Mathematics and the Built Environment Volume 5 Series Editors Michael Ostwald , Built Environment, University of New South Wales, Sydney, NSW, Australia Kim Williams, Kim Williams Books, Torino, Italy Edited by Kim Williams and Michael Ostwald. Throughout history a rich and complex relationship has developed between mathematics and the various disciplines that design, analyse, construct and maintain the built environment. This book series seeks to highlight the multifaceted connections between the disciplines of mathematics and architecture, through the publication of monographs that develop classical and contemporary mathematical themes – geometry, algebra, calculation, modelling. These themes may be expanded in architecture of any era, culture or style, from Ancient Greek and Rome, through the Renaissance and Baroque, to Modernism and computational and parametric design. Selected aspects of urban design, architectural conservation and engineering design that are relevant for architecture may also be included in the series. Regardless of whether books in this series are focused on specific architectural or mathematical themes, the intention is to support detailed and rigorous explorations of the history, theory and design of the mathematical aspects of built environment. More information about this series at http://www.springer.com/series/15181 Alberto Lastra Parametric Geometry of Curves and Surfaces Architectural Form-Finding Alberto Lastra Departamento de Física y Matemáticas Universidad de Alcalá Alcalá de Henares, Spain ISSN 2512-157X ISSN 2512-1561 (electronic) Mathematics and the Built Environment ISBN 978-3-030-81316-1 ISBN 978-3-030-81317-8 (eBook) https://doi.org/10.1007/978-3-030-81317-8 Mathematics Subject Classification: 53A04, 53A05, 00A67 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. 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Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com, by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland A mis abuelos Preface The parametric aspects of curves and surfaces have been studied from the point of view of differential geometry through history. Indeed, many different studies have been developed since the nineteenth century on this discipline, which can be found in detail in texts such as do Carmo (1976), Tapp (2016), Umehara et al. (2017). Apart from the theoretical relation of a curve (or surface) to any of its parametrizations, one can go a step further and describe it from a practical point of view. The geometric scheme of a curve or a surface has provided inspiration for numerous works of art and architecture. These creations not only respond to physical needs such as certain acoustic properties, lighting, etc., but also to a human desire to create structures with simple geometric shapes. This books describes the classical theory of parametric tools in the geometry of curves and surfaces with an emphasis on applications to architecture. This book is based on a decade of teaching a class on geometry in architectural studies at Universidad de Alcalá (Spain). This class, “Drawing Workshop II”, combined architectural design and mathematics. Nevertheless, it can also be used as a text on differential geometry for mathematics students, or a basic reference on mathematics for architects and designers (especially those working with CAD). I hope that the latter will find this text useful and interesting, shedding light on the theoretical aspects of their work, as well as on the applicability to architecture. The techniques used in the examples provided in the text serve as the mathematical realization of many geometric tools used in CAD programs such as the construction of an helix, extrusions, revolution or ruled surfaces, projections, and many others. I also provide algorithms related to some of the geometric objects and show how different actions on the parametrizations change the nature of the geometric object itself. These mathematical tools are important to understand the structure of a geometric object and to know how to modify it consciously. The structure of the book is as follows. The first chapter is devoted to the study of parametrization of plane curves, with special focus on conics. In this chapter, the implicitation and approximation of curves is also illustrated with geometric examples. vii viii Preface Fig. 1 Structure of the book The second chapter describes a parallel theory on space curves, and the appear- ance of such curves in architecture. Geometric transformations are performed on a space curve, making explicit the mathematics behind usual actions in curve design. The third and fourth chapters consider, respectively, general surfaces and other particular classes of surfaces which are of widespread use. The examples range from classic surfaces in architecture to the parametrization of such families or the construction of other which remain of particular interest regarding their properties. More precisely, we focus on some curves lying on surfaces and on the intersection of curves. Surfaces such as quadrics, ruled surfaces, surfaces of revolution, minimal and developable surfaces are also studied and applied to architectural elements. The structure of the book is illustrated in Fig. 1. The mathematical prerequisites for this book are first courses in topology, linear algebra and calculus (both single and multi- variable), as amply covered in the books Salas et al. (2003), Marsden and Tromba (2012), and Lang (1986); Strang (1993). For completeness, we have included two appendices covering knowledge that will be useful for understanding the material. The figures of geometric objects have been created with Geogebra software. I want to express my gratitude to everyone who was involved in this class, specially to Prof. Manuel de Miguel, who introduced me to the world of architecture. I also want to express my gratitude to Remi Lodh, who has guided me on its publication with high professionalism and also to Kim Williams for her enthusiasm, professionalism and effort in the revision of the manuscript, and also giving relevant and interesting details. Suggested Further Reading The following sources are suggested to interested readers seeking additional material (AAG 2008, 2010, 2013, 2014, 2016, 2018; Bridges 2003, 2004, 2008, 2011, 2012, 2014, 2016, 2018). Alcalá de Henares, Spain Alberto Lastra 2021 Contents 1 Parametrizations and Plane Curves ....................................... 1 1.1 PlaneCurvesandParametrizations.................................... 2 1.2 SomeClassicCurvesinArchitecture.................................. 9 1.3 SomeElementsofRegularPlaneCurves.............................. 15 1.4 Conics................................................................... 26 1.5 SomeConicsinArchitecture........................................... 38 1.6 OntheImplicitationandParametrizationofCurves.................. 41 1.7 ApproximationandInterpolationofCurves........................... 51 1.8 Suggested Exercises .................................................... 55 2 Parametrizations and Space Curves ....................................... 59 2.1 SpaceCurvesandParametrizations.................................... 59 2.2 SomeElementsofRegularSpaceCurves............................. 66 2.3 SomeClassicSpaceCurvesinArchitecture .......................... 87 2.4 Rigid Transformations in R3 ........................................... 90 2.5 SomeTransformationsona Helix ..................................... 96 2.6 Suggested Exercises .................................................... 101 3 Parametrizations and Regular Surfaces ................................... 103 3.1 Surfaces and Parametrizations ......................................... 103 3.2 Some Classic Surfaces in Architecture ................................ 116 3.3 Projections of Surfaces onto Planes ................................... 123 3.4 Curves in Surfaces and Intersection of Surfaces ...................... 126 3.5 Suggested Exercises .................................................... 136 4 Special Families of Surfaces ................................................ 139 4.1 Ruled Surfaces .........................................................