surfaces of revolution
definition
→→ rotation of a profile (in general 3D curve) around an axis
→→ special case of swept surfaces
green → profile
black dash-dot → axis of rotation
blue → meridians
red → crater circles
orange → neck & equator circles
circles of latitude
architecture, geometry and computers 2011
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architecture, geometry and computers 2011
split hyperboloid of
revolution
paraboloid of revolution
oblate ellipsoid of revolution
prolate ellipsoid of revolution
single hyperboloid of
revolution
(revolving a hyperbola)
anuloid (torus)
sphere
single hyperboloid of
revolution
(revolving a line)
straight circular cone
straight circular blunted
cone
straight circular cylinder
surfaces of revolution
examples
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surfaces of revolution
architecture i
Epidauros Theatre, Greece, 300BC [photo by Jan Foretnik, 1998]
architecture, geometry and computers 2011
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surfaces of revolution
architecture ii
Agia Sofia, Mystra,
Greece, 1400AD
[photo by Jan Foretnik, 1998]
architecture, geometry and computers 2011
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surfaces of revolution
architecture iii
Taj Mahal, Agra,
Uttar Pradesh, India,
Usad Ahmad, 1648
[photo by Adrien Mortini, 1995, en.structurae.de]
architecture, geometry and computers 2011
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surfaces of revolution
architecture iv
Atomium, Brussel, Belgium, André Waterkeyn, 1958 [photo by Jan Foretník, 2006]
architecture, geometry and computers 2011
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surfaces of revolution
architecture v
Biosphère Montréal, Canada, Richard Buckminster Fuller, 1967 [photo by Cédric Thévenet, 2005, commons.wikimedia.org]
architecture, geometry and computers 2011
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surfaces of revolution
architecture vi
Pavilion Z, Brno Trade Fair, Ferdinand Lederer, 1959 [photo by Jan Foretník, 2005]
architecture, geometry and computers 2011
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surfaces of revolution
architecture vii
Duxford Imperial War Museum, Duxford, England, Norman Foster, 1997
[photo Yoshito Isono, 2002 and Foster+Parners, en.structurae.de, fosterandpartners.com]
architecture, geometry and computers 2011
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surfaces of revolution
architecture viii
Reichstag dome, Berlin, Germany, Norman Foster, 1999 [photo by Jan Foretník, 2001]
architecture, geometry and computers 2011
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surfaces of revolution
architecture ix
Swiss Re, London,
Norman Foster, 2004
[photo by Elekhh, 2005, commons.wikimedia.org]
architecture, geometry and computers 2011
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surfaces of revolution
architecture x
Renaissance Center,
Detroit, USA,
John Portman, 1977
[photo by Kjetil r, 2006, commons.wikimedia.org]
architecture, geometry and computers 2011
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helicoid surfaces
definition
→→ sweeping movement of a profile (in general 3D curve) along helix
→→ special case of swept surfaces or special case of twist surfaces
→→ terminology
▪▪ open/close helicoid
▪▪ profile and path
▪▪ cross profile
▪▪ meridian
▪▪ neck helix
▪▪ equator helix
→→ path ↔ helix of any point
→→ profile ↔ cross profile or meridian
architecture, geometry and computers 2011
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architecture, geometry and computers 2011
helicoid vault
Archimedes' serpentine
wreathed column
cyclic helicoid surfaces
↓
closed helicoid surfaces
↓
right angled
oblique
bisecant
open helicoid surfaces
↓
right angled
oblique
bisecant
helicoid surfaces
examples
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helicoid surfaces
architecture i
VitraHaus, Vitra Design Museum,
Weil am Rhein, Germany,
Herzog & de Meuron, 2010
[photo Jan Foretník, 2010]
architecture, geometry and computers 2011
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helicoid surfaces
architecture ii
Louvre’s pyramid, Paris, France, Ieoh Ming Pei, 1989 [photo by Jan Foretník, 2001]
architecture, geometry and computers 2011
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helicoid surfaces
architecture iii
Turning Torso, Malmö, Sweden,
Santiago Calatrava, 2005
[photo by Sixtiz, 2005, commons.wikimedia.org]
architecture, geometry and computers 2011
185/247
helicoid surfaces
architecture iv
Pest Column,
Olomouc, Czech Republic,
Václav Render, 1723
[photo by Jan Foretník, 2009]
architecture, geometry and computers 2011
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ruled surfaces
definition
→→ sweeping movement of a line along two (in general 3D) curves, some has
additional conditions (like parallel plane for conoids, or third line path for
conusoids).
→→ single (lines in one direction) ruled surfaces
→→ double (lines in two directions) ruled surfaces; the only double ruled
surfaces are:
▪▪ hyberbolic paraboloid
▪▪ single hyperboloid of revolution
▪▪ a plane
→→ developable surfaces are:
▪▪ cylindrical
▪▪ conical
▪▪ swept surfaces, where the profile is a tangent to path
→→ skew surface is the opposite of developable
architecture, geometry and computers 2011
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architecture, geometry and computers 2011
general oblique cone
single hyperboloid of
revolution
(double ruled surface)
general oblique cylinder
oblique circular cylinder
möbius strip
oblique circular cone
general straight cylinder
straight circular blunted
cone
hyperbolic paraboloid
(double ruled surface)
straight circular cylinder
straight circular cone
circular conoid
ruled surfaces
examples
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ruled surfaces
architecture i
TV Transmitter and Hotel Ještěd,
Liberec, Czech Republic,
Karel Hubáček, 1973
[photo archive of Josef Kutman, Fenomén Ještěd]
architecture, geometry and computers 2011
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ruled surfaces
architecture ii
Kobe Port Tower, Kóbe, Japan,
Nikken Sekkei, 1963
[photo by 663highland, 2006, commons.wikimedia.org]
architecture, geometry and computers 2011
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ruled surfaces
architecture iii
Santa Maria del Fiore Dome, Firenze, Italy, Filippo Brunelleschi, 1436 [photo by Jan Foretník, 1992]
architecture, geometry and computers 2011
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ruled surfaces
architecture iv
Domkirkeruinen/Hamardomen,
Hedmarksmuseet, Hamar, Norway,
Kjell Lund and Nils Slaatto, 1998
[photo by Jan Foretník, 2000]
architecture, geometry and computers 2011
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ruled surfaces
architecture v
The Catalano House, Raleigh, USA, Eduardo Fernando Catalano, 1954, destroyed 2001
[photo archive of www.trianglemodernisthouses.com]
architecture, geometry and computers 2011
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ruled surfaces
architecture vi
Los Manantiales Restaurant,
Xochimilco, Mexico City,
Felix Candela, 1958
[photo archive of www.arch.mcgill.ca]
architecture, geometry and computers 2011
194/247
ruled surfaces
architecture vii
La Sagrada Família,Barcelona, Spain,
Francisco de Paula del Villar, Antoni Gaudí and others, since 1882 [photo by Jan Foretník, 1997]
architecture, geometry and computers 2011
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ruled surfaces
architecture viii
School by La Sagrada Família,Barcelona, Spain, Antoni Gaudí i Cornet, 1909, rebuilt 2002
[photo by Robert Ferréol, 2006, www.mathcurve.com and Doctor Casino, 2008, www.flickr.com]
architecture, geometry and computers 2011
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ruled surfaces
architecture ix
Sandvika railway station, Norway,
Arne Henriksen, 1994
[photo by Jan Foretník, 2000]
architecture, geometry and computers 2011
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ruled surfaces
architecture x
Trúba, Štramberk,
Czech Republic,
Kamil Hilbert, 1903
[photo by Jan Foretník, 2009]
architecture, geometry and computers 2011
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ruled surfaces
architecture xi
Abbesses subway entrance, Paris,
Hector Guimard, 1910
[photo by Jan Foretník, 2002]
architecture, geometry and computers 2011
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ruled surfaces
architecture xii
TGV Terminal Liège-Guillemins, Belgium, Santiago Calatrava, 2007 [photo by Jan Foretník, 2007]
architecture, geometry and computers 2011
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ruled surfaces
architecture xiii
Science Centre NEMO, Amsterdam, Renzo Piano, 2005 [photo by Jan Foretník, 2007]
architecture, geometry and computers 2011
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ruled surfaces
architecture xiv
National Library, Astana, Kazakhstan, BIG, competition 2009 [photo archive of www.archdaily.com, 2007]
architecture, geometry and computers 2011
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ruled surfaces
architecture xv
Möbius Strip II, M. C. Escher, woodcut 1963 [photo archive of www.mcescher.com]
architecture, geometry and computers 2011
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geometrically defined surfaces
cad i
Montpelliers’ arch in Rhinoceros
geometric definition (black), Sweep2 (blue) EdgeSrf (red)
architecture, geometry and computers 2011
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geometrically defined surfaces
cad ii
→→ in Rhinoceros:
▪▪ ExtrudeCrv, ExtrudeSrf → cylindrical surfaces
▪▪ ExtrudeCrvAlongCrv, ExtrudeSrfAlongCrv → extruded
surfaces
▪▪ ExtrudeCrvTapered, ExtrudeSrfTapered → tapered surfaces
▪▪ ExtrudeCrvToPoint, ExtrudeSrfToPoint → conical surfaces
▪▪ Sweep1, Sweep2 → swept surfaces
▪▪ Revolve → surfaces of revolution
▪▪ RailRevolve → surfaces of pseudorevolution
▪▪ Pipe → pipe and channel surfaces
▪▪ Plane, PlanarSrf → planar surfaces
▪▪ Ribbon
▪▪ NetworkSrf, Loft, SrfPtGrid → all other cases as an interpolation
surface
→→ creates a surface or polysurface object
architecture, geometry and computers 2011
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geometrically defined surfaces
cad iii
→→ in AutoCAD:
▪▪ EXTRUDE → cylindrical surfaces, tapered surfaces
▪▪ REVOLVE → surfaces of revolution
▪▪ SWEEP → swept surfaces (one path only, uniformly scaling profile, twisted
surfaces)
▪▪ LOFT (or SURFNETWORK since ACAD2011)→ all other cases as an
interpolation surface
→→ creates a surface object, with closed profiles can create a solid object
architecture, geometry and computers 2011
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geometrically defined surfaces
architecture, geometry and computers 2011
cad task
207/247