This document discusses different techniques for representing 3D solid objects in computer modeling, including boundary representation (B-rep), constructive solid geometry (CSG), and spatial partitioning. B-rep describes objects by their surface boundaries like vertices and faces. CSG uses Boolean operations on primitives. Spatial partitioning divides space into uniform cells. Each technique has strengths and weaknesses in terms of accuracy, domain of representable objects, uniqueness of representation, and efficiency.

1. “In the name of God”
Solid
Modeling

2. What is solid modeling ?!
A mathematical technique for
representing solid objects

3. OK…what’s its usage ?
• Represent solid interiors of objects
Surface may not be described explicitly
Visible human

4. OK…what’s its usage ?
• Some acquisition methods generate solids
Example: CAT scan

5. OK…what’s its usage ?
• Some applications require solids
Example: CAD/CAM
(computer-aided design/computer-aided manufacturing)
Computer systems used to design and manufacture products

6. What concerns do we have for
representing?
• Domain ---> large enough
• Unambiguous ---> one & only one , (!what is this)
• Unique ---> encode in only one way -> (testing)
• Accurate ---> without approximation

7. What concerns do we have for
representing?
• Closure ---> union-intersection-rotation-translation
• Compact ---> save space
• Allow using efficient algorithms.

8. REGULARIZED BOOLEAN SET
OPERATIONS

9. REGULARIZED BOOLEAN SET
OPERATIONS
But what we want :

10. REGULARIZED BOOLEAN SET
OPERATIONS

11. Representation solid modeling
• Primitive instancing
• Sweeps
• B-Reps
• Spatial partitioning (cell-decomposition,spatial
occupancy enumeration,octrees,BSP trees)
• CSG

12. Representation solid modeling
(Primitive instancing)
• Define a set of primitive 3D solid shapes
• Primitive instances are similar to parameterized
objects
• A family with few differences in members
• Relatively complex objects
• Without combining objects

13. Representation solid modeling
(Sweep)
• sweeping a 2D surface in 3D space to create
an object
• Translational sweep
• Rotational sweep
The object being swept does not need to be
2D

14. Representation solid modeling
(Sweep)
• General sweeps ---> (difficult to model efficiency)

15. Representation solid modeling
(Sweep)
• It is difficult to apply regularized Boolean set
operations
• Even simple sweeps are not close under
regularized Boolean set operations

16. Representation solid modeling
(Boundary)
• Describe an object in terms of its surface boundaries :
vertices , edges , and faces
Determining what constitutes a face can be difficult if curved
surfaces are allowed
• So some b-reps are restricted to planar , polygon , etc

17. Representation solid modeling
(Boundary)
• Polyhedra and Euler’s Formula
• The Winged-Edge Representation
• Boolean Set Operations
• Nonpolyhedral b-Reps
Because b-Reps tile an object’s surface, they
do not provide a unique representation of a
solid

18. Representation solid modeling
(Boundary - Polyhedra and Euler’s Formula)
• A polyhedron is solid that is bounded by a set
of polygon (and some additional constraints)
• The b-Reps of a simple polyhedron satisfies
Euler’s formula (also with curved edges and nonplanar faces)

19. Representation solid modeling
(Boundary - Polyhedra and Euler’s Formula)
• States necessary but not sufficient conditions
• Additional constraints are needed

20. Representation solid modeling
(Boundary - Polyhedra and Euler’s Formula)

21. Representation solid modeling
(Boundary - Polyhedra and Euler’s Formula)
Tweak operators do not affect number of vertices
, edges , etc…

22. Representation solid modeling
(Boundary – “Winged-Edge” Representation)
• What is simple representation? (advantages and
disadvantages?!)
• The better way but more complex is
“Winged-Edge” representation(handles only faces
without holes)

23. Representation solid modeling
(Boundary – Boolean Set Operation)
• Inspect the polygons in both objects , splitting
them if necessary

24. Representation solid modeling
(Boundary – Nonpolyhedral)
• Poly-rep can only approximate Nonpolys

25. Representation solid modeling
(Spatial-Partitioning)
• A solid is decomposed into a collection of
adjoining nonintersecting solids. (more primitive)
• 1- Cell Decomposition
• 2- Spatial-Occupancy Enumeration
• 3- Octrees
• 4- Binary Space-Partitioning Trees

26. Representation solid modeling
(Spatial-Partitioning – “Cell Decomposition”)
• Defines a set of primitive cells
• Differs from primitive instancing(“Glue” operation)
• Unambiguous but not necessarily unique
Difficult to validate

27. Representation solid modeling
(Spatial-Partitioning – “Spatial-Occupancy” Enumeration)
• Voxels
Unique and unambiguous
But approximate
!save space

28. Representation solid modeling
(Spatial-Partitioning – “Spatial-Occupancy” Enumeration)
• Often used in biomedical application (such as
Computerized Axial Tomography(CAT) scans)

29. Representation solid modeling
(Spatial-Partitioning –“Octrees”)
• A hierarchical variant of spatial-occupancy enumeration
• Full – partially full - empty

30. Representation solid modeling
(Spatial-Partitioning –“Octrees”)
The “Octree” is similar to the quadtree
(LUF,LUB,LDF,LDB,RUF,RUB,RDF,RDB)

31. Representation solid modeling
(Spatial-Partitioning –“Octrees”)
• Boolean set operations and transformations

32. Representation solid modeling
(Spatial-Partitioning –“Octrees”)
• Neighbor finding
• Linear notations
– 00X,010,011,020,…..

33. Representation solid modeling
(Constructive Solid Geometry)
CSG
• Operators at the internal nodes and simple
primitives at the leaves

34. Representation solid modeling
(Constructive Solid Geometry)
CSG
• Not unique
• Deleting-adding replacing-modifying subtrees-etc

35. Just a moment, let
compare them

36. Comparison of Representations
• Accuracy :
Spatial-partitioning & polygon B-Rep ---> approximate
CSG & curved surfaces B-Rep & primitive instancing ---> maybe better
• Domain :
Primitive instancing & sweeps ---> limited
B-rep ---> a very wide classes of objects
Spatial-partitioning ---> represent any solid
• Uniqueness :
only “Octree” and spatial-occupancy-enumeration approaches guarantee
the uniqueness of a representation !!

37. Comparison of Representations
• Closure :
primitive instancing ---> cannot be combined at all
sweeps ---> are not closed
• Compactness and efficiency :
CSG ---> compactness and the ability to record Boolean operations

38. Thank you