HafenCity Universität Hamburg Structural and Architectural Engineering/ Facade Systems and Building Envelopes December, 13 and 15, Hamburg, Germany Formfindung von Freiformflächen Formfindung von Freiformflächen Dr Stavric Milena, DI arch Institute of Architecture and Media Graz University of Technology, Austria
Content: Content: Curves Algebraic Curves Conic section Curves 2. Order Free Form Curves Bezier Curves B Spline Curves NURBS Curves Surfaces Traditional Surfaces Developable Surfaces Double curved Surfaces Geometrical properties Surfaces with negative Gaussian curvature Surfaces with positive Gaussian curvature Free Form Surfaces Bezier Surfaces B Spline Surfaces and NURBS Surfaces Curvature of Surfaces Developable Surfaces How to make Free Form Surfaces buildable Shape optimization Geometric Optimization Discrete form Discrete Free Form Surfaces Grid Generation Discrete component Discretization into curved Components Discretization into planar Components Triangulation Triangle meshes Quadrilateral Meshes with Planar Faces More edges Meshes with planar Faces Supporting structure Layer Structure Offset and Parallel Meshes
Free Form Surfaces Bézier surfaces B- Spline Surfaces Nurbs Surfaces
Free Form Surfaces Bézier surfaces B- Spline Surfaces Nurbs Surfaces
Free Form Surfaces Bézier surfaces B- Spline Surfaces Nurbs Surfaces
Free Form Surfaces Bézier surfaces B- Spline Surfaces Nurbs Surfaces
Free Form Surfaces Bézier surfaces Bézier surfaces of degree (2,2)
Free Form Surfaces Bézier surfaces Bezier surfaces of degree (1,1)
Free Form Surfaces Bézier surfaces Bezier surfaces of degree (2,2)
Free Form Surfaces Bézier surfaces Bezier surfaces of degree (1,1)
Free Form Surfaces Free Form Surfaces NURBS Surfaces
Why NURBS? Why NURBS? NURBS curves and surfaces are useful for a number of reasons: They offer one common mathematical form for both standard analytical shapes (e.g., conics) and free form shapes. They provide the flexibility to design a large variety of shapes. Theyreduce the memoryconsumption when storing shapes (compared to simpler methods). They can be evaluated reasonably quickly by numerically stable and accurate algorithms.
Free Form Surfaces NURBS Surface parametrisation P (u, v)
Free Form Surfaces Curvature Surfaces properties: tangent plane principal curvatures
Free form Surfaces Curvature Gaussian curvature
Optimisation Howtomakefreeform buildable? 1. Optimisation mathematical optimisation geometrical optimisation geometric shape optimisation 1. Geometric divison into more patches 2. Discretisation of the surfaces panelisation and planarisation Panelisation is a mainaspect of the desing process
Optimisation Howtomakefreeform buildable? 1. Optimisation 2. Geometric divison into more surfaces 3. Discretisation of the surfaces panelisation and planarisation Panelisation is a main aspect of the desing process Optimisation: Shape optimisation: 1. approximation i with ihruled ldsurfaces (surface S is ruled ldif through hevery point of S there is a straight line that lies on S) 2. approximation with developable surfaces 3. approximation with translational ti lsurfaces Geometric optimisation Discrete form
PLANARIZATION Ti Triangulation
PLANARIZATION Quad meshing
NURBS Curvature of surfaces
Optimisation p Approximation i byruled surfaces Staatsgalerie Stuttgart, James Stirling, 1984
Optimisation Approximation i by buildable developable surfaces Bézier Surfaces Space curve and associated developable surfaces
Optimisation Developable surfaces Bezier surfaces Walt Disney Concert Hall, Frank Gehry
Optimisation Developable surfaces Bezier surfaces
Optimisation Approximation i bytranslational lsurfaces Discretization quadrilaterial meshes Planar pannels
Optimisation Approximation i bytranslational lsurfaces
Optimisation Approximation i bytranslational lsurfaces offset problem
Non Standard Architektur Non Standard Architektur Mur Turm, Steiermark, arch. Leonhard, 2010
Non Standard Architektur Non Standard Architektur Mur Turm, Steiermark, arch. Leonhard, 2010 Problems Complex knots Complex knots Many edges Torsion Offset
Non Standard Architektur Non Standard Architektur Problems Complex knots Complex knots Many edges Torsion Offset
Architecture Free form surface Burnham Pavillon, Chicago Zaha Hadid, 2009,
Architecture Burnham Pavillon, Chicago Zaha Hadid, 2009, Architecture Free form surface
Architecture Burnham Pavillon, Chicago Zaha Hadid, 2009, Architecture Free form surface
Architecture Burnham Pavillon, Chicago Zaha Hadid, 2009, Architecture Free form surface
Architecture Burnham Pavillon, Chicago Zaha Hadid, 2009, Architecture Free form surface
Architecture Free form surface Burnham Pavillon, Chicago Zaha Hadid, 2009,
Architecture Free form surface Burnham Pavillon, Chicago Zaha Hadid, 2009,
Architecture Free form surface Burnham Pavillon, Chicago Zaha Hadid, 2009,
Architecture Free form surface Burnham Pavillon, Chicago Zaha Hadid, 2009,
Architecture Freeforms Discretization into curved segments Kunsthaus Friendly Alien, Graz Peter Cook and Colin Fournier, 2003
Architecture Freeforms Kunsthaus Friendly Alien, Graz Peter Cook and Colin Fournier, 2003
Architecture Freeforms Kunsthaus Friendly Alien, Graz Peter Cook and Colin Fournier, 2003
Architecture Freeforms Kunsthaus Friendly Alien, Graz Peter Cook and Colin Fournier, 2003
Architecture Triangulation Discretization into planar segments BMW Welt, München Coop Himmelb(l)au, 2007
Architecture From idea to realisation...
Free Form Surfaces NURBS Heydar Aliyev Centre, Baku, Azerbaijan Zaha Hadid
Free Form Surfaces NURBS Heydar Aliyev Centre, Baku, Azerbaijan Zaha Hadid
Free Form Surfaces NURBS Heydar Aliyev Centre, Baku, Azerbaijan Zaha Hadid
Free Form Surfaces NURBS Heydar Aliyev Centre, Baku, Azerbaijan Zaha Hadid
Free Form Surfaces NURBS
Free Form Surfaces NURBS
Free Form Surfaces NURBS Heydar Aliyev Centre, Baku, Azerbaijan Zaha Hadid
Free Form Surfaces NURBS Heydar Aliyev Centre, Baku, Azerbaijan Zaha Hadid
Free Form Surfaces NURBS Heydar Aliyev Centre, Baku, Azerbaijan Zaha Hadid
Free Form Surfaces Free Form Surfaces NURBS
Free Form Surfaces NURBS Heydar Aliyev Centre, Baku, Azerbaijan Zaha Hadid
Free Form Surfaces Free Form Surfaces NURBS
Special curves Art and City Museum by MAD architects, Ordos, China, 2011
Special curves
Special curves Art and City Museum by MAD architects, Ordos, China, 2011
Special curves Geodesics Art and City Museum by MAD architects, Ordos, China, 2011
Special curves Special curves Geodesics
Special curves Special curves Art and City Museum by MAD architects, Ordos, China, 2011
Special curves Special curves Art and City Museum by MAD architects, Ordos, China, 2011
Special curves Geodesics Art and City Museum by MAD architects, Ordos, China, 2011
Architecture Free form surface or not? Yas Hotel, arch. Asymptote, Abu Dhabi,2009
Architecture Yas Hotel, arch. Asymptote, Abu Dhabi,2009
Architecture Yas Hotel, arch. Asymptote, Abu Dhabi,2009
Architecture Yas Hotel, arch. Asymptote, Abu Dhabi,2009
Architecture Yas Hotel, arch. Asymptote Abu Dhabi,2009
Architecture Yas Hotel, arch. Asymptote Abu Dhabi,2009
Free Form Surfaces Nurbs Tori Tori Restaurant Rojkind Architectosmexico City, 2011
Free Form Surfaces Nurbs Tori Tori Restaurant Rojkind Architectosmexico City, 2011
Free Form Surfaces Nurbs Tori Tori Restaurant Rojkind Architectos mexico City, 2011
Discrete Form Pi Principali curvatures lines
Special Geodesics curves
Special curves Geodesics
Special curves p Geodesics Geodesic Lines on Free Form Surfaces Optimized Grids for Timber Rib Shells Ecole Polytechnique Fédérale de Lausanne (EPFL)
Special Geodesics curves
Special curves Geodesics Centre Pompidou Metz France, 2010 by Shigeru Ban
Speci Geodesics al Centre Pompidou Metz France, 2010 by Shigeru Ban curve s
Speci Geodesics al Centre Pompidou Metz France, 2010 by Shigeru Ban curve s
Special curves Geodesics HAESLEY NINE BRIDGE CLUB HOUSE, 2010 by Shigeru Ban
PLANARIZATION 2D Ornament > double curved surface uv parameter lines
PLANARIZATION Why tangent planes method? no Problems Complex knots Many edges Torsion Offset Because of aestethic reasons 2D Ornaments > > on a curved surface in space We can vary the tangent planes
Vary the points
Hyperbolic surface points Butterfly panels
EXAMPLES
EXAMPLES
EXAMPLES
EXAMPLES