Reinaldo Togores Fernández, César Otero González, José Andrés Díaz Severiano, Cristina Manchado del Val, Rubén Arias Fernández.
EGICAD Research Group, Geographic Engineering and Graphic Expression Techniques Department.
University of Cantabria.
University of Cantabria.
Abstract. This paper describes a research line that deals with the integration of Computational Geometry methods in a procedure which implements the geometric design of Space Structures that approximate the sphere and other quadric surfaces. As is well known, the classic procedures for the design of the so called Geodesic Structures resort to the subdivision of one face of the selected base polyhedron (usually an icosahedron) and the Gnomonic Projection of the resulting tessellation on the sphere, subsequently propagating this tessellation to the rest of the sphere through symmetry operations. The possibility of applying this single procedure to the design of Lattice (Fuller’s model), Plate (Wester’s model) and Geotangent (Yacoe-Davies’ model) Structures and generating from them a variety of complex non-spherical polyhedral shape combinations lead to a wide range of formal resources in which the Architect can find sources for inspiration. The outcome of this research has been presented in the Voronoi Diagram – Evolutionary Space Art Exhibition, Seoul, 2006.
Fuller arrives to essentially the same solution thirty years later following a path analogous to that of Bauersfeld. Since the late thirties he is interested in alternative representations of the earth globe. The projection of the globe on a flat surface entails problems similar to those which Bauersfeld had to solve: transferring a spherical (at least in appearance) surface to a mosaic of planar surfaces, in this case the faces of a Cuboctahedron (Figure 2, left). In this map, that Fuller named “Dymaxion”, a correspondence is established between the location of a point on the sphere and its position on the plane by referring it to a grid of greater circle arcs whose gnomonic projection on the corresponding planar face of the polyhedron is a straight line. If we compare the triangular grid included in the Dymaxion Map  patent documentation (Figure 2, right) and the Jena planetarium structure (Figure 1, left), the similarity is surprising. The Bauersfeld dome had been patented, but only as a shell structure, with no mention to the reinforcement’s geometry. This allowed Fuller to assume in 1951 the paternity of “a building framework of generally spherical form in which the main structural elements are interconnected in a geodesic pattern of approximate great circle arcs intersecting to form a three-way grid defining substantially equilateral triangles”. By this time Fuller had abandoned the Cuboctahedron as the base polyhedron, using, as Bauersfeld had done, the Icosahedron as his structure’s starting point.
Figure 2. Dymaxion Map, in its original version using the Cuboctahedron as base polyhedron, and the tessellation proposed by Fuller for the projection on one of the triangular faces of the Cuboctahedron.
Geodesic Structures were conceived by Fuller during his stay as a visiting professor at Black Mountain College  during the summers of 1948 – 1949. Fuller describes in U.S. Patent 2,682,235  a way to calculate the edge lengths which is adequate enough for his art school students:
“One way of determining the strut lengths is to construct a paperboard hemisphere to a scale of, say, 1 inch to 1 foot, and lay out the vertexes of one of the faces of a spherical icosahedron on its surface. These vertexes are next connected by drawing great circle lines (spherical straight lines) therebetween. The edges of the triangle defined by these lines are next divided equally into the number of units represented by the selected grid frequency. The division points are then connected by drawing great circle lines in the manner clearly shown by Figs. 1 and 2. (Note that the points along one edge are connected to every second point on another edge.)Such a procedure, which obviously satisfies their needs, lacks the precision demanded by real-world building techniques. This is where Computer Science comes in as a new instrument in the Architect and Engineer’s toolbox. The initial effort towards the systematization of the procedures for the design of geodesic tessellations of the sphere was undertaken under the leadership of Joseph D. Clinton in the Southern Illinois University (Carbondale Campus). These studies, that took place between 1964 and 1971, were sponsored by NASA on the belief that this type of structures had possible applications in future space missions. As a result, Clinton published  eight methods (with their corresponding computer programs) that could be applied to two different topologies for which he proposed the names of Class I and Class II. The methods described by Clinton can also be applied to Regular Polyhedra with triangular faces other than the Icosahedron.
We have now a completed three-way grid pattern. Finally the length of the chordal struts is measured directly with the use of ordinary draftsman’s dividers, allowance being made for the strut fastenings.”
Figure 3. Geodesic Dome used for sheltering the DEW Line’s radar antennae.
All of these methods start with a subdivision of the edges of one of the base polyhedron’s faces (or a part of this face) to obtain a tridimensional mesh the vertices of which are transferred by means of a Gnomonic Projection to the surface of the sphere. These methods differ in:
- The topology, that may be of Class I (mesh directions parallel to the face edges) or Class II (directions perpendicular to the edges).
- The tessellation being calculated upon the whole face or upon its Schwartz Triangle (a sixth part of the face, 1/120 of the sphere in the case of the Icosahedron.
- The division of the edge into equal lengths or into unequal lengths derived from equal central angles in the polyhedron (obtaining this way greater circle arcs equal in length).
Figure 4. Class I (left), Class II (center) and Class III (right) tessellations.
Figure 5. Left: Geodesic Lattice Structures. Center: Their dual Panel Structures. Right: a Geotangent Polyhedron.
The Geotangent Structures proposed by Yacoe and Davies in 1986  can have two types of faces that are regular polygons on the equator and on the north pole of the sphere), and 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 the face’s plane being a circle inscribed in that face (Figure 5, right). Each one of these circles is tangent to those sphere section circles inscribed in adjacent polygonal faces. The procedure used for the calculation of the form an dimensions of the faces that is described in the patent document is very complex, reaching a solution through a lengthy iterative process.
Figure 6. Stereographic and Gnomonic Projections according to Coxeter.
Reexamining Coxeter’s “Introduction to Geometry” , we find a figure (see Figure 6) that can be revealing of the differences between the approach we are dealing with in our research and the classic method for the design of spherical meshes. Two paradigms are here compared: Stereographic versus Gnomonic projections. The figure on the left represents plane σ and the sphere σ’. This sphere results from the inversion of this plane with respect to an inversion sphere (not shown in the drawing) of radius NS centered in N. For each point in plane σ there is a corresponding a point on the sphere σ’ that is the inverse of plane σ with respect to the sphere of radius NS. The inverse of point N is the point at the infinite. In this case we are considering a Stereographic Projection.
In the Gnomonic Projection represented on the right side of Figure 6, instead of projecting from N, the projection is done from the center O of the sphere. In this case each point on the plane is represented by two points on the sphere, P1 and its antipodal P2. The circles on the sphere are inverted in the Stereographic Projection as circles on the plane and viceversa. In a Gnomonic Projection greater circles are projected on the plane as straight lines.
Figure 7 represents the same Stereographic Projection shown in Coxeter’s illustration, but this time upside down. The sphere Cai with respect to which the inversion is made (omitted in Figure 6) is now a sphere of radius 1, centered in the coordinate system’s origin. The z=1 plane (and consequently points P1, P2, P3, … Pn) are inverted on the sphere C of radius ½, tangent to plane z=1 on its north pole, that has its south pole in the coordinate system’s origin.
Figure 7. Inversion of the set of points contained in plane z=1 on a sphere of radius ½.
The alternative procedure we are presenting assumes a set of points in plane z=1 that are transferred to the radius = ½ sphere through a Stereographic Projection. In this case, one single plane is capable of representing the sphere as a whole, the South Pole being the point at infinity. The mathematical treatment can be found in references , , .
- The Circumsphere on which all of the vertices lie.
- The Insphere, which all the faces touch.
- The Midsphere, which all of the edges touch.
- The Fuller (Geodesic Structure) Model to the Circumsphere.
- The Wester (Panel Structure) Model to the Insphere.
- The Yacoe (Geotangent Structure) Model to the Midsphere.
- In a Geodesic Structure (Fuller’s Model), the points (lattice nodes) will be considered as the vertices of the triangular faces of the approximating polyhedron.
- In a Panel Structure (Wester’s Model), they will represent those points where the polyhedron’s faces touch the Insphere.
- In a Geotangent Structure (Yacoe’s Model) they will correspond to the points where the polyhedron’s edges touch the Midsphere.
Figure 8. Left: Schlegel Diagram for an Icosahedron. Right: Inversion of a Delaunay Triangulation on the sphere.
The Wester Model.
The Wester Model.
Figure 9. Wester Model generation procedure and model created from a random set of points on plane z=1.
Figure 10. Circle packing and the Power Diagram derived from it.
The tangency points T between the radical axes and the circles are transformed into the points T’ on the sphere E centered in (0, 0, ½) and radius = ½.
Edges a’1, a’2 and a’3, tangent to the sphere in points T’1, T’2 and T’3 (points which define a plane α that intersects the sphere), meet in point V’ (Figure 11, right). These edges are the generatrices of a cone circumscribed to the sphere, and this cone’s vertex (at the same time a vertex of the approximating polyhedron) is the pole of plane α with respect to the sphere E.
Figure 11. Inversion of the circle packing and determination of the vertices.
This way we demonstrate that, by means of the procedure we have proposed, the Delaunay Triangulation, the Voronoi Diagram and the Power Diagram generated from a circle packing, will generate polyhedra approximating the sphere that will in turn reproduce the Fuller, Wester and Yacoe structural models.
P’’=MHOM *MINV *PGeneration of the Hyperboloid and Ellipsoid of Revolution. A procedure such as described that includes the inversion of a set of points in plane z=1 and the projective transformation MHOM may also be applied  to a sphere whose radius is not equal to ½. The south pole of this sphere is (0, 0, 2zc – 1). If inversion is applied from here, the inverse points of plane z=1 fall on the sphere CG. If the projective transformation MHOM is then applied the sphere will be transformed into the Quadric of Revolution CG. The nature of the resulting surface (Figure 12) depends on the projection center’s position:
- When zc =½, CG will be a Paraboloid (Figure 12, left).
- When zc <½, CG will be a 2 Sheet Hyperboloid (Figure 12, center).
- When zc >½, CG will be an Ellipsoid (Figure 12, right).
Figure 12. Paraboloid, Hyperboloid and Ellipsoid generated by this procedure.
- its radius restricted to being equal to ½.
- its center restricted to being on the OZ axis
Figure 13. Skewed quadric surfaces: 2 Sheet Hyperboloid and Ellipsoid.
The substitution of Euclidean metrics by Elliptic metrics has been experimented with certain success  in defining the point layout for the base triangle, distribution which can then be extended to the rest of the plane through Möbius transformations (Figure 14).
Figure 14. Point layout and its propagation through Möbius transformations.
But the non-spherical surface approximating polyhedra generated from the same set of points in z=1 that will produce an acceptable tessellation of the sphere will display a great variety in face sizes. This lack of regularity is usually deemed as a disadvantage from both structural and aesthetic reasons. A line of research we are pursuing at this moment deals with the possibility of defining geometric distributions of the generator points that will result in an increased regularity of the non-spherical quadrics approximating polyhedra compensating in length and area its faces. Through the use of Hyperbolic Geometry reproducing the well-known layout of Poincaré’s Disk (Figure 15, left) a possible approach has been demonstrated. Although the goal of reducing the size and area variation in faces is approximated, further studies are needed to reduce the number of different polygons generated (Figure 15, right).
Figure 15. Paraboloid structure generated using the vertices of a Poincaré Disk.
The real radius spheres in pseudo-Euclidean space have the appearance of one sheet hyperboloids, those with purely imaginary radius are represented as hyperboloids of two sheets and those with zero radii take the form of cones of revolution. In addition to Euclidean and Minkowskian spaces , semi-Euclidean spaces can also be considered, in which circular and hyperbolic cylinders stand for spheres centered on the origin of coordinates. Applying the stereographic projection of the sphere associated to them will produce the planar models (Figure 16) proposed by Poincaré for the study of the Cayley-Klein geometries.
Figure 16. Planar models of the Cayley-Klein geometries generated by Stereographic Projection.
This possibility coupled with the ability to generate a variety of skewed quadric surfaces points at a field of research in which mathematicians, architects and artists can join forces in the creation of up to now unsuspected structures, both utilitarian and aimed at the pure joy of the spirit. The Voronoi Diagram – Evolutionary Space  Art Exhibition held in Seoul in June 2006, in which along others, the outcome of this line of research was featured, served as a showcase for the wide spectrum of possibilities Computer Science offers for the generation of Architectural forms through Computer Science.
Figure 18. Complex structures derived from the combination of polyhedral patches.