurebe applied to starts the case by of selectingthe Voronoi a diagrams, set of points, where it S’, is necessary in the plane to project (called the points sites), of z from = 1 onwhich the Delaunaythe triangulation sphere and then (or calculate Voronoi their diagram) respective polar is calculated. planes. By changing The triangulation the points for the is planes, stereographically the projectedduality onto theof the sphere, problem in so 3D that space it is createsestablished. an A inscribedconventional (or projection circumscribed) from the origin polyhedron of the that approximatescoordinates this surface of the vertices [15]. of each Voronoi cell onto their respective polar plane generates the vertices of the faces belonging to the circumscribed polyhedron (refer to the examples shown in the section The projectionabout Study Cases). is strictly Taking stereographic into consideration for the triangulations, properties of Voronoi where diagrams, each the site, stereographic P’i, of the plane is transformedprojection to a point, and polarity Pi, on can the be sphere. combined These in a single points expression must be that linked directly with projects respect the vertices to the connectivityof of the originalthe cells triangulation onto their finalin positionz = 1, in giving the space birth [16]. to an inscribed polyhedron. However, this cannot be applied to the case of the Voronoi diagrams, where it is necessary to project the points of z = 1 on the sphere and then calculate their respective polar planes. By changing the points for the planes, the duality of the problem in 3D space is established. A conventional projection from the origin of the coordinates of the vertices of each Voronoi cell onto their respective polar plane generates the vertices of the faces belonging to the circumscribed polyhedron (refer to the examples shown in the section about Study Cases). Taking into consideration the properties of Voronoi diagrams, the stereographic projection and polarity can be combined in a single expression that directly projects the vertices of the cells onto their final position in the space [16]. Symmetry 2018, 10, 356 4 of 10

Symmetry 2018, 10, x FOR PEER REVIEW 4 of 10 The disposition of the sites chosen in z = 1 corresponds to the stereographic projection of the vertices of the The fundamental disposition of gridthe sites [17 ]chosen deﬁned in z on = 1 the corresponds sphere (Figure to the stereographic3). Until this projection moment, of itthe was not vertices of the fundamental grid [17] defined on the sphere (Figure 3). Until this moment, it was not possible to deﬁne these sites directly on the plane. The method to obtain them will be shown in the Symmetrypossible 2018 to define, 10, x FOR these PEER sites REVIEW directly on the plane. The method to obtain them will be shown in4 of the 10 followingfollowing section. section. The disposition of the sites chosen in z = 1 corresponds to the stereographic projection of the vertices of the fundamental grid [17] defined on the sphere (Figure 3). Until this moment, it was not possible to define these sites directly on the plane. The method to obtain them will be shown in the following section.

Figure 3. The definition of fundamental grid is cutting using parallel trihedra procedure [17]. Figure 3. The deﬁnition of fundamental grid is cutting using parallel trihedra procedure [17]. Rotations Rotations of great circles AM and BM around axis OE (perpendicular to OAB through O), each one of greatwith circles an AMangular and value BM aroundα (angle that axis must OE (perpendicular divide AOB in equal to OAB parts) through and opposite O), each directions, one with defines an angular

value αa(angle bi-directional that must grid in divide the spherical AOB in triangle equal ABM parts) as a and result opposite of the intersection directions, of the defines two families a bi-directional of grid in theFigurearcs. spherical This 3. Theis called triangledefinition the fundamental ABM of fundamental as a result grid. ofFromgrid the isthis intersection cutting grid, itusing is ofpossible parallel the two to trihedra familiesoptimally procedure of discretize arcs.This [17]. theis called the fundamentalRotationsinterior of of a grid. sphericalgreat Fromcircles patch thisAM by and grid, applying BM it around is symmetry possible axis OEoperations to (perpendicular optimally in space. discretize to OAB through the interior O), each of aone spherical with an angular value α (angle that must divide AOB in equal parts) and opposite directions, defines patch by applying symmetry operations in space. 2.2. Transformationsa bi-directional grid Procedure in the spherical triangle ABM as a result of the intersection of the two families of arcs. This is called the fundamental grid. From this grid, it is possible to optimally discretize the Let us consider the known the projections on the plane (A’, B’, C’) of the vertices (A, B, C) that 2.2. Transformationsinterior of a Procedure spherical patch by applying symmetry operations in space. define the spherical triangle (Figure 4), as well as the projection of the equatorial great circle Let2.2.(intersection us Transformations consider of the the Proceduresphere known with the the projectionsplane z = 0.5). on In the stereographic plane (A’, B’,projection, C’) of thegreat vertices circles and (A, B, C) that defineconcentric the spherical small circles triangle are transformed (Figure4 ),into as elliptic well aslines the and projection hyperbolic of pencils the equatorial of coaxal circles, great circle respectivelyLet us consider [14]. the known the projections on the plane (A’, B’, C’) of the vertices (A, B, C) that (intersectiondefine ofthe the spherical sphere triangle with the(Figure plane 4),z as= 0.5).well as In the the projection stereographic of the projection,equatorial great great circle circles and concentric(intersection small circles of the sphere are transformed with the∙ plane into z = elliptic0.5). In the lines stereographic∙ and̅ hyperbolic projection, pencils great circles of coaxal and circles, = or = (1) respectivelyconcentric [14]. small circles are transformed∙ into elliptic lines ∙and̅ hyperbolic pencils of coaxal circles, respectively [14]. This method makes use of the Möbiusa·z + b transformations (Equationa·z + b 1), consisting of rational linear 0 = 0 = functions of one complex variablez ∙ z, where a,or b, c, andz ∙ d ̅are complex constants necessary for (1) = c·z + d or = c· z + d (1) characterizing the transformation∙ [14]. These functions are∙ both̅ circular and conformal, so that they Thisreproduce method on makes the plane use theof transfor the Möbiusmations defined transformations on the sphere (Equation (rotations, reflections, (1)), consisting and spatial of rational linear functionsinversions),This method of both one directmakes complex (Equation use of variable the (1), Möbius leftz), and wheretransformati oppositea, b,ons (Equationc, and(Equationd are(1), 1),right) complex consisti [18].ng constants of rational necessary linear for characterizingfunctions the of transformationone complex variable [14]. z These, where functions a, b, c, and are d bothare complex circular constants and conformal, necessary so for that they characterizing the transformation [14]. These functions are both circular and conformal, so that they reproduce on the plane the transformations defined on the sphere (rotations, reflections, and spatial reproduce on the plane the transformations defined on the sphere (rotations, reflections, and spatial inversions),inversions), both directboth direct (Equation (Equation (1), (1), left) left and) and oppositeopposite (Equation (Equation (1), (1), right) right) [18]. [18].

Figure 4. Stereographic projection of the fundamental grid deployed on the spherical triangle. Great circles and concentric small circles are transformed into elliptic lines and hyperbolic pencils of coaxal

Figure 4. Stereographic projection of the fundamental grid deployed on the spherical triangle. Great Figure 4. Stereographic projection of the fundamental grid deployed on the spherical triangle. Great circles and concentric small circles are transformed into elliptic lines and hyperbolic pencils of coaxal circles and concentric small circles are transformed into elliptic lines and hyperbolic pencils of coaxal circles, respectively, in the inversive plane. The limiting points of those pencils (P´ in the ﬁgure) will coincide with the projection of the points where OE intersects the sphere. Symmetry 2018, 10, x FOR PEER REVIEW 5 of 10

Symmetrycircles,2018 respectively,, 10, 356 in the inversive plane. The limiting points of those pencils (P´ in the figure) will5 of 10 coincide with the projection of the points where OE intersects the sphere. The correct definition of a Möbius transformation, by means of the complex constants a, b, c, and d The correct definition of a Möbius transformation, by means of the complex constants a, b, c, and of Equation (1), requires the identification of three points and their images, because either they preserve d of Equation (1), requires the identification of three points and their images, because either they the cross ratio of four points, or they change it by their complex conjugation [18]. The correspondence preserve the cross ratio of four points, or they change it by their complex conjugation [18]. The between the points and images depends on the transformation to be implemented, as will be explained correspondence between the points and images depends on the transformation to be implemented, in the next paragraphs. as will be explained in the next paragraphs. Figure5 shows the homologous transformation of a rotation on the sphere around the polar axis Figure 5 shows the homologous transformation of a rotation on the sphere around the polar axis PQ. The endpoints of this axis (points P and Q) are projected onto the plane (P’ and Q’), which are PQ. The endpoints of this axis (points P and Q) are projected onto the plane (P’ and Q’), which are transformed onto themselves after the rotation (P’→P’; Q’→Q’). Another third point (A’) as well as its transformed onto themselves after the rotation (P’→P’; Q’→Q’). Another third point (A’) as well as transformed (B’), define the magnitude of the rotation to be performed (A’→B’). its transformed (B’), define the magnitude of the rotation to be performed (A’→B’).

Figure 5. Homologous transformation of a rotation on the sphere around the polar axis PQ. The Eq. Figure 5. Homologous transformation of a rotation on the sphere around the polar axis PQ. The Eq. circle is the projection of the equatorial great circle (intersection of the sphere with the plane z = 0.5). circle is the projection of the equatorial great circle (intersection of the sphere with the plane z = 0.5). Similarly, Figure 6 shows the homologous transformation of an inversion on the sphere, with respectSimilarly, to a reference Figure 6circle shows c [18] the passing homologous through transformation points A and ofB. anThis inversion curve is onprojected the sphere, into c’ with and thoserespect points to a reference into A’ circleand B’, c [ 18which] passing are throughself-inverse points points A and (A’ B.→ ThisA’, B’ curve→B’). is The projected constant into of c’ andthe inversionthose points is established into A’ and by B’, setting which arethe self-inverse correspondence points of (A’ any→ pointA’, B’→ withB’). its The transformed. constant of theWith inversion respect tois establishedany circle on by the setting sphere, the the correspondence polar points (P of and any Q) point of the with axis its (PQ) transformed. perpendicular With to respect the plane to any of suchcircle a on circle the are sphere, transformed the polar into points each (P other and Q)(P’ of→Q’). the axis (PQ) perpendicular to the plane of such a circleParticularly, are transformed an inversion into each on other the (P’sphere,→Q’). with respect to a great circle, is equivalent to a mirror symmetry,Particularly, with respect an inversion to the plane on the of sphere, the circle, with therefore respectto the a greatreflection circle, symmetry, is equivalent with to respect a mirror to ansymmetry, elliptic line, with will respect be characterized to the plane ofin thethe circle,same thereforeway as the the inversion. reﬂection symmetry, with respect to an elliptic line, will be characterized in the same way as the inversion.

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Figure 6. Homologous transformation of an inversion on the sphere with respect to a circle c. Figure 6. Homologous transformation of an inversion on the sphere with respect to a circle c. Correspondence of points: A’→A’; B’→B’; P’→Q’. The Eq. circle is the projection of the equatorial Correspondence of points: A’→A’; B’→B’; P’→Q’. The Eq. circle is the projection of the equatorial great circle (intersection of the sphere with the plane z = 0.5). great circle (intersection of the sphere with the plane z = 0.5). 2.3. Fundamental Grid and Propagation of Sites 2.3. Fundamental Grid and Propagation of Sites The following procedure will be used to propagate a fundamental grid in the projection of a The following procedure will be used to propagate a fundamental grid in the projection of a regular spherical triangle (see Figures 3 and 7). The starting point is the definition of the points A’, regular spherical triangle (see Figures3 and7). The starting point is the definition of the points A’, B’, C’, and O’ on the inversive plane by using stereographic projection, as well as the equatorial great B’, C’, and O’ on the inversive plane by using stereographic projection, as well as the equatorial circle. great circle. • Figure 7a: •1. DrawFigure side7a: A’B’ (there are two possible arches, keep the one at the opposite side point O’). 1.2. ObtainDraw side projections A’B’ (there P’ and are twoQ’ from possible the endpoints arches, keep of the onepolar at axis the oppositeperpendicular side point to the O’). plane of 2. theObtain great projections circle containing P’ and Q’A and from B the on endpointsthe sphere. of These the polar points axis are perpendicular the intersection tothe of the plane lines of perpendicularthe great circle to containing the side A’B’ A and from B onA’ theand sphere.B’. These points are the intersection of the lines 3. Divideperpendicular side A’B’ to theinto side n equal A’B’ parts from A’(points and B’. N’i) by intersection with the elliptic lines dividing 3. angleDivide A’P’B’ side into A’B’ n into equal n equalparts (these parts (pointslines belong N’i) to by an intersection elliptic pencil with of thecircles elliptic through lines P’ dividing and Q’). • Figureangle A’P’B’7b: into n equal parts (these lines belong to an elliptic pencil of circles through P’ and 4. DrawQ’). sides A’C’ and B’C’ and locate the middle point (M’) of the projected spherical triangle by • intersectionFigure7b: of the medians of both sides. 4.• FigureDraw sides7c: A’C’ and B’C’ and locate the middle point (M’) of the projected spherical triangle by 5. Defineintersection the Möbius of the medians transformation of both sides.that performs the rotations of consecutive division points • (N’i)Figure of7 A’B’.c: Apply, in the triangle A’M’B’, such rotations to side B’M’ (direction B’→A’) and to 5. sideDefine A’M’ the Möbius(direction transformation A’→B’). The that intersections performs theof rotationsthe rotated of consecutivesides define division the points points of (N’i) the fundamentalof A’B’. Apply, grid in the(which triangle belong A’M’B’, to a hyperbolic such rotations pencil to of side circles B’M’ [18]). (direction B’→A’) and to side A’M’ (direction A’→B’). The intersections of the rotated sides define the points of the fundamental • Figure 7d: grid (which belong to a hyperbolic pencil of circles [18]). 6. Propagate the fundamental grid to the whole projection of the spherical triangle by means of • Figure7d: reflection symmetry, with respect to lines A’M’ and B’M’. 6.7. PropagatePropagate the fundamentalsites of the whole grid toprojection the whole of projectionthe spherical of the triangle spherical to the triangle adjacent by meansones, by of meansreflection of symmetry,reflection withsymmetry, respect with to lines respect A’M’ andto its B’M’. three sides. Unknown vertices of the 7. projectionsPropagate the of sitesthe new of the spherical whole projection triangles ofare the obtained spherical by triangle inversion to the (for adjacent example, ones, vertex by means D’ is theofreflection inverse of symmetry, A’, with respect withrespect of side toB’C’). its three sides. Unknown vertices of the projections of the new spherical triangles are obtained by inversion (for example, vertex D’ is the inverse of A’, with respect of side B’C’).

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Figure 7. Propagation of the fundamental grid in the inversive plane. The Eq. Circ. is the projection Figure 7. Propagation of the fundamental grid in the inversive plane. The Eq. Circ. is the projection of of the equatorial great circle (intersection of the sphere with the plane z = 0.5). All elliptic lines intersect the equatorial great circle (intersection of the sphere with the plane z = 0.5). All elliptic lines intersect the Eq. Circ. in antipodal points. the Eq. Circ. in antipodal points.

2.4. Generalization of Models by Means of Power Diagrams 2.4. Generalization of Models by Means of Power Diagrams The design procedure outlined in the previous section allows to define both the inscribed and The design procedure outlined in the previous section allows to define both the inscribed and circumscribed polyhedra, depending on the actions applied to the points in plane z = 1. However, circumscribed polyhedra, depending on the actions applied to the points in plane z = 1. However, computational geometry has proved that it is possible to obtain solutions that are more generic when computational geometry has proved that it is possible to obtain solutions that are more generic when the following occurs: the following occurs: • The sites are substituted by circles with variable diameter. • The sitesVoronoi are substituteddiagrams and by Delaunay circles with triangulations variable diameter. are substituted by power diagrams (planar • Thesubdivision Voronoi by diagrams radical andaxes) Delaunay and their triangulationsdual graphs [1]. are substituted by power diagrams (planar subdivision by radical axes) and their dual graphs [1]. In this case, each cell of the power diagram will lead to a face of the polyhedron approximating the quadricIn this case,surface, each which cell of can the cut, power or not, diagram such a willsurface lead (Figure to a face 8, ofright). the polyhedron If this happens, approximating the section theproduced quadric by surface, the intersection which can is cut, stereographically or not, such a surfaceprojected (Figure onto8 the, right). plane If as this one happens, of the circles the section of the producedpower diagram. by the intersection is stereographically projected onto the plane as one of the circles of the powerFor diagram. those situations that require more symmetrical configurations, it is possible to select the pointsFor from those the situations fundamental that require grid, using more symmetricalthis grid as a configurations, design template it is to possible define tothe select generators the points (in fromthis case, the fundamentalcircles) of the grid, power using diagram this grid (Figure as a design8, right). template to define the generators (in this case, circles) of the power diagram (Figure8, right).

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Figure 7. Propagation of the fundamental grid in the inversive plane. The Eq. Circ. is the projection of the equatorial great circle (intersection of the sphere with the plane z = 0.5). All elliptic lines intersect the Eq. Circ. in antipodal points.

2.4. Generalization of Models by Means of Power Diagrams The design procedure outlined in the previous section allows to define both the inscribed and circumscribed polyhedra, depending on the actions applied to the points in plane z = 1. However, computational geometry has proved that it is possible to obtain solutions that are more generic when the following occurs: • The sites are substituted by circles with variable diameter. • The Voronoi diagrams and Delaunay triangulations are substituted by power diagrams (planar subdivision by radical axes) and their dual graphs [1]. In this case, each cell of the power diagram will lead to a face of the polyhedron approximating the quadric surface, which can cut, or not, such a surface (Figure 8, right). If this happens, the section produced by the intersection is stereographically projected onto the plane as one of the circles of the power diagram. For those situations that require more symmetrical configurations, it is possible to select the Symmetrypoints from2018, 10the, 356 fundamental grid, using this grid as a design template to define the generators8 of (in 10 this case, circles) of the power diagram (Figure 8, right).

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Figure 8. Left Left:: Projection Projection of of a a power power diagram diagram on on zz = 1 1 for defining deﬁning the most generic polyhedron

approximating the sphere. Right:: The The fundamental fundamental grid grid used used as as a a template template permits permits selecting selecting a a set set of of sites to locate the circles of the diagram, so that it leads to solutions that areare moremore symmetrical.symmetrical.

3. Study Study Cases Cases The procedureprocedure outlined outlined in thein the previous previous section section has been has conceived been conceived for the definition for the ofdefinition polyhedral of polyhedralstructures approximating structures approximating the sphere. Therefore,the sphere. it Therefor is not restrictede, it is not to restricted the field of to structural the field of engineering, structural engineering,and it can also and be it appliedcan alsoto be other applied fields to other such fields as those such cited as those in Section cited 1in. Additionally,Section 1. Additionally, it has been it hasstated been that stated it is alsothat possibleit is also topossible extend to the extend catalogue the catalogue of lattices of in lattices two directions. in two directions. The first first one is the the use use of of cutting cutting by by parallel parallel tr trihedraihedra procedure procedure [17], [17], which which allows allows for for generating generating two kinds ofof surfacessurfaces muchmuch moremore specialized, specialized, namely: namely: the the inflated inflated and and the the flattened flattened surfaces surfaces shown shown in inFigure Figure9 (left) 9 (left) and and Figure Figure9 (right), 9 (right), respectively. respectively.

Figure 9. SurfacesSurfaces resulting resulting from from the the aggregation aggregation of of Para Parallelllel Trihedra Trihedra [17], [17], according according to an an icosahedral icosahedral symmetry: inflated inflated ( (leftleft),), unaltered unaltered ( (middlemiddle)) and and flattened flattened ( (rightright).). It It is is also possible to obtain these configurationsconfigurations by the juxtaposition of congruentcongruent spherical triangles with non-aligned centers.centers.

The second one deals with the application of a projective transformation on any of the polyhedra The second one deals with the application of a projective transformation on any of the polyhedra associated to the sphere. This procedure allows for generating new meshes approximating other associated to the sphere. This procedure allows for generating new meshes approximating other elliptic quadrics, both lattice and panel structures, which can present revolution symmetry elliptic quadrics, both lattice and panel structures, which can present revolution symmetry (CR-tangent (CR-tangent meshes [15]) or not (C-tangent meshes [19]), as can be shown in Figure 10. meshes [15]) or not (C-tangent meshes [19]), as can be shown in Figure 10.

Figure 10. CR-tangent mesh, application to a paraboloid of revolution (left) and inflated mesh, application to a hyperboloid of revolution (right).

After obtaining the polyhedron approximating the sphere, it is possible to apply a specific homological transformation to the sphere (or to the obtained polyhedron) in such a way that the result is an elliptic quadric surface (or the approximating polyhedron). Figure 10 (left) shows an application to a paraboloid of revolution, while Figure 10 (right) depicts an application to a hyperboloid of revolution (globular).

Symmetry 2018, 10, x FOR PEER REVIEW 8 of 10

Figure 8. Left: Projection of a power diagram on z = 1 for defining the most generic polyhedron approximating the sphere. Right: The fundamental grid used as a template permits selecting a set of sites to locate the circles of the diagram, so that it leads to solutions that are more symmetrical.

3. Study Cases The procedure outlined in the previous section has been conceived for the definition of polyhedral structures approximating the sphere. Therefore, it is not restricted to the field of structural engineering, and it can also be applied to other fields such as those cited in Section 1. Additionally, it has been stated that it is also possible to extend the catalogue of lattices in two directions. The first one is the use of cutting by parallel trihedra procedure [17], which allows for generating two kinds of surfaces much more specialized, namely: the inflated and the flattened surfaces shown in Figure 9 (left) and Figure 9 (right), respectively.

Figure 9. Surfaces resulting from the aggregation of Parallel Trihedra [17], according to an icosahedral symmetry: inflated (left), unaltered (middle) and flattened (right). It is also possible to obtain these configurations by the juxtaposition of congruent spherical triangles with non-aligned centers.

The second one deals with the application of a projective transformation on any of the polyhedra associated to the sphere. This procedure allows for generating new meshes approximating other Symmetryelliptic2018, 10 quadrics,, 356 both lattice and panel structures, which can present revolution symmetry 9 of 10 (CR-tangent meshes [15]) or not (C-tangent meshes [19]), as can be shown in Figure 10.

Figure 10. CR-tangent mesh, application to a paraboloid of revolution (left) and inflated mesh, Figure 10. CR-tangent mesh, application to a paraboloid of revolution (left) and inﬂated mesh, application to a hyperboloid of revolution (right). application to a hyperboloid of revolution (right). After obtaining the polyhedron approximating the sphere, it is possible to apply a specific Afterhomological obtaining transformation the polyhedron to the sphere approximating (or to the obtained the sphere, polyhedron) it ispossible in such a toway apply that the a speciﬁc result is an elliptic quadric surface (or the approximating polyhedron). Figure 10 (left) shows an homological transformation to the sphere (or to the obtained polyhedron) in such a way that the result application to a paraboloid of revolution, while Figure 10 (right) depicts an application to a is an elliptichyperboloid quadric of revolution surface (or (globular). the approximating polyhedron). Figure 10 (left) shows an application to a paraboloid of revolution, while Figure 10 (right) depicts an application to a hyperboloid of revolution (globular).

4. Discussion The proposed procedure assures the correct generation of meshes for any distribution of points in plane z = 1. Therefore, an adequate modiﬁcation of the distribution of the sites would allow for emulating the solutions proposed by other researchers (Fuller, Clinton, Wester, etc.). Until now, the comparison of all of these models has been carried out using only graphical methods. When the distribution of the sites matches the points of a fundamental grid, the result is an optimal mesh [17], both for the lattice or the panel conﬁgurations (because of the duality between Delaunay triangulations and Voronoi diagrams). The projective transformation generates polygons whose sizes are larger when they are farther from the origin of projection. Improved knowledge of the Möbius transformations is just one of the ways of ensuring that the edges of the polygons obtained in this way are more similar among them.

5. Conclusions It is possible to design polyhedral structures approximating the sphere by means of procedures in the bi-dimensional plane. The application of Möbius transformations allows for replicating the operations (symmetry, rotation, and inversion) in the plane performed in the three-dimensional space. Therefore, an appropriate selection of sites can be made using computational geometry to generate (both lattice and panel) space structures with an optimal size and shape. As a result, computational geometry becomes a useful tool capable of providing new ideas for the development of design software and structure calculation, giving support to many different ﬁelds of science and technology. By means of performing projective transformations, the polyhedra that approximate the sphere can become the polyhedra approximating any elliptic quadrics, although the optimal shape and size of the lattice or panels cannot be maintained. However, these kinds of transformations open a wide new catalogue of original spatial structures.

6. Further Research Some interesting questions can still be asked from a structural point of view, namely: “Would it be possible to calculate the spatial mesh in the inversive plane?”, “How could we dimension geometrical Symmetry 2018, 10, 356 10 of 10

and mechanical parameters to perform those calculations?”, and “How could we introduce the loads and boundary conditions to the so-deﬁned structure?”. From a different point of view, now taking into consideration the design and fabrication process, “Would it be possible to represent on the drawings the three-dimensional components (e.g. bars, nodes, and panels), usually represented symbolically (e.g. lines, points, and polygons) in a realistic manner?”.

Author Contributions: The authors contributed equally to this work, which comes integrally from J.A.D.-S.’s doctoral thesis, supervised by C.O. The methodology and supervision of article and thesis, C.O.; investigation and writing (original draft), J.A.D.-S.; software development for spatial space models and computational geometry, C.M.; writing, reviewing, and editing final manuscript, V.G.-J. Funding: This research received no external funding. Acknowledgments: Thanks to INGEGRAF for their contribution and financial support in the publishing process of this article. Conflicts of Interest: The authors declare no conflict of interest.

References

1. Aurenhammer, F. Power Diagrams: Properties, Algorithms and Applications. SIAM J. Comput. 1987, 16, 78–96. [CrossRef] 2. Diaz, J.A.; Otero, C.; Togores, R.; Manchado, C. Power diagrams in the design of Chordal Space Structures. In Proceedings of the 2nd International Symposium on Voronoi Diagrams in Science and Engineering, Seoul, Korea, 10–13 October 2005; pp. 93–104. 3. Wester, T. A Geodesic Dome-Type Based on Pure Plate Action. Int. J. Space Struct. 1990, 5, 155–167. [CrossRef] 4. Fuller, R.B. Building Construction. U.S. Patent US2682235, 29 June 1954. 5. Bauersfeld, W. Projection Planetarium and Shell Construction. Proc. Inst. Mech. Eng. 1957, 171, 75–80.[CrossRef] 6. Fuller, R.B. Cartography. U.S. Patent US2393676, 29 January 1946. 7. Snyder, J.P. An equal-area map projection for polyhedral globes. Cartographica 1992, 29, 10–21. [CrossRef] 8. Coxeter, H.S.M. Virus macromolecules and geodesic domes. In A Spectrum of Mathematics; Butcher, J.C., Ed.; Auckland University Press: Auckland, New Zealand, 1972; pp. 98–108. 9. Clinton, J.D. A topological model of the Buckminsterfullerene family of molecules and other clusters having icosahedral symmetry. Unpublished work, 1992. 10. Clinton, J.D. Advanced Structural Geometry Studies. Part I: Polyhedral Subdivision Concepts for Structural Applications; NASA-CR-1734; NASA: Washington, DC, USA, 1971. 11. Wenninger, M.J. Spherical Models; Cambridge University Press: Cambridge, UK, 1979. 12. Yacoe, J.C. Polyhedral Structures that Approximate a Sphere. U.S. Patent US4679361, 14 July 1987. 13. Turrin, M.; von Buelow, P.; Stouffs, R. Design explorations of performance driven geometry in architectural design using parametric modeling and genetic algorithms. Adv. Eng. Inform. 2011, 25, 656–675. [CrossRef] 14. Coxeter, H.S.M. Introduction to Geometry; John Wiley and Sons: New York, NY, USA, 1969. 15. Otero, C.; Gil, V.; Alvaro, J.I. CR-Tangent Meshes. IASS 2000, 41, 41–48. 16. Rey Pastor, J.; Santaló, L.A.; Balazant, M. Geometría Analítica, 5th ed.; Kapelusz: Buenos Aires, Argentina, 1969; pp. 481–484. 17. Alvaro, J.I.; Otero, C. Designing optimal spatial meshes: Cutting by parallel trihedra procedure. IASS 2000, 41, 101–110. 18. Schwerdtfeger, H. Geometry of Complex Numbers. Circle Geometry, Moebius Transformation, Non-Euclidean Geometry; Dover Publications: New York, NY, USA, 1979. 19. Otero, C.; Togores, R. Computational Geometry and Spatial Meshes. In Computational Science-ICCS 2002; Sloot, P.M.A., Hoekstra, A.G., Tan, C.J.K., Dongarra, J.J., Eds.; Springer: Heidelberg, Germany, 2002; pp. 315–324.

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