page 143
V – E + F = 2
where,
V = number of vertices
E = number of edges
F = number of faces
V – E + F – ( L – F) – 2( S – G) = 0
where,
L = number of edge loops
S = number of shells
G = genus of solid (holes)
• When developing solid modelers we can use the Euler operations to ensure that the model stays topographically valid at all times.
DQ:16.2.5 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
page 144
•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.
DQ:16.2.6 Tessellated Models
• Space is broken down as a regular/irregular grid.
This evenly tessellated space is divided down into 256 squares
This space is broken down into 28 squares for an equivalent representation
•locations in space are marked as occupied/empty/partially filled.
•this method is most common when using scanners such as CAT and MRI that collect data in
page 145
voxels (these are small rectangular volumes)
DQ:16.2.7 Features
•The designer would simply define a part in terms of fundamental manufacturing features, such as chamfers, through slots, blind slots, etc.
•Very high level, but can complicate additions of unanticipated features, like a ridge in a car hood.
•Advantages,
-very intuitive and easy to use
-can simplify other aspects of CIM (eg. If a standard feature is used there will be a standard process plan to make that feature).
-emphasizes the use of standard components.
•Disadvantages,
-restrictive when dealing with nonstandard features
-interaction of features can be hard to estimate
-a complete set of all possible features would be very large
•There are two levels of features commonly used in these systems,
-micro
-macro
•A set of standard features for rotational parts might be,
•Macro Features,
-cylinder
-taper
•External Features
-rotational fillet
-thread
-square neck
-chamfer
-shoulder
-external radius
-key seat
-spline
-flat
-thread
•Internal Features
-internal taper
-internal slot]