1. COMPUTER AIDED DESIGN -(CAD)-3

2. Geometrical Modelling Of Parts
OVERVIEW • A model is used to mathematically capture some aspect of the real world so that it may be manipulated, studied, tested, etc. • To computerize a design we will capture a very explicit model of a part in a computer program. • Geometry is by far, the most important part of any design. It is used as a reference for tolerances, processes, functions, materials, assemblies, etc.

4. Three major functions in geometric modelling

5. • There are a number of methods of representing geometry, 1D, 2D, 2
• There are a number of methods of representing geometry, 1D, 2D, 2.5D, 3D
2D - All in one dimension, such as a side view of a part
2.5D - Points have an x, y coordinate, and height. This is a special case which can describe
raised surfaces like buildings.
3D - Full spatial dimensions
• 2D models are quickly becoming obsolete, 2.5D models were an early compromise, but 3D models are quickly becoming the standard.

6. Geometric Models
• There are many ways to model a part, the major categories are,
- Elemental (using lines and points like drafting)
- Surfaces (such as polygons used in ES 206)
- Solids - Swept
- Solids - BRep
- Solids - CSG
- Features
• If describing a block with a hole in it, each of the methods above will result in different descriptions

9. • Which is best???? all of them in the right situations.
• Each method has its particular advantages, and disadvantages.
• The best software and hardware supports a combination of all methods.
• It is assumed that other information is used to describe the geometries above, like,
- Position
- Orientation
- Dimensions
• The geometries can also be used to associate other information,
- Materials
- Tolerances
- Finishes
- Inspection information

10. Elemental Depiction:
• Depicted with the simplest of details (lines, points arcs, etc.)
• Advantages,
- very easy to store and alter
- well suited to line based problems
- does not require a powerful computer
- easy to perform traditional drafting
• Disadvantages,
- not capable of carrying complex information
- ambiguous
- complex items require long time to model
- requires repetition
- very hard to connect to programs for FEM, etc.
• Typically used in older CAD systems like AUTOCAD, CADKEY, etc.
• A classic demonstration of the arbitrary nature is shown below,

12. Surface Description
• The geometry is described with polygons which should represent an entire surface of an object.
• Generally these polygons do not indicate which side a volume lies on, but inside/outside is
defined with tricks like defining polygon vertices so that counter-clockwise is out.
• STL is a good example of an engineering use of this surface representation.
• This method is also used in computer games where speed is important, and the overhead of the
Which shape this wire frame drawing produces is arbitrary !
full solid information is not desired.
• Advantages,
- gives appearance of solids.
- well known, and fast software and hardware for drawing.
• Disadvantages,
- because objects are not solid, they may be subject to ambiguities
- hard to pass data to other systems, like FEA
- not well suited to CAD
- polygon selection is problematic
• Commonly used in graphics packages like HOOPS, PHIGS, CORE, etc. Also acts as the basis
for the SGI computer graphics.

13. • An example of the polygon meshes is given below.
• We can also define these geometries using edge meshes.

14. Solid – Swept
• A profile is created in 2D, and then swept along a path to create a volume, or to cut a volume.
• The path may be straight, rotating about an axis, rotation along a helix, following a curved twisting path.
• Advantages,
- Can make very complex parts quickly
• Disadvantages,
- Requires a powerful computer
- Some operations difficult

15. Solid - B-Rep (Boundary Representation)
• This still bears a remote resemblance to Surface Modelling.
• Major differences are that,
- inside/outside is defined for each surface
- the edges, and vertices of touching faces are defined
• Advantages,
- can store very complex geometries
- easy to propagate changes to faces, edges and vertices
- can easily generate and store complex surfaces
- many systems support this method, such as PARASOLIDS, ACIS, etc.
- can be used to mimic CSG
• Disadvantages,
- high Level information is still not present in model
- requires a powerful computer
- hard to recognize some simple features like a block

16. • A BRep object is pictured below,
• Each feature in a B-Rep object can be varied independently
• Geometry is kept in parallel with the object topology.
One example of a data structure is seen to the wright.

17. • A common data focus uses the edges of an object to define the shape (vertices and faces can also be used)
Winged-edge data structure
• Euler operations can be used to build an object.
• We can check to see if the solid model is valid using the basic Euler equation, or the more involved Euler-Poincare tologoical equation. The above equations must be satisfied for the models to be valid

18. Solid - CSG
• Does not calculate lines/vertices/faces when storing part geometries
• Uses primitive shapes such as planes, blocks, spheres, cylinders, wedges, torii, etc. to model shapes
• The primitives can be rescaled to meet requirements
• Uses a basic set of operators to combine or cut with the primitives,
Union - Both primitives are joined into one (boolean OR)
Intersection - The part of the primitives which overlaps (boolean AND)
Not - The inverse of a shape
Assemble - Parts may overlap without being joined
Difference - The area of one primitive is removed from another
• Basic common primitives are,
- blocks
- cylinders
- wedges
- tetrahedrons
- spheres
- torii
- cones

19. • Used in systems like PADL2, Romulus, Build, etc.
• Advantages,
- Very compact representation
- Primitive shapes match human though processes
- Very fast when creating parts with standard geometrical features
• Disadvantages,
- Slow because all interpretation is done at once
- may be difficult to incorporate irregular surfaces
• Used in systems like PADL2, Romulus, Build, etc.
• CSG designs can be stored in trees
• Various types of CSG operators are possible based on closure of sets. In particular we can consider two boxes that touch, but don’t overlap.
• Halfspaces can be used for defining boundaries of an object