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Advanced Geometric Modeling and Dependency Relations

This comprehensive resource explores advanced geometric modeling techniques, focusing on various geometric shapes including disks, rectangles, circles, and more. It details the relationships between different geometric elements, specifically emphasizing dependency relations and containment structures. The document includes mathematical representations, transformation techniques, and boundary relations which are critical for fields such as computer graphics, engineering design, and mathematical visualization. This guide is essential for professionals looking to enhance their understanding of geometric configurations and their interrelations.

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Advanced Geometric Modeling and Dependency Relations

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  1. TEMPLATE: disk LHKF F-cell disk LHKF Q E K F-cell circle LHKF F H L F-cell S D Q E K F H L rectangle DEQS polyline DEQS rectangle DEQS polyline DEQS rectangle DEQS S D B-cell B-cell B-cell B-cell B-cell disk LHKF circle LHKF disk LHKF disk LHKF point K arc KFL half-disk KFL point L segment KL F-cell F-cell F-cell F-cell T-cell T-cell T-cell T-cell T-cell Boundary relations

  2. 0D cells E Q K F H L 1D cells S D 2D cells Boundary relations polyline DEQS rectangle DEQS B-cell B-cell circle LHKF disk LHKF point F F-cell F-cell T-cell

  3. 0D cells E Q K F H L 1D cells S D 2D cells polyline DEQS rectangle DEQS B-cell B-cell circle LHKF disk LHKF point F F-cell F-cell T-cell

  4. 1D cells 1D cells 2D cells 2D cells Boundary relations Containment relations disk LHKF circle LHKF rectangle DEQS polyline DEQS circle LHKF rectangle DEQS polyline DEQS disk LHKF B-cell F-cell F-cell B-cell B-cell F-cell B-cell F-cell

  5. E Q 1D cells K S D F H L 2D cells rectangle DEQS polyline DEQS circle LHKF disk LHKF Containment relations B-cell B-cell F-cell F-cell

  6. Q Q Q Q E E E E disk LHKF … rectangle DEQS … K K K K … RadiusParameter … Transform.translation.x F H H H H L L L L S S S S D D D D F F F

  7. point M point O segment OM segment OM rectangle DEQS disk LHKF circle LHKF disk LHKF rectangle DEQS polyline DEQS 0D cells B-cell F-cell F-cell B-cell F-cell B-cell P-cell P-cell P-cell P-cell Q E O K F H L 1D cells M S D 2D cells

  8. disk LHKF segment OM circle LHKF disk LHKF segment OM F-cell F-cell P-cell F-cell P-cell O M Implicit dependency relations K F H L Explicit dependency relation Dependency relations

  9. point P segment OM polyline DEQS segment ON point M circle LHKF rectangle DEQS disk LHKF Segment MP point N point O segment PN B-cell F-cell F-cell B-cell T-cell P-cell T-cell T-cell P-cell T-cell T-cell P-cell 0D cells O Q E P M K N F H 1D cells L S D 2D cells

  10. half-disk KLF segment KL point K point M arc KLF circle LHKF disk LHKF rectangle DEQS point O polyline DEQS segment OM point L B-cell T-cell F-cell F-cell B-cell T-cell T-cell P-cell P-cell T-cell P-cell T-cell O Q E M K F H L S D 0D cells R 1D cells 2D cells

  11. half-disk KLF circle LHKF disk LHKF rectangle DEQS polyline DEQS point L segment OM arc KLF segment KL segment RM point R point M point K point O P-cell T-cell P-cell B-cell T-cell T-cell B-cell F-cell T-cell T-cell F-cell T-cell P-cell T-cell O Q E M K F H L S D R 0D cells 1D cells 2D cells

  12. O O Q Q E E M M K K F F H H L L S S D D R R

  13. Q E DEQS B-cell LHKF F-cell K F H L S D Q E H K polyline DEQS circle LHKF F L F-cell B-cell S D rectangle DEQS B-cell rectangle DEQS disk LHKF B-cell disk LHKF disk LHKF F-cell F-cell F-cell

  14. Q E K F H L S D Q E H K F L Combined T-cell S D

  15. G2 K G3 F H G1 L G2 E Q G3 G1 0D cells S D 1D cells G2 G3 G1 A 2D cells gear bnd. G3 gear G3 gear G1 gear bnd. G2 gear bnd. G1 gear G2 B-cell B-cell F-cell F-cell B-cell B-cell Boundary relations point A T-cell

  16. 0D cells 1D cells G2 G3 G1 A 2D cells gear G1 gear bnd. G3 gear G3 gear bnd. G1 gear bnd. G2 gear G2 B-cell F-cell F-cell B-cell B-cell B-cell Boundary relations point A T-cell

  17. 0D cells B 1D cells G2 A G3 G1 2D cells gear bnd. G3 gear G1 gear G3 gear bnd. G2 gear G2 gear bnd. G1 B-cell B-cell B-cell F-cell B-cell F-cell Boundary relations point A point B T-cell T-cell

  18. Q E H G 1D cells K D S F 2D cells Boundary relations rectangle DEQS polyline DEQS B-cell B-cell rectangle FGHK polyline FGHK F-cell F-cell

  19. Q E V S D 0D cells U G H K F 1D cells 2D cells rectangle DEQS polyline DEQS Boundary relations B-cell B-cell polyline FGHK rectangle FGHK line UV point V point U F-cell F-cell T-cell T-cell T-cell

  20. Q E S D 0D cells U G H K F 1D cells 2D cells rectangle DEQS polyline DEQS Boundary relations B-cell B-cell polyline FGHK rectangle FGHK line UV point V point U F-cell F-cell T-cell T-cell T-cell

  21. Q E S D 0D cells U G H K F 1D cells 2D cells polyline DEQS rectangle DEQS Boundary relations B-cell B-cell polyline FGHK rectangle FGHK point U F-cell F-cell T-cell

  22. tN-1<t<tN → t>tN t1<t<t2 → t>t2 t2<t<t3 → t>t3 … S1 S2 SK ti-1<t<ti → t>ti

  23. tN-1<t<tN → t>tN … … KN K1 K2

  24. Kj KN K1 K2 KN-1 … …

  25. … t , , , , K1 K2 KN Kj K2 K1 K1 Kj K2 K1 … PKj TN PK2 T2 T1 PK1 t , , , , … t , , ,

  26. K1 K2 Kj K1 … t1=0.5 t1=0.5 t , , , , Local time is 0.5 but referring to different moments within the global timescale

  27. Pull Retrieve valid parameter Pull Cell 4 Cell 2 Cell 3 Cell 1 Retrieve valid parameter Parameter1 Parameter1 Parameter1 Pull Parameter1 Parameter2 Parameter2 Parameter2 Retrieve valid parameter Dependency Dependency Cell 2 Cell 3 Cell 1 Cell 4 Dependency

  28. Kj Ki … … TKi TKj t ,

  29. K1(0) K2 Q E Q E Q Q Q E E E K K K K K F H F H H H H F F F L L L L L S D S D S S S D D D K1 K3 K3 t

  30. Geometry Initial Geometry Geometry T-cell Cylinder Cylinder Deformedcylinder B/F-cell B/F-cell Deformed cylinder T-cell

  31. K1 K2 K3 T1 T2 T3 t , ,

  32. disk LHKF segment OM circle LHKF F-cell P-cell F-cell O M K F H L Dependency relations

  33. point O segment OM polyline DEQS circle LHKF disk LHKF rectangle DEQS point M F-cell P-cell P-cell P-cell B-cell F-cell B-cell O Q E M 0D cells K F H L S D 1D cells 2D cells

  34. point P point O segment OM segment OP disk LHKF circle LHKF rectangle DEQS Segment MP polyline DEQS point M T-cell F-cell F-cell B-cell B-cell P-cell P-cell T-cell T-cell P-cell P 0D cells O Q E M K F H 1D cells L S D 2D cells

  35. rectangle DEQS point R disk LHKF circle LHKF segment OM segment ON polyline DEQS Segment MP point M point N point O point P segment PN P-cell T-cell P-cell T-cell T-cell T-cell B-cell P-cell F-cell T-cell F-cell T-cell B-cell E O P 0D cells Q R M K N H F L 1D cells S D 2D cells

  36. P O O O O O O O O E Q Q Q Q Q Q Q Q E E E E E M M M M M M M M K K N N K K K K K K H H H F F F H F F F F F H H H H K5 K4 K2 K1 K3 L L L L L L L L P N S S S S S S S S D D D D D D D D E E P P R R R

  37. Probably also need to add “hook” objects using boundary relations Handle Handle Barrel Barrel Rope Hook Hook Rope Convolution skeleton (not visible) Dependency relation “Material” Conv. “Blend” “Material” F-cell F-cell F-cell

  38. Handle Barrel Handle Barrel Rope Hook Rope Skeleton (not visible) Dependency relation “Material” Conv. “Blend” “Material” F-cell F-cell F-cell

  39. Probably also need to add “hook” objects using boundary relations Rope2 Barrel2 Rope1 Handle Handle Barrel1 Rope1 Barrel2 Barrel1 Rope2 Dependency relation Skeleton (not visible) “Material” Skeleton “Blend” “Material” F-cell F-cell F-cell

  40. 1 2 3 4 1 2 3 4 4 3 1 2 4 3 1 2

  41. Dependencies Cells 1 2 3 4 1 2 3 4 5 6 5 6 4 5 4 5 4 5 3 3 6 3 6 6 1 1 1 2 2 2

  42. Dependencies Cells 1 2 3 4 1 2 3 4 5 6 5 6 4 5 4 5 4 5 3 3 6 3 6 6 1 1 1 2 2 2

  43. Dependencies Cells 1 2 3 4 1 2 3 4 5 6 5 6 4 4 3 5 3 1 6 5 2 1 6 2

  44. 4 3 5 Concurrent list 1 Concurrent list 2 Concurrent list 3 4 5 3 1 6 2 1 6 2 4 3 5 1 6 2 Concurrent list 1 Concurrent list 2 Concurrent list 3 4 5 3 1 6 2

  45. p2<p1 p1<p2 p1 evaluate cell 1 using previous state of cell 2 evaluate cell 2 using previous state of cell 1 1 2 1 2 p1 p2 p2 evaluate cell 2 using up-to-date state of cell 1 evaluate cell 1 using up-to-date state of cell 2 2 1 p2<p1 p1<p2 evaluate cell 1 using previous state of cell 2 evaluate cell 2 using previous state of cell 1 1 2 p1 p2 evaluate cell 2 using up-to-date state of cell 1 evaluate cell 1 using up-to-date state of cell 2 2 1 p2 p1 evaluate cell 1 using up-to-date state of cell 2 evaluate cell 2 using up-to-date state of cell 1 1 2

  46. 4 5 Concurrent list 1 Concurrent list 2 Concurrent list 3 Concurrent list 4 3 6 4 5 3 6 1 2 1 2

  47. 2 4 Concurrent list 1 Concurrent list 2 Concurrent list 3 5 3 2 4 5 3 1 1

  48. 1 2 3 4 5 1 2 3 4 2 4 Concurrent list 1 Concurrent list 2 Concurrent list 3 5 3 2 4 5 3 1 1

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