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. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. 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 ...... 139 4.2 Some Subfamilies of Ruled Surfaces ...... 148 4.3 Parametrization of Some Ruled Surfaces...... 154 4.4 Surfaces of Revolution ...... 160 4.5 Quadric Surfaces...... 169 4.6 QuadricsRevisited:SomeExamplesinArchitecture...... 187

ix x Contents

4.7 Curvature: Minimal and Developable Surfaces ...... 190 4.7.1 FinalComments ...... 200 4.8 Suggested Exercises ...... 201

A Coordinate Systems ...... 205 B Mathematical Tool Kit...... 213 B.1 Introduction to Linear Algebra...... 213 B.1.1 SystemsofLinearEquations...... 213 B.1.2 Vector Spaces ...... 214 B.1.3 Euclidean Vector Spaces...... 215 B.1.4 Diagonalization: Eigenvalues and Eigenvectors...... 215 B.2 RealFunctionsofOneVariable...... 216 B.3 FunctionsofSeveralRealVariables...... 219 B.4 DifferentialEquationsandSystemsofDifferentialEquations...... 221 C Solution to the Suggested Exercises ...... 223 C.1 Chapter1...... 223 C.2 Chapter2...... 235 C.3 Chapter3...... 245 C.4 Chapter4...... 249 References...... 267 Index ...... 273 About the Author

Alberto Lastra has a Ph.D. in Mathematics by the University of Valladolid. He is associate professor at the University of Alcalá (Spain). He has been teaching mathematics in the degree of Architecture and Fundamentals in Architecture and Urbanism at the University of Alcalá since 2011, in subjects under the point of view of innovation and interdisciplinary thinking in Architecture. His research interests do not only go in the previous direction, but also in the study of asymptotic analysis of functional equations in the complex domain and related topics, symbolic computation, or orthogonal polynomials. He is a member of the research groups ECSING-AFA of the University of Valladolid and ASYNACS (CT-CE2019/683) of the University of Alcalá. He has also been a visitor at foreign research centers during the last decade, such as the University of Lille (France), the University of Warsaw (Poland), the University of La Rochelle (France), Universidade Federal de Minas Gerais (Brazil), among others.

xi List of Symbols

AT Transpose matrix of the matrix A C∞(U) Set of scalar or vector functions which are differentiable for every degree of differentiation in the open set U d(P,Q) Euclidean distance from a point P ∈ Rn to Q ∈ Rn dX (v) X : U ⊆ R2 → R3 (u ,v ) ∈ U (u0,v0) differential of at 0 0 , evaluated at v ∈ R2 D(P, r) Disc centered at the point P and radius r>0 d dx derivative with respect to the variable ∂ ∂ ∂ ∂x , ∂y , ∂z,. . . Partial derivative with respect to x, y, z,... Mm×n(K) Set of m × n matrices with coefficients in a field K I(ω1,ω2) First fundamental form II(ω) Second fundamental form ∇f(P) Gradient of the function f , evaluated at the point P ⊥ Orthogonal rank(A) Rank of a matrix A ∼ Asymptotic equivalence ·, · or · Inner product in Rn × Cross product in R3 [·, ·, ·] Scalar triple product · Euclidean norm in Rn C Set of complex numbers Q Set of rational numbers R Set of real numbers R Set of real numbers, except from the origin. R \{0} Z Set of integer numbers NZ\{0, −1, −2,...} PQ vector from the point P to the point Q of a Euclidean space v vector of an Euclidean space Im(f ) Range of a function, i.e. {f(x): x ∈ X}, whenever f : X → Y Ker(f ) Kernel of a function f : X → Rn,i.e.{x ∈ X : f(x)= 0} v ||w The vectors v and w are parallel

xiii List of Figures

Fig. 1 Structure of the book ...... viii Fig. 1.1 Circle centered at (0, 0) and radius R = 3 (left), lemniscateofBernoulli(right) ...... 2 Fig. 1.2 Cardioid(left)andepicycloid(right) ...... 3 Fig. 1.3 Regularcurve...... 5 Fig. 1.4 Counterexample of regular curve in Example 1.1.10 ...... 8 Fig. 1.5 KimbellArtMuseum ...... 9 Fig. 1.6 Rolling circle producing a cycloid. QR Code 1 ...... 10 Fig. 1.7 Catenary, a = 1.QRCode2 ...... 11 Fig. 1.8 Catenaryarchs ...... 12 Fig. 1.9 Lemniscate ...... 13 Fig. 1.10 The high-speed train station Reggio Emilia AV Mediopadana,Italy ...... 14 Fig. 1.11 Spirals: Casa Batló, by Antoni Gaudí (left) and Staircase (right) ...... 14 Fig. 1.12 Logarithmic spiral. a = 1,b = 0.3 ...... 15 Fig. 1.13 Secant ...... 18 Fig. 1.14 Normalline ...... 19 Fig. 1.15 Successive approximations of the length of a curve by segments ...... 22 Fig. 1.16 Curvatureata point,I ...... 24 Fig. 1.17 Curvatureata point,II.QRCode3 ...... 25 Fig. 1.18 Conicsassectionsofa conebya plane,I ...... 27 Fig. 1.19 Conicsassectionsofa conebya plane,II ...... 28 Fig. 1.20 Orthogonal transformation of a coordinate system ...... 29 Fig. 1.21 Campidoglio square, Rome. Engraving: Étienne Dupérac, 1568 ...... 39 Fig. 1.22 AerialviewofSt.Peter’ssquare,inRome ...... 39 Fig. 1.23 Cathedral of Brasilia, by Oscar Niemeyer ...... 40 Fig. 1.24 Oceanogràfic by Félix Candela ...... 41 Fig. 1.25 QRCode4 ...... 44

xv xvi List of Figures

Fig. 1.26 Graph of y = x2 rolling around y =−x2 (symmetric parabola of y =−x2 withrespecttothetangentlines) ...... 44 Fig. 1.27 Rouletteofa pointdrawinga cissoid ...... 44 Fig. 1.28 Constructionofthecissoid ...... 45 Fig. 1.29 ConchoidofNicomedes ...... 47 Fig. 1.30 QR Code 5 (left) Deltoid with R/r = 3; QR Code 6 (right) Astroid with R/r = 4 ...... 48 Fig. 1.31 QR Code 7 (left) Cardioid with R/r = 1; QR Code 8 (center) Nephroid with R/r = 2; QR Code 9 (right) Epicycloid with R/r = 3 ...... 48 Fig. 1.32 A nephroidasanepicycloid ...... 49 Fig. 1.33 Deltoid (left) and astroid (right) as hypocycloids ...... 49 Fig. 1.34 StainedglasswindowfromSaint-ChapelleinParis ...... 50 Fig. 1.35 Boor–deCasteljaualgorithm ...... 52 Fig. 1.36 QRCode10 ...... 52 Fig. 1.37 Teatro popular in Niterói, Brazil, by Oscar Niemeyer ...... 54 Fig. 1.38 The Teatro popular with author’s overlay ...... 55 Fig. 2.1 Circle centered at P = (0, 0, 1) and radius R = 3, at height z = 1 (left);Helix(right) ...... 60 Fig. 2.2 Curves in Fig. 2.1 determined by the intersection of surfaces ...... 60 Fig. 2.3 Viviani’scurve(left),andsolenoidtoric(right) ...... 61 Fig. 2.4 α(t) = (t2,t3,t4), t ∈ (−2, 2) ...... 62 Fig. 2.5 CurveinExample2.1.5 ...... 63 Fig. 2.6 Parametrizations (I1,α1) and (I2,α2) associated to Viviani’scurve ...... 63 Fig. 2.7 Regularcurve ...... 64 Fig. 2.8 Secant lines ...... 68 Fig. 2.9 Approximation of a curve with polygonal chains ...... 71 Fig. 2.10 Frenettrihedronina spacecurve ...... 75 Fig. 2.11 LinesandplanesassociatedtotheFrenettrihedron ...... 76 Fig. 2.12 Positive torsion (left) and negative torsion (right) in Example2.2.27 ...... 78 Fig. 2.13 Localshapeofa curveata point ...... 82 Fig. 2.14 Exampleoflocalshapeofa curveata point ...... 83 Fig. 2.15 Someosculatingcirclesassociatedtoa curve ...... 84 Fig. 2.16 QRCode11 ...... 84 Fig. 2.17 Circularhelix ...... 87 Fig. 2.18 Projectionsofthecircularhelix ...... 88 Fig. 2.19 QRCode12 ...... 88 Fig. 2.20 Examplesofhelicesinarchitecture ...... 89 Fig. 2.21 Twisted cubic (2.26), for a = b = c = 1 ...... 90 Fig. 2.22 Differentprojectionsofthetwistedcubic ...... 90 Fig. 2.23 CapitalGateTowerinAbuDhabi...... 91 List of Figures xvii

Fig. 2.24 Translation of vector v = (1, 2, 3) of the curve in Example2.4.3.QRCode13 ...... 93 Fig. 2.25 Rotation of angle β = π/4 around {y = z = 0} of the curve (I, α)˜ inExample2.4.3.QRCode14 ...... 94 Fig. 2.26 Reflection with respect to the plane x = 0ofthecurveα˜ inExample2.4.3.QRCode15 ...... 95 Fig. 2.27 Circular helix (black) vs. (R,α0) (green) ...... 97 Fig. 2.28 Helix (R,α1) for ρ = 1 (black) vs. ρ = 3 (green) ...... 97 Fig. 2.29 Helix (R,α2) for ρ = 1, h = 1 (black) vs. h = 1/2 (green) ...... 98 (R,α ) ρ = h(t) = t h(t) = t + π Fig. 2.30 Helix 3 for 1, (black) vs. 2 (green) ...... 99 3 Fig. 2.31 Helix (R,α3) for ρ = 1, h(t) = t (black) vs. h(t) = t (green) ...... 100 Fig. 2.32 Helix (R,α3) for ρ = 1, h(t) = t (black) vs. h(t) = exp(t) (green) ...... 100 Fig. 2.33 Helix (R,α3) for ρ = 1, h(t) = t (black) vs. h(t) = sin(t/3) (green) ...... 101 Fig. 3.1 Sphere centered at (1, 0, 0) and radius R = 2 (left); hyperbolic paraboloid (right) ......  104 ∂x ∂y ∂z Fig. 3.2 { (u ,v ), (u ,v ), (u ,v ) ,  ∂u 0 0 ∂u 0 0 ∂u 0 0 ∂x ∂y ∂z ∂v (u0,v0), ∂v(u0,v0), ∂v(u0,v0) }...... 105 Fig. 3.3 Autointersection ...... 106 Fig. 3.4 Regularsurface ...... 106 Fig. 3.5 Localcoveringsoftheunitsphere ...... 107 Fig. 3.6 Localcoveringsoftheconeminusthevertex ...... 108 Fig. 3.7 Helixcontainedina cylinderinExample3.1.13 ...... 112 Fig. 3.8 Scheme of the construction of the regular curve contained ina regularsurface ...... 113 Fig. 3.9 Scheme of the construction of a regular plane curve from a regularcurvecontainedina surface ...... 113 Fig. 3.10 Tangent plane to z − exp((x − 1)2 + y) = 0atP = (1, 0, 1) .... 114 2 Fig. 3.11 Normal vector to X(u,v) = (u,v,e(u−1) +v) at P = (1, 0, 1) ...... 116 Fig. 3.12 Torus ...... 117 Fig. 3.13 Geometricconstructionofthetorus ...... 118 Fig. 3.14 Dubai’s Museum of the Future by Killa Design ...... 119 Fig. 3.15 QRCode16 ...... 120 Fig. 3.16 Detailoftheconstructionofa Möbiusband ...... 121 Fig. 3.17 Möbiusband ...... 121 Fig. 3.18 PhoenixInternationalMediabyShauWeipingofBIAD ...... 122 Fig. 3.19 Klein bottle. r = 3 ...... 123 Fig. 3.20 Orthogonal projection π ona plane ...... 124 xviii List of Figures

Fig. 3.21 National Center for the Performing Arts in Beijing by PaulAndreu ...... 125 Fig. 3.22 Approximation of the National Centre for the Performing Arts, in Beijing. Photograph transformed from the previous ...... 126 Fig. 3.23 Ellipsoid approximating of the National Centre for the Performing Arts, in Beijing ...... 127 Fig. 3.24 Curveinsurface ...... 129 Fig. 3.25 ReichstagDome,Berlin,byFoster+Partners ...... 130 Fig. 3.26 Approximation of the ramps in the dome of Reichstag building ...... 131 Fig. 3.27 CampAdventureTower,inHaslev,Denmark,byEffekt ...... 132 Fig. 3.28 Intersectionoftwocylinders,thebasisofa groinvault ...... 135 Fig. 3.29 Vibiani curve as the intersection of surfaces ...... 135 Fig. 3.30 Church of Kópavogur, Iceland ...... 137 Fig. 4.1 A ruledsurfaceandsomeofitsgenerators ...... 140 Fig. 4.2 Oloid.QRCode17 ...... 141 Fig. 4.3 Sphericon.QRCode18 ...... 141 Fig. 4.4 Conical surface with an ellipse at height z =−2as directrix, and vertex (1, 1, 2) ...... 142 Fig. 4.5 ConicalsurfaceinExample4.1.3 ...... 145 Fig. 4.6 Cylindrical surface of directrix given by one branch of a hyperbola at height z = 0, ω = (0, 3, 3) ...... 145 Fig. 4.7 ThecylindricalsurfaceinExample4.1.5 ...... 147 Fig. 4.8 ThetangentdevelopablesurfaceinExample4.1.6 ...... 147 Fig. 4.9 Helicoid. c = 0.2 ...... 149 Fig. 4.10 Example of a generalized helicoid based on a lemniscate. QRCode19 ...... 151 Fig. 4.11 Double lemniscate (left), and double staircase in Vatican Museum,Rome(right) ...... 151 Fig. 4.12 Example of a parabolic conoid ...... 153 Fig. 4.13 Sagrada Familia, Schools in Barcelona by Antoni Gaudí ...... 153 Fig. 4.14 The church of San Juan de Ávila in Alcalá de Henares, Spain,byEladioDieste ...... 155 Fig. 4.15 ThevaultoftheKimbellArtMuseum,byLouisKahn ...... 156 Fig. 4.16 Detail of the Church of Cristo Obrero in Atlántida, Uruguay, by Eladio Dieste ...... 157 Fig. 4.17 WallsofthechurchofCristoObrero ...... 158 Fig. 4.18 Scheme for a catenary arc in the vault of the church of CristoObrero ...... 159 Fig. 4.19 SchemeforthevaultofthechurchofCristoObrero ...... 160 Fig. 4.20 Scheme of the church of Cristo Obrero performed in Maple ...... 161 Fig. 4.21 Rodrigues’rotationformula ...... 162 Fig. 4.22 ThesurfaceofrevolutioninExample4.4.3 ...... 166 List of Figures xix

Fig. 4.23 Surfaceofrevolution ...... 167 Fig. 4.24 Torus; r = 1, R = 3 ...... 168 Fig. 4.25 Approximation of the Water tower in Fedala as a surface ofrevolution ...... 169 Fig. 4.26 Orthogonal transformation of a coordinate system ...... 170 Fig. 4.27 Ellipsoid ...... 172 Fig. 4.28 Sections of an ellipsoid in canonical form by the coordinateplanes ...... 173 Fig. 4.29 Hyperboloidofonesheet ...... 173 Fig. 4.30 Sections of a hyperboloid of one sheet in canonical form bythecoordinateplanes ...... 174 Fig. 4.31 Hyperboloidoftwosheets ...... 175 Fig. 4.32 Sections of a hyperboloid of two sheets in canonical form bythecoordinateplanes ...... 175 Fig. 4.33 Cone ...... 176 Fig. 4.34 Elliptic paraboloid ...... 177 Fig. 4.35 Sections of an elliptic paraboloid in canonical form at positive height (left) and with y = 0 (right) ...... 178 Fig. 4.36 Hyperbolicparaboloid ...... 178 Fig. 4.37 Sections of a hyperbolic paraboloid in canonical form at positive(left),negative(center)andnull(right)height ...... 178 Fig. 4.38 Sections of a hyperbolic paraboloid in canonical form with the planes x = 0andy = 0 ...... 179 Fig. 4.39 Quadric defined in Eq. (4.12) and parametrized by Eq.(4.13) ...... 185 Fig. 4.40 Sheratonhotel ...... 188 Fig. 4.41 Apple...... 188 Fig. 4.42 James S. McDonnell Planetarium by Gyo Obata ...... 189 Fig. 4.43 ScotiabankSaddledomebyGECArchitecture ...... 190 Fig. 4.44 Catenoid of Example 4.7.5, with a = 1 ...... 195 Fig. 4.45 OlympiastadioninMunichbyFreiOtto ...... 198 Fig. 4.46 AnEnnepersurface ...... 199 Fig. 4.47 QRCode20 ...... 202 Fig. 4.48 Exampleofcurveshifting.QRCode21 ...... 203 Fig. A.1 A Cartesiancoordinatesystem ...... 206 Fig. A.2 Polarcoordinatesystem ...... 207 Fig. A.3 Cartesian coordinates of the point (1, 2, 3) ...... 208 Fig. A.4 Lines in the Gran Via Capital Hotel, Spain by La Hoz Arquitectura ...... 209 Fig. A.5 Cylindrical coordinates of the point (1, 2, 3) ...... 210 Fig. A.6 Spherical coordinates of the point (1, 2, 3) ...... 210 Fig. A.7 Cloud Gate in Chicago by Anish Kapoor ...... 211 Fig. B.1 f(x)= sin(x) and some Taylor polynomials at x = 0 ...... 217 xx List of Figures

2 Fig. B.2 Area between f(x)= 2xex − 4x and OX from x = 0 and x = 1 ...... 219 Fig. C.1 Cusp.Exercise1.5 ...... 224 Fig. C.2 Tangent line in a point of a lemniscate. QR Code 22. Exercise1.6 ...... 225 Fig. C.3 Orthogonal lines to the symmetry axis of the parabola. Exercise1.14 ...... 228 Fig. C.4 Two secant lines. Exercise 1.16 ...... 229 Fig. C.5 Hyperbola.Exercise1.17 ...... 231 Fig. C.6 Ellipse. Exercise 1.18 ...... 232 Fig. C.7 Catenary arc and three Lagrange approximations in [0, 1], a = 1.Exercise1.24...... 235 Fig. C.8 Spacecurve.Exercise2.1 ...... 236 Fig. C.9 The conical spiral of Pappus. Exercise 2.1 ...... 237 Fig. C.10 Exampleofspacecurve.Exercise2.2 ...... 237 Fig. C.11 Example of a loxodrome contained in a torus. Exercise 2.3 ...... 239 Fig. C.12 Construction of a curve of constant curvature and torsion. Exercise2.7 ...... 241 Fig. C.13 Lamé surface for p = 1/2.Exercise3.7 ...... 248 Fig. C.14 Intersection of surfaces. Exercise 3.8 ...... 248 Fig. C.15 Intersectionoftwoparabolas.Exercise3.9 ...... 249 Fig. C.16 Cylindrical surface. Exercise 4.1 ...... 250 Fig. C.17 Conical surface. Exercise 4.2 ...... 251 Fig. C.18 Tangent developable surface. Exercise 4.3 ...... 252 Fig. C.19 Quadric.Exercise4.4 ...... 253 Fig. C.20 Geometricscheme.Exercise4.5 ...... 254 Fig. C.21 Geometricscheme.Exercise4.6 ...... 255 Fig. C.22 Geometricscheme.Exercise4.7 ...... 256 Fig. C.23 Elliptic torus. Exercise 4.11 ...... 259 Fig. C.24 Generalized elliptic torus. Exercise 4.12 ...... 261 Fig. C.25 Conoid structure. Exercise 4.13 ...... 261 Fig. C.26 Second conoid structure. Exercise 4.13 ...... 262 Fig. C.27 Pseudosphere. Exercise 4.14 ...... 263 Fig. C.28 QRCode23.Exercise4.15 ...... 265 Fig. C.29 Sweepingcurve.Exercise4.15 ...... 265 Photograph Credits

Figure 1.5 Source: By Photo: Andreas Praefcke—Self-photographed, Public Domain, Link: https://commons.wikimedia.org/w/index.php?curid=8382419

Figure 1.8 (left) Source: Mattancherry koonan kurish, Kochi, Kerala, India. Koonan Kurish Palli, Flickr. com. Link: https://flic.kr/p/V4vMDm

Figure 1.8 (right) Source: Photo by Johnson Liu on Unsplash. Link: https://unsplash.com/photos/C3SEO9ORkMg

Figure 1.10 Source: Photo by Luca Bravo on Unsplash Link: https://unsplash.com/photos/alS7ewQ41M8

Figure 1.11 (left) Source: Photo by Andrea Junqueira on Unsplash Link: https://unsplash.com/photos/mNoMLlDDJbg

Figure 1.11 (right) Source: Photo by Pavel Nekoranec on Unsplash Link: https://unsplash.com/photos/-qJlgvKXE1M

Figure 1.21 Source: Public domain, Link: https://commons.wikimedia.org/w/index.php?curid=37361

Figure 1.22 Source: Photo by Michael Martinelli on Unsplash. Link: https://unsplash.com/photos/jgESEijOorE

Figure 1.23 Source: Photo by ckturistando on Unsplash. Link: https://unsplash.com/photos/RWWHa5TUF8w

xxi xxii Photograph Credits

Figure 1.24 Source: Oceanogràfic, Wikipedia.com. De Felipe Gabaldón, CC BY 2.0, Link: https://commons.wikimedia.org/w/index.php?curid=12532971

Figure 1.34 Source: Photo by Stephanie LeBlanc on Unsplash Link: https://unsplash.com/photos/FiknH_A0SLE

Figue 1.37 Source: Photo stored in pxhere.com Link: https://pxhere.com/es/photo/556035

Figure 1.38 Source: Photo stored in sphere.com Link: https://pxhere.com/es/photo/556036

Figure 2.20 (left) Source: Photo by Yusuf Dündar on Unsplash Link: https://unsplash.com/photos/Sm2IjyvrzDk

Figure 2.20 (right) Source: Photo by Steven Jackson, Gaudi’s columns at Park Guell.Attribution 2.0 Generic (CC BY 2.0) On flic.kr Link: https://flic.kr/p/9zsUoE

Figure 2.23 Source: Photo stored in pxhere.com Link: https://pxhere.com/es/photo/555498

Figure 3.14 Source: Photo by Darcey Beau on Unsplash Link: https://unsplash.com/photos/q8D7WZc40eA

Figure 3.18 Source: Photo by Nico Villanueva on Unsplash Link: https://unsplash.com/photos/V89ZSyrExxs

Figures 3.21 and 3.22 Source: By Flickr user Hui Lan from Beijing, China— national theatre at Flickr, CC BY 2.0 Link: https://commons.wikimedia.org/w/index.php?curid=3334661

Figure 3.25 Source: Photo stored in pxhere.com Link: https://pxhere.com/es/photo/1355308

Figure 3.27 (left) Source: Photo by Robby McCullough on Unsplash Link: https://unsplash.com/photos/i7UsLKFX-Ms

Figure 3.27 (right) Source: Photo by Robby McCullough on Unsplash Link: https://unsplash.com/photos/DtzJFYnFPJ8 Photograph Credits xxiii

Figure 4.11 (right) Source: Photo by Jonathan Singer on Unsplash Link: https://unsplash.com/photos/Jda9U-CMc8c Source: Photo stored in pxhere.com

Figure 4.13 Source: By Unknown author—Maria Antonietta Crippa: Gaudí, Taschen, Köln, 2007, ISBN 978-3-8228-2519-8, Public Domain, Link: https://commons.wikimedia.org/w/index.php?curid=3560968

Figure 4.14 Source: Photograph of the church of San Juan de Ávila, in Alcalá de Henares, taken on the 13th of February, 2021, by the author.

Figure4.16Source:DeAndrésFranchiUgart...,CC BY-SA 3.0 Link: https://commons.wikimedia.org/w/index.php?curid=54386268

Figure 4.40 Source: Photo stored in pxhere.com Link: https://pxhere.com/es/photo/1042708

Figure 4.41 Source: Photo by Juliana Lee on Unsplash Link: https://unsplash.com/photos/xibAcLZDUTY

Figure 4.42 Source: By Original uploader was Colin.faulkingham at en.wikipedia—Transfered from en.wikipedia Transfer was stated to be made by User:jcarkeys., Public Domain, Link: https://commons.wikimedia.org/w/index.php?curid=3338724

Figure 4.43 Source: Photo stored in pxhere.com Link: https://pxhere.com/es/photo/707570

Figure 4.45 Source: Photo stored in pxhere.com Link: https://pxhere.com/es/photo/916411

Figure A.4 Source: Photo by Joel Filipe on Unsplash Link: https://unsplash.com/photos/RFDP7_80v5A

Figure A.7 Source: Photo by Mevlüt ?ahin on Unsplash Link: https://unsplash.com/photos/FevvUdyk-oE

Figure 3.30 Source: Wikipedia. Church of Kópavogur (icelandic: Kópavogskirk ja) being built ca. 1960 Link: https://is.m.wikipedia.org/wiki/Mynd:Kopavogur_church_ca1960.jpg

Figure C.25 Source: By M. Kocandrlova—Public Domain Link: https://commons.wikimedia.org/w/index.php?curid=4327175