SAH 2009 Laboratory of timber constructions An iterative surface model for timber construction

1. 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)

2. 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 Summary

3. 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 Problematics

4. Pottmann, Wallner,… Meshes approximating a continuous surface Discrete differential geometry tools Only usable for smooth surfaces Not usable in our context (folded structures) Existing methods

5. 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

6. Affine sum of two curves • Operator • Defined as a Minkowski sum • Inputs • Two curves and • Output • A surface , defined as follows :

7. Inputs : 2 polylines Output : a mesh Affine sum of curves, discrete case

8. Topology Quadrangular meshes Geometry Plane faces Meshes composed only by parallelograms Opposite edges similar Properties of obtained meshes

9. 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

10. Properties of central projections Planar points stay planar The parallelism is not preserved Interests Less restrictive model Needed properties are kept Use of projective geometry

11. w = 1 w < 1 w > 1 w > 1 w = 1 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

12. Sum of weighted polylines • Visualization of the weighted sum for a single surface element w y x

13. 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 The IFS model (Iterated Function System)

14. Interactive handling, by moving points in the space Control points (red) Global aspect Subdivision points (blue) Local aspect (smooth / rough) Curve handling

15. 2 input IFS curves defining the surface Parameters Control points and subdivision points of the 2 curves Surface handling

16. Modification of the weight of the control points

17. This model defines surface meshes For the construction, we need timber pannels, with the same thickness Application to construction

18. 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 Volumic elements modelling

19. Corner joints • Chamfered corners

20. Example of generated structure

21. 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 Conclusion

22. For further information, please visit: http://ibois.epfl.ch/ Fonds National Suisse (FNS)

24. Modification of the weight of the subdivision points