Download - SAH 2009 Laboratory of timber constructions An iterative surface model for timber construction
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An iterative surface model for timber constructionSAH - April 2009 – Gilles Gouaty , Ivo Stotz, Eric Tosan, Yves Weinand
SAH 2009
Laboratory of timber constructions
An iterative surface model for timber construction
Gilles Gouaty, Ivo Stotz, Eric Tosan, Yves Weinand
Fonds National Suisse (FNS)
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An iterative surface model for timber constructionSAH - April 2009 – Gilles Gouaty , Ivo Stotz, Eric Tosan, Yves Weinand
Summary
• Problematics• Quadrilateral planar meshes modelling
– Surface modelling by sum of two curves– Use of projective geometry
• Iterative modelling– The IFS model (Iterative Function System)– Iterative model for curves and surfaces– Interpretations in affine and projective geometry
• Application to construction• Conclusion
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An iterative surface model for timber constructionSAH - April 2009 – Gilles Gouaty , Ivo Stotz, Eric Tosan, Yves Weinand
Problematics
• Geometrical modelling for timber construction– Complex shapes generated by an iterative model : the
IFS model (Iterative Function System)– Physical construction by timber panels
• Constraints of construction– Surface meshes– Planar faces– Quadrangular faces
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An iterative surface model for timber constructionSAH - April 2009 – Gilles Gouaty , Ivo Stotz, Eric Tosan, Yves Weinand
Existing methods
• Pottmann, Wallner,…– Meshes approximating a continuous surface – Discrete differential geometry tools– Only usable for smooth surfaces
• Not usable in our context (folded structures)
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An iterative surface model for timber constructionSAH - April 2009 – Gilles Gouaty , Ivo Stotz, Eric Tosan, Yves Weinand
Planar quadrilateral meshes design
• We defined a particular surface mesh model
• Principle– Surface construction by sum of 2 curves– Use of projective geometry
• In order to extend the model
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An iterative surface model for timber constructionSAH - April 2009 – Gilles Gouaty , Ivo Stotz, Eric Tosan, Yves Weinand
Affine sum of two curves
• Operator – Defined as a Minkowski
sum
• Inputs– Two curves and
• Output– A surface , defined as
follows :
bGaG
( , ) ( ) ( )
( ) ( )
( ) ( )
a b
a b
a b
G s t G s G t
c cG s cG t
G s G t c
G
*
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An iterative surface model for timber constructionSAH - April 2009 – Gilles Gouaty , Ivo Stotz, Eric Tosan, Yves Weinand
Affine sum of curves, discrete case
– Inputs : 2 polylines– Output : a mesh
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An iterative surface model for timber constructionSAH - April 2009 – Gilles Gouaty , Ivo Stotz, Eric Tosan, Yves Weinand
Properties of obtained meshes
• Topology– Quadrangular meshes
• Geometry– Plane faces– Meshes composed only by parallelograms– Opposite edges similar
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An iterative surface model for timber constructionSAH - April 2009 – Gilles Gouaty , Ivo Stotz, Eric Tosan, Yves Weinand
Use of projective geometry
• Goal– Reduce some restrictions to the previous model
• Principle of the projective geometry– The 3D space (X,Y,Z) is replaced by a homogenous
space (w,x,y,z)– Equivalence between 3D and 4D points : divizion by
w• (X,Y,Z) = ( x/w , y/w , z/w )• Projection centered in 0 on the hyperplane w=1
– Interpretation : weighted 3D points• Weight = w coordinate
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An iterative surface model for timber constructionSAH - April 2009 – Gilles Gouaty , Ivo Stotz, Eric Tosan, Yves Weinand
Use of projective geometry
• Properties of central projections– Planar points stay planar
– The parallelism is not preserved
• Interests– Less restrictive model
– Needed properties are kept
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An iterative surface model for timber constructionSAH - April 2009 – Gilles Gouaty , Ivo Stotz, Eric Tosan, Yves Weinand
Projective sum of weighted polylines
• The operator stays unchanged, but operates in
• The 2 entry polylines are defined in– The extra coordinate w is the weight
4
4
w = 1
w = 1
w > 1
w < 1
w > 1
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An iterative surface model for timber constructionSAH - April 2009 – Gilles Gouaty , Ivo Stotz, Eric Tosan, Yves Weinand
Sum of weighted polylines
• Visualization of the weighted sum for a single surface element
x
y
w
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An iterative surface model for timber constructionSAH - April 2009 – Gilles Gouaty , Ivo Stotz, Eric Tosan, Yves Weinand
The IFS model (Iterated Function System)
• Mathematical model allowing to produce self-similar objects
• Variable figures according to following aspects:
• Geometrical aspect: smooth or rough• Topology: curves, surfaces, …• multi-resolution aspect: the mesh is more or
less subdivided into discretized elements
• Interactive object handling• Control points• Subdivision points
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An iterative surface model for timber constructionSAH - April 2009 – Gilles Gouaty , Ivo Stotz, Eric Tosan, Yves Weinand
Curve handling
• Interactive handling, by moving points in the space– Control points (red)
• Global aspect– Subdivision points (blue)
• Local aspect (smooth / rough)
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An iterative surface model for timber constructionSAH - April 2009 – Gilles Gouaty , Ivo Stotz, Eric Tosan, Yves Weinand
Surface handling
• 2 input IFS curves defining the surface
• Parameters– Control points and subdivision points of the 2
curves
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An iterative surface model for timber constructionSAH - April 2009 – Gilles Gouaty , Ivo Stotz, Eric Tosan, Yves Weinand
Modification of the weight of the control points
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An iterative surface model for timber constructionSAH - April 2009 – Gilles Gouaty , Ivo Stotz, Eric Tosan, Yves Weinand
Application to construction
– This model defines surface meshes
– For the construction, we need timber pannels, with the same thickness
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An iterative surface model for timber constructionSAH - April 2009 – Gilles Gouaty , Ivo Stotz, Eric Tosan, Yves Weinand
Volumic elements modelling
• For every element :– Base surface– Parallel surface, at a determined distance (the thickness)– Chamfered edges : bissectors planes with neighbour faces
• Problem : the 4 bissector planes generally don't intersect
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An iterative surface model for timber constructionSAH - April 2009 – Gilles Gouaty , Ivo Stotz, Eric Tosan, Yves Weinand
Corner joints
• Chamfered corners
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An iterative surface model for timber constructionSAH - April 2009 – Gilles Gouaty , Ivo Stotz, Eric Tosan, Yves Weinand
Example of generated structure
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An iterative surface model for timber constructionSAH - April 2009 – Gilles Gouaty , Ivo Stotz, Eric Tosan, Yves Weinand
Conclusion
• Quadrangular meshes
• Planar faces
• Smooth or rough shapes
• Surfaces controlled by 2 edges– Few parameters to handle– Indirect control of the 2 opposite edges
• Thickening not integrated into the model– Post-treatment
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An iterative surface model for timber constructionSAH - April 2009 – Gilles Gouaty , Ivo Stotz, Eric Tosan, Yves Weinand
For further information, please visit:
http://ibois.epfl.ch/
Fonds National Suisse (FNS)
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An iterative surface model for timber constructionSAH - April 2009 – Gilles Gouaty , Ivo Stotz, Eric Tosan, Yves Weinand
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An iterative surface model for timber constructionSAH - April 2009 – Gilles Gouaty , Ivo Stotz, Eric Tosan, Yves Weinand
Modification of the weight of the subdivision points