1. Sawing of Logs in Virtual Trees using 3D-Intersection Algorithms S. Despréaux, L.Biard, N.Szafran (LMC-IMAG-Grenoble) F.Blaise (INRIA-Lorraine)

2. 14-15 Oct. 2003PMA03 International Symposium Objectives:  Tree internal structure geometrical modelisation  Sawing logs

3. 14-15 Oct. 2003PMA03 International Symposium Tree Internal Structure Tree growing in 2 steps: Primary growth: creation of the growth unit Secondary growth: the stem growth by creation of 2 rings:  Earlywood  Latewood

4. 14-15 Oct. 2003PMA03 International Symposium   Year 1 Pith creation Stem growth

5. 14-15 Oct. 2003PMA03 International Symposium   Year 1 Pith creation +earlywood Stem growth

6. 14-15 Oct. 2003PMA03 International Symposium   Year 1 Pith creation +earlywood +latewood Stem growth

7. 14-15 Oct. 2003PMA03 International Symposium   Year 1 Pith creation +earlywood +latewood   Year 2 +earlywood +latewood Stem growth

8. 14-15 Oct. 2003PMA03 International Symposium Conic Tree modelisation  Works done by Daily in 1996 Barthélémy, Blaise, Fourcaud, Nicolini in 95 Reffy, Houllier,Blaise in 98 No branches, so no knots… Only conic rings representation

9. 14-15 Oct. 2003PMA03 International Symposium Objectives for tree modelisation Objectives:  Connect branches to see knots  Rings of all shapes: not necessarly circular

10. 14-15 Oct. 2003PMA03 International Symposium Branches Connection Problem

11. 14-15 Oct. 2003PMA03 International Symposium  Growth unit modelised by connected spline surfaces  Problem: how to connect branches in 3D to provide logs intersections directly 2D continuously connected ? 3D Explicit Approach

12. 14-15 Oct. 2003PMA03 International Symposium 3D Explicit Approach : bezier surfaces  Works begun by Bieth in 2000  Section modelized by cubic bezier curves  Surface computed by tensor product of bezier surfaces

13. 14-15 Oct. 2003PMA03 International Symposium 3D Explicit Approch: branches connection Method: projection of the branch section on the stem surface.

14. 14-15 Oct. 2003PMA03 International Symposium 3D Branches Connection: results  Advantages: 3D model enables analysis and computation. Board intersections are directly 2D continuously connected  Difficulties: Too close rings generate geometric incoherence Strong difficulties when branches ‘ size is too more different

15. 14-15 Oct. 2003PMA03 International Symposium 3D Implicit Approach A growth unit i is modelised by implicit functions : F[i] M belongs to the ring k of the growth unit i If and only if F k [i] (M)  0 and F k-1 [i] (M) < 0

16. 14-15 Oct. 2003PMA03 International Symposium If M does not belong to any ring k of the growth i But if the quantity where Then M is considered to belong to the ring k of the growth unit i 3D Implicit Approach: smooth correction

17. 14-15 Oct. 2003PMA03 International Symposium 3D Implicit Approach: cutting For each point of the image Compute the 3D associated point Find the oldest ring which contains it The associated colour is the colour of the found ring

18. 14-15 Oct. 2003PMA03 International Symposium 3D Implicit Approach: results Conic Tree with 6 branches

19. 14-15 Oct. 2003PMA03 International Symposium 3D Implicit Approach: results Cutting boards on Conic Tree

20. 14-15 Oct. 2003PMA03 International Symposium 3D Implicit Approach:results Polyhedron Tree with 4 branches

21. 14-15 Oct. 2003PMA03 International Symposium 3D Implicit Approach:results Sawing logs in polyhedron Tree