An iterative surface model for timber construction SAH - 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)

An iterative surface model for timber construction SAH - 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

An iterative surface model for timber construction SAH - 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

An iterative surface model for timber construction SAH - 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)

An iterative surface model for timber construction SAH - 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

An iterative surface model for timber construction SAH - 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 :

An iterative surface model for timber construction SAH - April 2009 – Gilles Gouaty, Ivo Stotz, Eric Tosan, Yves Weinand Affine sum of curves, discrete case –Inputs : 2 polylines –Output : a mesh

An iterative surface model for timber construction SAH - 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

An iterative surface model for timber construction SAH - 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

An iterative surface model for timber construction SAH - 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

An iterative surface model for timber construction SAH - 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 w = 1 w > 1 w < 1 w > 1

An iterative surface model for timber construction SAH - 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

An iterative surface model for timber construction SAH - 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

An iterative surface model for timber construction SAH - 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)

An iterative surface model for timber construction SAH - 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

An iterative surface model for timber construction SAH - April 2009 – Gilles Gouaty, Ivo Stotz, Eric Tosan, Yves Weinand Modification of the weight of the control points

An iterative surface model for timber construction SAH - 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

An iterative surface model for timber construction SAH - 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

An iterative surface model for timber construction SAH - April 2009 – Gilles Gouaty, Ivo Stotz, Eric Tosan, Yves Weinand Corner joints Chamfered corners

An iterative surface model for timber construction SAH - April 2009 – Gilles Gouaty, Ivo Stotz, Eric Tosan, Yves Weinand Example of generated structure

An iterative surface model for timber construction SAH - 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

An iterative surface model for timber construction SAH - April 2009 – Gilles Gouaty, Ivo Stotz, Eric Tosan, Yves Weinand For further information, please visit: Fonds National Suisse (FNS)

An iterative surface model for timber construction SAH - April 2009 – Gilles Gouaty, Ivo Stotz, Eric Tosan, Yves Weinand

An iterative surface model for timber construction SAH - April 2009 – Gilles Gouaty, Ivo Stotz, Eric Tosan, Yves Weinand Modification of the weight of the subdivision points