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History: CNC: ~1950 Mainframe Computers: ~1960’s BREP: 1970 (Baumgart) CSG: 1974 (Ian Braid)

Solid Modeling. History: CNC: ~1950 Mainframe Computers: ~1960’s BREP: 1970 (Baumgart) CSG: 1974 (Ian Braid). Computerized Drafting. Advantages: Saves on storage/retrieval; Easy modification, update; Shortcomings: Can’t analyse the strength, shape, geometry, weight

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History: CNC: ~1950 Mainframe Computers: ~1960’s BREP: 1970 (Baumgart) CSG: 1974 (Ian Braid)

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  1. Solid Modeling History: CNC: ~1950 Mainframe Computers: ~1960’s BREP: 1970 (Baumgart) CSG: 1974 (Ian Braid)

  2. Computerized Drafting Advantages: Saves on storage/retrieval; Easy modification, update; Shortcomings: Can’t analyse the strength, shape, geometry, weight center of mass, center of inertia Popular Commercial tools: AutoCAD, CADKEY…

  3. 3D and Solids Representation Wireframe Models Stores each edge of the object data: The vertices start point, end point The equation of the edge-curve

  4. Wireframe Problem: Ambiguity

  5. Other Methods Constructive Solid Geometry (CSG) Boundary Representation (BREP) Spatial Enumeration (voxels, octrees)

  6. Constructive Solid Geometry (CSG) Introduced: Ian Braid (Cambridge University, ~74) Concepts: Primitives: small set of shapes Transformations: scaling, Rotation, Translation Set-theoretic Operations Union, Intersection, Difference [Euler operators ] Combinations of these  Solid part

  7. Euler operators U* (regular union) -* (regular difference) ∩* (regular intersection) CSG Tree: Sequence of operators  design

  8. Z Z Z X X X - - * * U* U* Trans( 10, 0, 0) box( 3, 10, 10) Trans( 10, 0, 0) box( 3, 10, 10) Trans( 20, 12.5, 15) cylinder( 5, 3) Trans( 20, 12.5, 15) cylinder( 5, 3) U* U* box( 25, 25, 15) box( 25, 25, 15) Trans( Trans( 5 - - 5, 0, 5, 0, - - 5) box( 10, 25, 10) 5) box( 10, 25, 10) 2,0, 15) Rot(Y, 45) Trans(  Examples of CSG

  9. Questions: Can we use a different set of primitives ? Is the CSG representation unique ? [how to determine if two solids are identical ?]

  10. Regularized operators Is the set of 3D solids is closed with respect to ( U, -, ∩ )? closure of a set S: kS interior of a set S: iS A U* B = ki ( A U B) A -* B = ki ( A –B) A ∩* B = ki ( A ∩ B) Why is closure over operations important? uniform data structures

  11. Regularized Euler Operators Maintain solid as a regular 2-Manifold 2-Manifold regular solids Open neighborhood of each point is similar to an open disc Non 2-Manifold:

  12. Problems with CSG Non-Unique representation Difficulty of performing analysis for some tasks

  13. BREP (Boundary REPresentation) What entities define the Boundary of a solid ? Boundary of surfaces? Boundary of curves (edges) ? Boundary of points ?

  14. BREP Boundary of a solid… Boundary of surfaces… Boundary of curves (edges)…

  15. BREP: Polyhedral models

  16. Using a Boundary Model Compute Volume, Weight Compute Surface area Point inside/outside solid Intersection of two faces …

  17. An Edge-Based Model

  18. Edge-Based Models: inefficient algorithms Compute Surface Area: 1. Identify Loops 2. Compute area of each loop 3. Compute area of face

  19. The Winged-Edge Data Structure Efficient implementation of often-used algorithms Area of Face Hidden surface removal Find neighbor-faces of a face

  20. co-edges Face CCW convention => Each edge is once +ve, once -ve Observations 2-Manifold => Each edge is shared by exactly 2 faces

  21. BREP Example

  22. BREP Example

  23. BREP Example..

  24. BREP Example…

  25. BREP: Winged edge data structure

  26. BREP or CSG ? Using: CSG is more intuitive Computing: BREP is more convenient Modern CAD Systems: CSG for GUI (feature tree) BREP for internal storage and API’s

  27. BREP: non-polyhedral models? Same Data Structure, plus For each edge, store equation For each curved face, store equation Why do we need to learn all this ? (a) To anticipate when an operation will fail (b) To allow us to write API’s

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