Geometric Modeling in CAD

 
03- Model Representations
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Geometric modeling is done in 3 principle ways:
 
1- Wire frame modeling
 
2- Surface modeling
 
3- Solid modeling
 
In wire frame modeling the object is represented by its edges.
In the initial stages of CAD, wire
 
frame models were in 2-D. Subsequently 3-D
wire frame modeling software was introduced.
The wire frame model of a box is shown in Fig.1a. 
The object appears as if it is
made out of
 
thin wires
. Fig. 1(b), 1(c) and 1(d) show three objects which can have
the same wire frame
 
model of the box. Thus 
in the case of complex parts wire
frame models can be confusing.
 Some
 
clarity can be obtained through hidden
line elimination.
 
Though this type of modeling may
 
not provide unambiguous
understanding of the object, 
this has been the method traditionally
 
used in the
2-D representation of the object
,
 
where orthographic views like plan, elevation,
end view etc. are used to describe the object graphically.
 
W
I
R
E
 
F
R
A
M
E
 
M
O
D
E
L
I
N
G
 
A
m
b
i
g
u
i
t
y
 
i
n
 
W
i
r
e
 
F
r
a
m
e
 
M
o
d
e
l
i
n
g
 
A wireframe model with an ambiguous
 
orientation: the
necker cube
Which face is in front and which is in back?
 
Example of a wireframe model
lacking uniqueness
The same edge and vertex list can
describe different objects,
depending on
 
how the faces are
interpreted.
 
In this approach, a component is represented by its surfaces which in turn are
represented by their vertices and edges. 
For example, eight surfaces are put
together to create a box, as shown in Fig. 2. 
Surface modeling has been very
popular in aerospace product design and automotive design.
 Surface modeling
has been particularly useful in the development of manufacturing codes for
automobile panels and the complex doubly curved shapes of aerospace structures
and dies and moulds.
 
S
U
R
F
A
C
E
 
M
O
D
E
L
I
N
G
 
S
u
r
f
a
c
e
 
R
e
p
r
e
s
e
n
t
a
t
i
o
n
 
Surface modeling
 
techniques are available for interactive modeling
 
and
editing of curved surface geometry. Surfaces can be created through an
assembly of
 
polygonal meshes or using advanced curve and surface
modeling techniques like B-splines
 
or NURBS (Non-Uniform Rational B-
splines). Standard primitives used in a typical surface
 
modeling software
are shown in Fig. 3. Tabulated surfaces, ruled surfaces and edge
 
surfaces
and revolved are
 
simple ways in which curved geometry could be created
and
 
edited.
 
T
y
p
i
c
a
l
 
A
p
p
r
o
a
c
h
e
s
 
i
n
 
S
u
r
f
a
c
e
 
M
o
d
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l
i
n
g
 
Examples of surfaces
commonly used in
engineering design.
 
Surfaces can be created using a number of different techniques.
 
The technique
used is determined both by the shape
 
to be created and by the tools available in
the surface modeler.
Among the most popular methods for creating surfaces
 
are 
sweeping,
revolving,
 
lofting and creating patches with
 
curve boundaries or sets of
points (point clouds).
 
Sweeping
 
is a modeling technique that
allows you to
 
define surfaces by moving a
directrix along a generatrix
 
The 
directrix 
is typically a 2-D curve,
while the 
generatrix
 can be a line, planar
curve, or 3-D curve.
 
Figure 4 shows an oblique cylinder
 
being
created by moving a circle directrix
 
along a
straight line
 
generatrix. Figure 4 also shows
increasingly complex
 
directrix curves
 
being
swept out with straight-line
 
generatrixes to
created ruled surfaces.
 
Swept surfaces
Generating swept surfaces by sweeping generator
 
entities
along director entities.
 
Using curved generatrixes
 
allows for even
more complex surface generation.
 
Notice that a closed-curve directrix
 
creates
a tubelike, hollow surface model.
 
An alternative to defining a generatrix
directly is to
 
revolve
 the directrix about an
axis.
 
A directrix can be rotated about an axis
between 1 and
 
360 degrees.
 
Using a series of directrix curves to define multiple
 
intermediate points along the
generatrix path can create
 
more complex surfaces.
 
This technique, 
lofting
, allows
 
you to defi ne critical changes in the directrix
shape over
 
the surface.
 
Lofting to define a surface
Lofting uses two or more directrix curves to define a surface.
 
A Bezier bicubic surface patch
The patch consists of four connected
 
Bezier
curves and
 
12 control points.
 
In addition to straight lines and circular
 
curves, 
freeform curves such as B-splines and
Bezier
 
curves can be used to generate all or part of the curve
.
 
Freeform curves usually
provide controls that allow you
 
to both define the curve prior to surface generation and
 
edit
the resulting curve by redefining the original curves
 
used to generate the surface.
Freeform curves are regularly used to create surface
 
patches from boundary curves.
 
shows a surface patch made from four
Bezier
 
curves. To form a surface, the
boundary curves should
 
form a closed
path. Just as with creating polygons, there
need to be at least three boundary curves,
but the surface
 
can contain more than four.
The upper limit is typically a
 
practical
matter of managing the surface.
 
NURBS 
stands for
 
Non-Uniform Rational B-Splines. Rational B-splines can
define a wide variety of curves including linear, circular,
 
and conic curves.
This means that NURBS can define
 
the complete set of curves used in a surface
model and
 
rapidly deform, changing curve type on the fly as needed.
 
A bicycle frame defined with complex surface patches.
 
Solid models
It 
includes
 
Volumetric information
, that is, what is on the inside of the
 
3-D
model, as well as information about the surface of an
 
object. In
 
this case,
the surface of the model represents the
 
boundary between the inside and
outside of the object.
 
A complete solid is one which enables a point in space to be classified
relative to the object, if it is inside, outside, or on the object
A valid solid is the one that does not have dangling edges or faces
An unambiguous solid has one and only one interpretation
Solid modelling achieves completeness, validity, and unambiguity of
geometric models
 
Geometry and Topology
 
Geometry is the actual dimensions that define the entities of the object
Topology is the connectivity and associativity of the object entities
 
Solid Entities
 
All primitives can be
created using feature
base approach, that’s
why most of the CAD
software don’t provide
built-in primitives
 
Solid Entities (Contd….)
 
Modeling with primitives 
uses only a limited set of geometric
 
primitives; therefore,
only certain topologies can
 
be created. This is called primitive instancing.
 
A camera described with geometric primitives
Additive modeling with geometric primitives allows a variety
 
of objects to be
represented.
 
The three Boolean operations:
union, difference, and intersection
The three operations, using the same
primitives in the same locations,
create very different objects.
 
The effects of ordering of
operands in a difference
operation
Unlike the union operation, the
difference
 
operation is sensitive
 
to
the ordering of operations.
 
Boolean operations on adjoining
primitives
Only the union operation is effective
when
 
primitives are
 
adjoining but not
overlapping.
 
Properties of Solids
 
Rigidity
This implies that the shape of a solid model is invariant and does
not depend on the model’s location or its orientation in space
Homogenous three-dimensionality
Solid boundaries must be in contact with the interior. No isolated
or dangling boundaries should be permitted
Finiteness and finite describability
Finiteness means that the size of the solid is not infinite, while
finite describability ensures that a limited amount of information
can describe the solid
 
Properties of Solids
 
Closure under rigid motion and regularized Boolean operations
The property ensures that manuplating solids by moving them in
space or changing them via boolean operations must produce
other valid solids
Boundary determinism
The boundary of the solid must contain the solid and hence must
determine distinctivily
 
Solid Representation Schemes
 
1.
Half Spaces
2.
B-rep
3.
CSG
4.
Sweeps
 
Half-spaces
 
By combining half-spaces, using set operations in a block
fashion, various solids can be constructed.
Each geometric entity divides the representation space in two
infinite halves:
Filled with material
Empty
 
Half-spaces
 
For planner surface
For Cylindrical half-space
 
Spherical half-space
 
 
 
Half-spaces
 
Half-spaces
 
Is only useful for research purpose
Unbounded edges/faces cause system crash.
Modeling is cumbersome
Few software use half-space approach
 
Boundary Representation (B-rep)
 
A B-rep model of an object consists of edges, faces, vertices,
loops and handles
 
 
B-rep
 
Face :closed, orientable and bounded by edges
Edge :bounded by two vertices
Vertex : a point in E
Loop: hole in a face
Handle: through hole in a solid
 
Euler Equations
 
To validate the B-rep model, Euler equation is used which is
given below;
F – E + V – L = 2(B-G)
Where
F = faces
E = Edges
V = Vertices
L = Loops
B = Bodies
G = genus (hollow space inside the model. i.e. a through hole)
 
Euler Equations
 
Exp 9.7:
Check the validity of 3D models shown:
 
CSG
 
A physical model can be divided into a set of primitives that
can be combined in a certain order using Boolean operations to
form object.
The data of the solid model is stored in its database in a tree
called CSG Tree.
 
CSG
 
Exp 9.9:
Sketch the CSG tree for the solid S
2
 as shown
 
CSG
 
CSG Tree
 
A designer can use the tree in various ways
Trace the creation step
Edit a feature or a primitive
Prune the tree; moving an entire tree branch to a new location
 
Sweeps
 
Sketching a cross-section and sweeping it
Useful in creating a 2½D objects
Both extrusion and revolution is possible
Three type:
Linear - the sweeping path is linear (for extrusion) or circular (for
axisymmetric solid) vector
Nonlinear – the path is a curve described by a higher order
(quadratic, cubic or higher) equation
Hybrid – combines linear and nonlinear sweeps
 
Types of sweeps
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Geometric modeling in computer-aided design (CAD) is crucially done in three key ways: wireframe modeling, surface modeling, and solid modeling. Wireframe modeling represents objects by their edges, whereas surface modeling uses surfaces, vertices, and edges to construct components like a box. Each technique has its advantages and applications, from aerospace design to automotive manufacturing. Explore the nuances of these modeling methods to enhance your understanding of CAD.


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  1. 03- Model Representations

  2. Geometric modeling is done in 3 principle ways: 1- Wire frame modeling 2- Surface modeling 3- Solid modeling

  3. WIRE FRAME MODELING In wire frame modeling the object is represented by its edges. In the initial stages of CAD, wire frame models were in 2-D. Subsequently 3-D wire frame modeling software was introduced. The wire frame model of a box is shown in Fig.1a. The object appears as if it is made out of thin wires. Fig. 1(b), 1(c) and 1(d) show three objects which can have the same wire frame model of the box. Thus in the case of complex parts wire frame models can be confusing. Some clarity can be obtained through hidden line elimination. Though this type of modeling may not provide unambiguous understanding of the object, this has been the method traditionally used in the 2-D representation of the object, where orthographic views like plan, elevation, end view etc. are used to describe the object graphically. Ambiguity in Wire Frame Modeling

  4. Example of a wireframe model lacking uniqueness The same edge and vertex list can describe different depending on how the faces are interpreted. objects, A wireframe model with an ambiguous orientation: the necker cube Which face is in front and which is in back?

  5. SURFACE MODELING In this approach, a component is represented by its surfaces which in turn are represented by their vertices and edges. For example, eight surfaces are put together to create a box, as shown in Fig. 2. Surface modeling has been very popular in aerospace product design and automotive design. Surface modeling has been particularly useful in the development of manufacturing codes for automobile panels and the complex doubly curved shapes of aerospace structures and dies and moulds. Surface Representation

  6. Surface modeling techniques are available for interactive modeling and editing of curved surface geometry. Surfaces can be created through an assembly of polygonal meshes or using advanced curve and surface modeling techniques like B-splines or NURBS (Non-Uniform Rational B- splines). Standard primitives used in a typical surface modeling software are shown in Fig. 3. Tabulated surfaces, ruled surfaces and edge surfaces and revolved are simple ways in which curved geometry could be created and edited. Typical Approaches in Surface Modeling

  7. Examples commonly engineering design. of surfaces used in

  8. Surfaces can be created using a number of different techniques. The technique used is determined both by the shape to be created and by the tools available in the surface modeler. Among the most popular methods for creating surfaces are sweeping, revolving, lofting and creating patches with curve boundaries or sets of points (point clouds). Sweeping is a modeling technique that allows you to define surfaces by moving a directrix along a generatrix The directrix is typically a 2-D curve, while the generatrix can be a line, planar curve, or 3-D curve. Figure 4 shows an oblique cylinder being created by moving a circle directrix along a straight line generatrix. Figure 4 also shows increasingly complex directrix curves being swept out with straight-line generatrixes to created ruled surfaces. Swept surfaces Generating swept surfaces by sweeping generator entities along director entities.

  9. Using curved generatrixes allows for even more complex surface generation. An alternative to defining a generatrix directly is to revolve the directrix about an axis. Notice that a closed-curve directrix creates a tubelike, hollow surface model. A directrix can be rotated about an axis between 1 and 360 degrees.

  10. Using a series of directrix curves to define multiple intermediate points along the generatrix path can create more complex surfaces. This technique, lofting, allows you to defi ne critical changes in the directrix shape over the surface. Lofting to define a surface Lofting uses two or more directrix curves to define a surface.

  11. In addition to straight lines and circular curves, freeform curves such as B-splines and Bezier curves can be used to generate all or part of the curve. Freeform curves usually provide controls that allow you to both define the curve prior to surface generation and edit the resulting curve by redefining the original curves used to generate the surface. Freeform curves are regularly used to create surface patches from boundary curves. shows a surface patch made from four Bezier curves. To form a surface, the boundary curves should form a closed path. Just as with creating polygons, there need to be at least three boundary curves, but the surface can contain more than four. The upper limit is typically a practical matter of managing the surface. A Bezier bicubic surface patch The patch consists of four connected Bezier curves and 12 control points.

  12. NURBS stands for Non-Uniform Rational B-Splines. Rational B-splines can define a wide variety of curves including linear, circular, and conic curves. This means that NURBS can define the complete set of curves used in a surface model and rapidly deform, changing curve type on the fly as needed. A bicycle frame defined with complex surface patches.

  13. Solid models It includes Volumetric information, that is, what is on the inside of the 3-D model, as well as information about the surface of an object. In this case, the surface of the model represents the boundary between the inside and outside of the object. A complete solid is one which enables a point in space to be classified relative to the object, if it is inside, outside, or on the object A valid solid is the one that does not have dangling edges or faces An unambiguous solid has one and only one interpretation Solid modelling achieves completeness, validity, and unambiguity of geometric models

  14. Geometry and Topology Geometry is the actual dimensions that define the entities of the object Topology is the connectivity and associativity of the object entities

  15. Solid Entities All primitives can be created using base approach, why most of the CAD software don t provide built-in primitives feature that s

  16. Solid Entities (Contd.)

  17. Modeling with primitives uses only a limited set of geometric primitives; therefore, only certain topologies can be created. This is called primitive instancing. A camera described with geometric primitives Additive modeling with geometric primitives allows a variety of objects to be represented.

  18. The three Boolean operations: union, difference, and intersection The three operations, using the same primitives in the same locations, create very different objects.

  19. The operands operation Unlike the union operation, the difference operation is sensitive to the ordering of operations. effects of ordering a difference of in

  20. Boolean primitives Only the union operation is effective when primitives are adjoining but not overlapping. operations on adjoining

  21. Properties of Solids Rigidity This implies that the shape of a solid model is invariant and does not depend on the model s location or its orientation in space Homogenous three-dimensionality Solid boundaries must be in contact with the interior. No isolated or dangling boundaries should be permitted Finiteness and finite describability Finiteness means that the size of the solid is not infinite, while finite describability ensures that a limited amount of information can describe the solid

  22. Properties of Solids Closure under rigid motion and regularized Boolean operations The property ensures that manuplating solids by moving them in space or changing them via boolean operations must produce other valid solids Boundary determinism The boundary of the solid must contain the solid and hence must determine distinctivily

  23. Solid Representation Schemes 1. Half Spaces 2. B-rep 3. CSG 4. Sweeps

  24. Half-spaces By combining half-spaces, using set operations in a block fashion, various solids can be constructed. Each geometric entity divides the representation space in two infinite halves: Filled with material Empty

  25. Half-spaces = ( , , : ) z 0 H x y z For planner surface For Cylindrical half-space = + 2 2 2 ( , , : ) z H x y x y R Spherical half-space ( ) = + + 2 2 2 2 , , : H x y z x y z R

  26. Half-spaces

  27. Half-spaces Is only useful for research purpose Unbounded edges/faces cause system crash. Modeling is cumbersome Few software use half-space approach

  28. Boundary Representation (B-rep) A B-rep model of an object consists of edges, faces, vertices, loops and handles

  29. B-rep Face :closed, orientable and bounded by edges Edge :bounded by two vertices Vertex : a point in E Loop: hole in a face Handle: through hole in a solid

  30. Euler Equations To validate the B-rep model, Euler equation is used which is given below; F E + V L = 2(B-G) Where F = faces E = Edges V = Vertices L = Loops B = Bodies G = genus (hollow space inside the model. i.e. a through hole)

  31. Euler Equations Exp 9.7: Check the validity of 3D models shown:

  32. CSG A physical model can be divided into a set of primitives that can be combined in a certain order using Boolean operations to form object. The data of the solid model is stored in its database in a tree called CSG Tree.

  33. CSG Exp 9.9: Sketch the CSG tree for the solid S2 as shown

  34. CSG

  35. CSG Tree A designer can use the tree in various ways Trace the creation step Edit a feature or a primitive Prune the tree; moving an entire tree branch to a new location

  36. Sweeps Sketching a cross-section and sweeping it Useful in creating a 2 D objects Both extrusion and revolution is possible Three type: Linear - the sweeping path is linear (for extrusion) or circular (for axisymmetric solid) vector Nonlinear the path is a curve described by a higher order (quadratic, cubic or higher) equation Hybrid combines linear and nonlinear sweeps

  37. Types of sweeps

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