an iterative surface model for timber construction sah - april 2009 – gilles gouaty, ivo stotz,...

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)


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


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


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)


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


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

*


Affine sum of curves, discrete case

– Inputs : 2 polylines– Output : a mesh


Properties of obtained meshes

• Topology– Quadrangular meshes

• Geometry– Plane faces– Meshes composed only by parallelograms– Opposite edges similar


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


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


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


Sum of weighted polylines

• Visualization of the weighted sum for a single surface element

x

y

w


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


Curve handling

• Interactive handling, by moving points in the space– Control points (red)

• Global aspect– Subdivision points (blue)

• Local aspect (smooth / rough)


Surface handling

• 2 input IFS curves defining the surface

• Parameters– Control points and subdivision points of the 2

curves


Modification of the weight of the control points


Application to construction

– This model defines surface meshes

– For the construction, we need timber pannels, with the same thickness


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


Corner joints

• Chamfered corners


Example of generated structure


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


For further information, please visit:

http://ibois.epfl.ch/

Fonds National Suisse (FNS)


Modification of the weight of the subdivision points

an iterative surface model for timber construction sah - april 2009 – gilles gouaty, ivo stotz,...

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