Federico Fallavollita (Università degli Studi di Bologna, Italy)

Copyright: © 2016
|Pages: 21

DOI: 10.4018/978-1-5225-0029-2.ch010

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TopThis paper describes the use of the synthetic method through digital generative algorithms in the study of geometry. In particular, this paper describes the construction of a ruled surface that rests on three skew lines in the method of mathematical representation. The study is part of a bigger picture that has as its goal the renewal of the study of geometry according to the synthetic method, in other words through the use of drawing in the virtual laboratory.

The drawing in this sense is a logical tool to understand the geometry and the problems in space. The possibility of constructing algorithms to generate and control geometric shape allows having a greater control on both the final form and, above all, on the generative process of form itself. This is particularly important for those who have a synthetic approach in the study of geometry. It refers to the ability to view the entire creative process of the shape and the ability to alter some parameters to improve the result. The representation of the connections and the algorithms that structure shape allows having both an overview and a detailed view of the entire creative process.

In computing environment, there are several software platforms to create algorithms and scripting. Scripting is becoming more and more an integral part of the design process. “Scripting is a rather loose term by any definition and in this primer can be taken to mean computer programming at several levels. For the novice dabbling at the more accessible end of the user spectrum, scripting is the capability offered by almost all design software packages that allows the user to adapt, customize or completely reconfigure software around their own predictions and modes of working. At its most demanding for the emerging connoisseur, scripting can refer to higher-level computer programming where, in the ‘open-source’ environment, ‘libraries’ of functions can be combined with preconfigured routines (algorithms) as a means to produce manufacturer-independent digital design capability. At its simplest, therefore, affords a significantly deeper engagement between the computer and the user by automating routine aspects and repetitive activities, thus facilitating a far greater range of potential outcomes for the same investment of time (Burry, 2011, p. 8).”

Today designer, as an architect who designs buildings or as a researcher who studies geometric shapes in space, has available software platforms like *Grasshopper* which have a completely synthetic approach to problems of generative form. This allows designers to be the creators of the algorithms that generate and control shape. This aspect of the culture of digital design is crucial. It is no longer necessary to be a computer wizard to approach these instruments and the combination of the generative potential of these new tools and knowledge of designers can give interesting results. Generative design through use of digital algorithms allows strengthens the heuristic potential of design as a research tool and better explains how the object is constructed rather than describing its shape. What matters it is not only the final shape but how you get to that form. The logic of the construction becomes part of the representation of the form itself. A representation that takes place directly in the space of three-dimensional virtual laboratory.

A virtual laboratory is intended by the author as a *bottega virtuale* where you can experiment and draw geometric shapes directly in space. In other words any environment of computer graphics. As is well known the methods of digital representation used in the virtual laboratory can be of two types: the mathematical representation method and numerical representation method. The first method, the mathematical representation, is the description of three-dimensional forms made by means of the mathematical equations or by means of the NURBS equations; the numerical representation uses instead lists of coordinates of points, the connections between these points and faces formed by the connections. The former is continuous, the latter is discrete. This study adopted both digital methods. The main difference is how they are used. In this sense there is no difference from the classical methods of descriptive geometry. To continue the comparison we can say that the use of the mathematical method is the same as the method of Monge and the use of the numeric representation is the analogue of the method of perspective. In fact, the former is used to design, construct and measure the shapes in space; the latter, instead, is used to visualize and formally control forms in space.

Virtual Laboratory: A virtual laboratory is intended as a bottega virtuale where you can experiment and draw geometric shapes directly in space. In other words any environment of computer graphics.

Elliptic Hyperboloid: Three skew curves identify a ruled surface. When the three curves become three straight lines, the surface is a quadric ruled surface. As is well known, there are two cases: 1) if the three lines are parallel to a plane (director plan) but not to each other, the surface that is generated is a hyperbolic paraboloid (special case); 2) instead, if the three skew lines identify in pairs non-parallel planes, the surface is a elliptic hyperboloid (general case). The elliptic hyperboloid is a quadric surface with three distinct principal axes. The elliptic hyperboloid of one sheet is a doubly ruled surface.

Numerical Representation: The numerical representation uses instead lists of coordinates of points, the connections between these points and plane faces formed by the connections. This representation is discrete.

Hyperbolic Paraboloid: Three skew curves identify a ruled surface. When the three curves become three straight lines, the surface is a quadric ruled surface. As is well known, there are two cases: 1) if the three lines are parallel to a plane (director plan) but not to each other, the surface that is generated is a hyperbolic paraboloid (special case); 2) instead, if the three skew lines identify in pairs non-parallel planes, the surface is a elliptic hyperboloid (general case). The hyperbolic paraboloid is a quadric surface with one distinct principal x-axis. The other two axes y and z are at infinity; the surface has, all the same, three orthogonal principal planes. The hyperbolic paraboloid is a doubly ruled surface.

Ruled Surface: A ruled surface is a surface that can be generated by moving a line in space. The Gaussian curvature on a ruled regular surface is everywhere non-positive.

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