l30 solidmod basics
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Solid Modeling
Concepts
AML710 CAD LECTURE 30
Evolution of Geometric Modeling A wireframe representaion of an object is done using edges (linescurves) and vertices. Surface representation then is the logicalevolution using faces (surfaces), edges and vertices. In thissequence of developments, the solid modeling uses topologicalinformation in addition to the geometrical information to represent theobject unambiguously and completely.
Wireframe Model Solid Model
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Manifold Vs Non-manifoldTwo Manifold Representations:
Two manifold objects are well bound, closed and homomorphic to atopological ball.
Non-manifold Representations:When restrictions of closure and completeness are removed from the soliddefinition, wireframe entities can coexist with volume based solids.
Two-Manifold Object
Danglingface
Non-ManifoldRepresentation
Danglingedge
Disadvantages of Wireframe Models
Ambiguity Subjective human interpretation Complex objects with many edges
become confusing Lengthy and verbose to define Not possible to calculate Volume
and Mass properties, NC toolpath, cross sectioning etc
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Definition of a Solid Model A solid model of an object is a more complete representation than itssurface (wireframe) model. It provides more topological informationin addition to the geometrical information which helps to representthe solid unambiguously.
Wireframe Model Solid Model
History of Geometric Modeling
Univ. of Rochester, Voelcker & RequichaPAP, PADL-I,PADL2
1972
Baumgart, Stanford Univ., USAEuler OpsWinged Edge,B-rep
1975
Evans and Sutherland, 1 st commercialRomulus1981
Eastmans Group in CMU, USAGLIDE-I1975
Hokkaido University, JapanTIPS-I1973
Braids CAD Group inCambridge, UK
Build-IBuild-II
1973
Developer Modeler Year
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Geometric Modeling - Primitives
Primitives and Instances
PRIMITIVES
INSTANCES
Instancing: An instance is a scaled
/ transformed replica of its original.The scaling may be uniform ordifferential. It is a method ofkeeping primitives in their basicminimum condition like unit cubeand unit sphere etc.
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Solid Primitives: Ready starting objects
Origin of MCS: P Orientation Topology and shape
Dimension (size) L,B,H.D,R Perspective on/off
X
Y
Z
L
R
Y
X
Z
R
Z
BH
DX
Y
P
Y
X
Z B
H
D
P
SPHEREWEDGE
CYLINDERBLOCK
Solid Primitives: Ready starting objects
Ro
P
R i
RX
Y
Z
R
H
Z
X
Y
TORUS
CONE
PERSPECTIVE OFF PERSPECTIVE ON
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Operation on Primitives A desired solid can be obtained by combining two or more solidsWhen we use Boolean (set) operations the validity of the third(rusulting) solid is ensured
UNION: BLOCK CYLINDER A B
A
B
A
B
3 Dimensional
Operation on Primitives A desired solid can be obtained by combining two or more solids
When we use Boolean (set) operations the validity of the third(rusulting) solid is ensured
UNION: BLOCK CYLINDER A B
A
B
A
B
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Operation on Primitives
INTERSECTION:
BLOCK CYLINDER
A B
A
B
A
B
A
B
DIFFERENCE:
BLOCK - CYLINDER
A - B
B - A
Eulers formula gives the relationamongst faces, edges and vertices of asimple polyhedron.
V-E+F=2 . This formula tells us thatthere are only 5 regular polyhedra
Consider polyhedra with every facehaving h edges, k edges emanatingfrom each vertex and each edgesharing exactly 2 faces, then we canwrite an invariant for the solid as:
hF=2E=kV Substituting this in Eulersformula we get
Simple Polyhedron Eulers Formula
21111
222
+==+k h E h
E E
k E
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We notice that for a polyhedron h,k 3 What happens if h=k=4 ? If we take h=3, then 3 k 5 Then by symmetry, if k=3 then 3 h 5 Thus the possib le polyhedra can be symbolic ally given as:(h,k,E)=(3,3,6), (3, 4, 12), (4,3,12), (5, 3, 30) and (3, 5, 30);
tetrahedron, oc tahedron, cube, dodecahedron, icosahedron
21111
0222
+=
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The following example will illustrate the conceptsof interior , boundary , closure and exterior . Consider a unit sphere
Its interior is Its Boundary is Solid sphere is
Its closure
Whatever is outside of this closure is exterior
Interior, Exterior and Closure
1
1
1
222
222
222
++
=++
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Rigidity: Shape of the solid is invariant w.r.tlocation/orientation
Homogeneous 3-dimensionality : The solidboundaries must be in contact with the interior.No isolated and dangling edges are permitted.
Properties of Solid Models
=
Danglingedge
Danglingface
Non-ManifoldRepresentation
Finiteness and finite describability: Size isfinite and a finite amount of information candescribe the solid model.
Closure under rig id motion and regularisedBoolean operations : Movement and Booleanoperations should produce other valid solids.
Boundary determinism: The boundary mustcontain the solid and hence determine the soliddistinctly.
Any valid so lid must be bounded , closed,regular and semi-analytic subsets of E 3
Properties of Solid Models
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The regularized point sets which representthe resulting solids from Booleanoperations are called r-sets (regularizedsets .
R-sets can be imagined as curvedpolyhedra with well behaved boundaries .
Properties of Solid Models
The schemes operate on r-sets or point sets to produce thedesired solid model, which is automatically a valid solid
For unambiguous, complete and unique representation thescheme maps points form point set (Euclidian) into a validmodel to represent the physical object.
Properties of Solid Modeling Schemes
Unambiguous,complete, unique
O P
O P 1 P 2
O1 O 2P
Unambiguous,complete, non-unique
Amb iguous ,incomplete, non-unique
Unl imited Po in t set , W Mod el space , S
UCU
UCN
AINRepresentation
Scheme
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Domain : The type of objects that can be represented or thegeometric coverage.
Validity : The resulting solid model when subjected toalgorithms should succeed as valid solid.
Completeness and Unambiguousness .
Uniqueness : Positional and Permutational uniqueness.
Common Features of Modeling Schemes
Positionally non-unique
A B C AC B
a
b ba
Permutationally non-unique
S=A U B U C S=C U B U A
Conciseness, ease of creation and efficiency inapplication.
Conciseness means compact database torepresent the object.
Ease of operation implies user-friendl iness.Most of the packages use CSG approach tocreate the solids
The representation must be usable for lateroperations like CAM / CNS etc
Desirable Properties of ModelingSchemes
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Classification of Solid Modeling
Boundary based or Volume based Object based or spatially based Evaluated or unevaluated
Evaluated
Unevaluated
Boundarybased
VolumeBased
SpatialbasedObjectbased
E v al u a t e d
Un ev al u a t e d
OctreeHalf spaceSpatial
Non-parametic
BoundaryRep
Object
Cell EnumBoundaryCell Enum
Spatial
CSGEuler OpsObject
Boundary | Volume
1. Half Spaces.
2. Boundary Representation (B-Rep)
3. Constructive Solid Geometry (CSG)
4. Sweeping
5. Analytical Solid Modeling (ASM)
6. Cell decomposition
7. Spatial Ennumeration
8. Octree encoding
9. Primitive Instancing
The 3 most popular schemes: B-rep, CSG,Sweeping
Major Modeling Schemes
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Some representations can be converted to otherrepresentations.
CSG to B-REP (but not B-rep to CSG)
B-Rep
2. Sweep CSGCell Decomposition
Conversion Among Representations
Some Solid Modelers in Practice
BREP+CSGBREPPARAMETRICPRO-E
BREP+CSGBREPINTERGRAPHSOL. MOD. SYS
BREP+CSGCSGMcDONELLDOUGLAS
UNISOLIDS /UNIGRAPHICS
BREP+CSGBREPCOMPUTERVISION
SOLIDESIGN
CSGCSGCORNELL UNI.PADL-2
HYPERPATCHES+CSG ASMPDA ENGG.PATRAN-G
BREP+CSGBREPSDRC/EDSGEOMOD /I-DEAS
BREP+CSGCSGIBMCATIA
User InputPrimaryScheme
Developer Modeler
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Three types of algorithms are in use in manysolid modelers according to the kind of inputand output
1. a; data -> representation
most common build, maintain, managerepresentations
Algorithms in Solid Modelers
Data Representation
Algori thm
Some algorithms operate on existing models toproduce data.
2. a; representation -> data
E.g.. Mass property calculation
Algorithms in Solid Modelers
DataRepresentation
Algori thm
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Some representations use the availablerepresentations to produce anotherrepresentation.
3. a: representation -> representation
E.g. Conversion from CSG to B-rep
Algorithms in Solid Modelers
Representation
Algori thm
Representation
Closure and Regularized setOperations
The interior of the solid is geometrically closed by itsboundaries
S bS iS =U
bS iS kiS S U==
If the closure of the interior of a given set yields that samegiven set then the set is regular
Mathematically a given set is regular if and only if:
)(*);(*
)(*);(*
cQkiQcQPkiQP
QPkiQPQPkiQP
bS iS kiS S
==
==
== U
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