Under Graduate Course on
Computer Aided Design & Manufacturing
( ME 502 )
Geometric
Modeling
Shibendu Shekhar Roy
Robotics & A.I Lab. (RAIL)
Department of Mechanical Engineering
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Geometric
Modeling
Concept
CAD
Computer
Graphics
Algorithm
Design & Analysis
Theories/ Tools
…To Automate the Design Process
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Geometric Modeling
‰ Geometric modeling plays a crucial role in the overall application of CAD-CAM-CAE
system
Requirements of Geometric Modeling
‰ Information entered through geometric modeling is utilized in number of downstream
applications like:
Design Analysis
¾ Evaluation of centroid, area (cross-sectional & surface) & volume
¾ Estimate the mass & inertia properties
¾ Interference checking in assemblies
¾ Analysis of tolerance build-up in assemblies
¾ Kinematic/ Dynamic analysis & simulation
¾ Finite element analysis for stress, vibration, thermal & optimization
Drafting
¾ Automatic 2D view generation
¾ Automatic planar cross-sectioning
¾ Automatic dimensioning
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Geometric Modeling
Manufacturing
¾ Part classification & Process planning
¾ NC tool path generation & verification
¾ Manufacturing process simulation
¾ Robot program generation
Production &
Industrial
Engineering
¾ Preparation of Bill of Material (BOM)
¾ Material requirement planning
¾ Manufacturing resource requirement
¾ Scheduling
¾ Marketing
Computer-Aided Inspection & Quality control
etc…..
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Geometric Modeling
‰ Hence, It is important that the geometric model generated should be as clear &
comprehensive as possible so that the other modules of CAD-CAM-CAE system are
able to use this information in the most optimal way.
Two Dimensional
(2-D)
Geometric model
Wireframe Modeling
Three Dimensional
(3-D)
Surface Modeling
Solid Modeling
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Geometric Model
Two Dimensional (2-D)
¾ Utility of 2-D model lies in many of the low end drafting packages which is required
for preparing manufacturing drawings
¾ Their utility is limited because of their inherent difficulty in representing complex
objects.
Three Dimensional (3-D)
¾ The 3-D geometric modeling has the ability to provide all the information required for
CAD-CAM-CAE applications
Basic Requirement
¾ A 3-D geometric model should be an unambiguous representation of an object.
¾ A 3-D model should be complete to all engineering function from documentation
(drafting & shading) to engineering analysis to manufacturing.
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3
Wireframe Modeling
¾ In this method the complete object is represented by number of lines, points, arcs &
curves and their connectivity relationships
Advantages
9 The construction of a wireframe model is simple
9 It does not require much computer time & memory.
9 It can be used for simple NC tool path generation
Disadvantages
¾ It can not be used for calculation of mass, inertia properties
¾ The interpretation of wireframe models having many edges is very difficult
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Surface Modeling
¾ The surface model is constructed essentially from surfaces such a s planes, rotated
curved surfaces & even very complex synthetic surfaces.
¾ Surface creation on existing CAD system usually requires wireframe entities as a
start
¾ Surface & wireframe form the core of all existing CAD system
Advantages
9 Surface model of an object is a relatively more complete & less ambiguous representation
than its wireframe model
9 This method is very much useful for specific non-analytical surfaces ( free-form surface/
sculptured surfaces) such as those used for modeling automobile & airplane bodies &
turbine blades etc.
9 From an application point of view, surface models can be utilized in Finite Element
Modeling, NC tool path generation, sectioning & interference detections.
Disadvantages
¾ The calculation of mass & inertia properties would be difficult
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Solid Modeling
¾ Solid model of an object is a more complete representation than surface
model, as all the information required for engineering analysis &
manufacturing can be obtained with this technique.
Advantages
9 Solid modeling produces accurate design,
9 provides complete 3D definition
9 Improves the quality of design
9 Improves visualization
9 has potential for functional simulation of the system
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Wireframe Modeling
Point
Lines
Analytic Entities /
Curves
Arcs
Circles
Ellipse
Conics
Wireframe entities
Parabolas
Hyperbolas
<Geometric Primitives>
Cubic spline
Synthetic Curves
Bezier curves
B-spline
NURBS (Non-Uniform Rational B-spline)
Analytic Curve
- are described by analytic equations
Synthetic Curve
- are described by a set of data points (i.e.
control points)
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Analytic Curve
¾ are defined as those that can be described by analytic equations such as lines, circle,
conics etc.
¾ provide very compact forms to represent shapes & simplify the computation of
related properties such as areas & volume.
¾ Analytic curves are usually not sufficient to meet today’s geometric design
requirements of complex mechanical parts like automobile bodies, aeroplane wings,
propeller blades, bottles etc.
That require synthetic curves & surfaces (free-form surfaces)
Synthetic Curve
¾ are defined as those that can be described by a set of data points (i.e. control points)
such as Splines, Bezier curve etc.
¾ Synthetic curves provide designers with greater flexibility & control of a curve shape
by changing the positions of the one or more data points or control points.
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Synthetic Curve
The need for synthetic curves in design arises on two occasions:
i) when a curve is represented by a collection of measured data points (in case of
Reverse Engineering) [graphical visualization of experimental data]
ii) when an existing curve must change to meet new design requirements.
Synthetic Curve Construction Techniques :
Interpolation technique
Approximation technique
Curve passes through the data points
Curve do not passes through the data points
Mathematically, synthetic curves represent a Curve-fitting problem to construct a smooth curve
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Interpolation Technique
Approximation Technique
- Curve resulting from this technique pass
through the given data points; curve itself is
called Interpolant
- Produce curves that do not pass through the
given data points. Instead, these points are
used to control the shape of the resulting
curves
- Ex. : Cubic Spline
-Ex. : Bezier Curve
P7
P1
P5
P3
P2
P9
P8
P6
P11
P1
P12
P0
P2
P4
P0
P2
P10
P0
P3
P1
P3
Most often, approximation techniques are preferred over interpolation techniques in freeform curve design due to the added flexibility & the additional intuitive feel provided by
approximate technique.
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Mathematical Representation of Curves
Explicit form
Non-parametric equation
Curve can be described mathematically by
Implicit form
Parametric equation
Non-parametric equation : Explicit form
If the co-ordinates ‘y’ & ‘z’ of a point on the curve are expressed as two separate functions of the third coordinate ‘x’ <independent variable>. This curve representation is known as Non-parametric Explicit form.
Non-parametric Explicit form of a general 3-D curve
Y
Position vector of a point ‘P’ on the curve = P = [x y z]T= [x f(x) g(x) ]T
P (x,y,z)
P
Z
X
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Non-parametric equation : Implicit form
If the co-ordinates ‘x’, ‘y’ & ‘z’ are related together by two functions, a non-parametric implicit form results
Non-parametric Implicit form of a general 3-D curve
f1 (x,y,z) = 0
f2 (x,y,z) = 0
Parametric representation of curves
Y
P (x,y,z)
umax
In parametric form, each point on a curve is expressed as a function
of a parameter “u”. This parameter acts as a local co-ordinate for
points on the curve.
P’(u)
umin
P(u)
Z
Parametric form of a general 3-D curve
X
Position vector of a point ‘P’ on the curve = P(u) = [x(u) y(u) z(u) ]T
Where, umin ≤ u ≤ umax
The parametric curve is bounded by two parametric variable values umin and umax.
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Parametric Representation of Curves
Advantages :
¾In case of commonly used curves (such as circle, conics), these
equations are polynomials rather than equations involving roots.
Hence, the parametric form is not only more general but it is also
well suited to computation in geometric modeling.
¾Parametric geometry can be easily expressed in terms of vectors &
matrices which enables the use of simple computation techniques
to solve complex analytic geometry problem.
¾To check whether a given point lies on the curve or not, reduces to
finding the corresponding ‘u’ values & checking whether that value
lies in the stated ‘u’ range.
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¾Blending is used to construct composite curve. Blending of two
curves implies the joining of two curves subjected to the
satisfaction of continuity condition.
¾Various Continuity requirements can be specified at data points
to impose various degrees of smoothness of the resulting curve.
¾The order of continuity becomes important when a complex
curve is modeled by several curve segments pieced together
end-to-end.
Order of Continuity
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Order of Continuity
Zero-order
Continuity
First-order
Continuity
Second-order
Continuity
(C0 - continuity)
(C1 - continuity)
(C2 - continuity)
Slope Continuous
curve
Curvature Continuous
curve
Ex.: Cubic Spline
Ex.: B-Spline
Position Continuous
curve
C1 – Continuous is “Smoother” than C0 – Continuous at the joining point
C2 – Continuous is “Smoother” than C1 – Continuous at the joining point
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Synthetic Curves
Cubic Splines
¾Splines are used to interpolate to given data i.e. based on
Interpolation technique.
¾A spline is a piecewise parametric representation of the geometry
of a curve with a specified order of continuity.
¾Cubic splines use a parametric equations of 3rd degree with the
first order continuity maintained at the intersection point of the
curve.
¾Name from the traditional drafting tool called “Splines” or “French
Curves”
P0.8
P1
P.0.4
P0
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Cubic Splines
¾ Cubic splines use cubic polynomial.
The parametric equation of a cubic spline segment is given by
Vector form
Matrix form
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G
P(u) = ∑ Ci u i
i=0
G
P(u) = C0 + C1u + C2 u 2 + C3u 3
G
P(u) = U T C
Where,
⎡1⎤
⎢u⎥
U = ⎢ 2⎥
⎢u ⎥
⎢u3 ⎥
⎣ ⎦
u= parameter 0 ≤ u ≤ 1
Ci= Polynomial Coefficients
⎡ C0 ⎤
⎢C ⎥
C = ⎢ 1⎥
⎢C 2 ⎥
⎢C ⎥
⎣ 3⎦
C= Coefficients
Vector
¾ Cubic polynomial has four coefficients & thus requires four conditions to evaluate .
Cubic Spline
Hermite Cubic Spline
- Uses four data points
P0.8
P1
- Uses two data points at its ends & two tangent
vectors at these points
P.0.4
P0
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Bezier Curves
¾ Based on approximation techniques.
¾ Developed by P. Bezier, Designer of French car firm Regie Renault (1962).
¾ Used in his Software system (UNISURF) to define the outer panels of several
Renault cars .
¾ Bezier curve uses the vertices of Bezier Characteristic polygon (or
control polygon) as control points for approximating the generated
curve. The curve will pass through the first & last point with all other
points acting as control points. The curve is also always tangent to
the first & last polygon segment.
Control Points
Control Polygon
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Bezier Curves
¾The degree of Bezier curve is related to the number of data points;
(n+1) points define an nth degree Bezier curve.
¾ If no. of data points is 4 [n+1=4], Then n=3 i.e degree of curve =3 (cubic)
Control Points
Control Polygon
Cubic Bezier Curve
¾ This is widely used for the design of aesthetic surfaces. The flexibility of the
curve becomes more with more control points.
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Bezier Curves
Control Points
Control Polygon
Mathematically, for (n+1) control points, the Bezier curve is defined by polynomial
of degree n:
P(u) is a point on the curve, Pi is a control point
Bi,n(u) are the Berntein polynomials
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Bezier Curves
C(n,i) is the binomial coefficient
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Bezier Curves
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Major difference between the Bezier Curve & Cubic Spline Curve
i.
The degree of Bezier curve is variable & is related to the number of data points;
(n+1) points define an nth degree curve which permits higher-order continuity.
This is not the case for cubic splines where the degree is always 3 <cubic> for a
spline segment.
ii.
The shape of Bezier curve is controlled by its data points only. First derivatives
are not used in the curve development as in the case of Hermite cubic spline.
iii.
The Bezier curve is smoother than the cubic spline because it has higher order
derivative.
¾ Bezier curve is considered as a single curve controlled by all the control points. As a
result, with an increase in the number of control points, the degree of the polynomial
representing the curve increases & this provides for global modification effect rather
than local.
¾ To reduce this complexity, the curve is broken down into more segments with better
control excised with individual segments, while maintaining a simple continuity
between the segments.
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B-Spline Curves
¾ Single piecewise parametric polynomial curve through any number of control
points with the degree of polynomial selected by Designer.
¾ It provides the ability to add control points without increasing the degree of the
curve.
¾ B-spline exhibit a local control of the curve shape i.e. whenever a single vertex is
moved, only those vertices around that will be affected while rest remains the
same.
Interpolate a given set of data points
Useful in displaying design & engineering
analysis results such as Stress/displacement
distribution
Approximate a given set of data points
Useful for modeling free-form surfaces
B-Spline Curves
¾ In contrast to Bezier curve, the theory of B-spline curve separates the degree of
the resulting curve from the no. of the given control points.
¾ Four control points can always produce a cubic Bezier curve but four control
points can produce linear, quadratic or cubic B-spline curve.
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B-Spline Curves
Mathematically, B-spline curve defined by (n+1) control points Piis given by
Where ‘k’ controls the degree (k-1) of the resulting polynomial & also continuity of
the curve.
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B-Spline Curves
Local control of B-spline curve
¾ It provides the ability to add control points without increasing the degree of the
curve.
¾ B-spline exhibit a local control of the curve shape i.e. whenever a single vertex is
moved, only those vertices around that will be affected while rest remains the
same.
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NURBS
Non-Uniform Rational B- Spline
¾ are generalization of the curve & surface theories.
¾ are almost exclusively used by modern CAD-CAM-CAE systems
to provide a unified approach to formulate & represent curves &
surfaces.
¾ provide a convenient design tool to create smooth curves &
surfaces interactively.
9 Rational Curve: is defined by algebraic ratio of two polynomials while a
non-rational curve is defined by one polynomial.
n
Rational form of Bezier Curves
G
P(u) =
∑Pw B
i =0
n
i
i
i,n
(u)
∑ w i Bi,n (u)
;
0≤u≤1
i=0
where wi is the weighting factor for each of the vertex.
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NURBS
Advantages
9 NURBS are considered a unified representation that can be define both synthetic
(like Bezier, B-spline etc.) & analytic (i.e. circle, conics etc.) curves & surfaces.
Any curve or surface can be formulated using NURBS.
9 It can represent all curves, surfaces, & solid entities, allowing unification &
conversion from one CAD system to another via exchange standards (like IGES,,
STEP etc.). Their related algorithm are stable & accurate.
9 This unified representation also have the advantage of reducing the database
complexity & the number of procedures required in CAD system for display &
manipulation of geometric entities.
Disadvantages
¾ Simple curves (like arcs, circles, conics) require more data to define as NURBS
than traditional way.
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¾ The effective use of analytic & synthetic curves in a design environment depends
mainly on their manipulation.
Displaying
Blending
Segmentation
Curve
Manipulations
Trimming
Intersection
Transformation
Translation
Rotation
Scaling
Mirror/ Reflection
Evaluating points on curves
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Curve Manipulations
Displaying
¾ provides the Designer with a means of visualizing geometric models.
Blending ¾ is used to construct composite curves from various types of curve
segments. Two curve segment should be continuous at the joint.
Segmentation ¾ replacing one existing curve by one or more curve segments of the
same curve type such that the shape of the composite curve is
identical to that of the original curve.
Trimming ¾ is a very useful function which can truncate or extend a curve
Intersection ¾ the intersection point of two curve in space requires the solution of
the parametric equations of curves.
Translation
Transformation
Rotation
Scaling
Mirror/ Reflection
Evaluating points on curves
¾ Curve parametric eqn. is used to evaluate points on it.
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Surface Modeling
Plane surface
Analytic Surface
Entities
Ruled surface/ Lofted surface
Surface of revolution
Tabulated cylindrical surface
Surface entities
Bi-cubic Hermite spline surface
Synthetic Surface
Entities
B-Spline surface
Bezier surface or patch
Coons patch
Curve segment : is the fundamental building block for curve entities
Surface patch : is the fundamental building block for surfaces
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Plane Surface
¾ is the simplest surface which requires three non-coincident
points to define a plane.
¾ The plane surface can be used to generate cross-sectional
view by intersecting a surface model with it.
Ruled Surface
¾ is a surface constructed by transitioning between two or more curves by using
linear blending between each section of the surface
¾ It interpolates linearly between two boundary curves that define the surface.
Lofted Surface
¾ is a surface constructed by transitioning between two or more
curves by a smooth i.e. higher order blending between each
section of the surface.
¾ Used for modeling engine manifolds, turbine blades etc.
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Surface of Revolution
¾ is an axi-symmetric surface that can model axisymmetric objects.
¾ is generated by a rotating a planar wireframe entity
in space about the axis of symmetry a certain
angle.
Tabulated Cylindrical Surface
¾ is a surface generated by translating a planar curve
a certain distance along a specified direction (axis of
cylinder).
¾ Plane of the curve is perpendicular to the axis of the
cylinder.
¾ is used to generate surfaces that have identical
curved cross-sections.
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Bi-cubic Hermite Spline Surface
¾ It connects four corner data points & utilizes a bi-cubic equation.
Bezier Surface
¾ is a synthetic surface that approximates given input data i.e. it doesnot pass
through all given data points.
¾ Allows only global control of the surface.
B-spline Surface
¾ is a synthetic surface that can either approximate or interpolate given input data.
¾ Permits local control of the surface.
Coons patches / surface
¾ Coons patch or surface is obtained by blending four boundary curves.
¾ is used to create surfaces using curves that form closed boundaries.
¾ the single patch can be extended in both the directions by adding further
patches.
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Displaying
Blending
Segmentation
Trimming
Surface
Manipulations
Intersection
Transformation
Projection
Evaluating points & curves on surfaces
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Solid Modeling
¾ Solid model are known to be informationally more complete, valid &
unambiguous representation of objects than its wireframe or
surface model.
¾ The completeness & unambiguity of a solid model are attributed to
the fact that CAD database stores both its geometry & topology.
¾ Solid models are complete & unambiguous but they are not unique
because same object may be constructed in various ways.
¾ Solid modeling has been acknowledged as the technological
solution to automating & integrating design & manufacturing
functions like…
Design / Engineering Analysis
Drafting
CAPP, CNC tool path generation
MRP
Computer-Aided Inspection & Quality control
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¾ A solid model of an object consists of both the geometrical & topological
data of the object.
Geometry
- Geometry of an object defines the actual dimensions of its entities.
Topology
- Topology of an object defines the connectivity & associativity of the entities.
- it determines the relational information between entities.
From user point of view,
Geometry is visible.
Topology is considered to be non-graphical relational
information that is stored in solid model databases &
are not visible to users.
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Geometry
The geometry that defines the object is the
- the length of lines L1, L2, L3
- the angle between the lines
- radius ‘R’ & the centre ‘P’ of semi-circle
Topology
The topology of the object can be stated as
- L1 shares a vertex (point) with L2 & C1
- L2 shares a vertex with L1 & L3
- L3 shares a vertex with L2 & C1
- L1 & L3 donot overlap
- ‘P’ lies outside the object
L1
C1
L2
R
P
L1
R
L2
P
C1
L3
L1
L3
R
R
L2
L1
L2
P
P
C1
C1
L3
L3
Same Topology but Different Geometry
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Same Geometry but Different Topology
Solid Modeling
Modern CAD systems offer two approaches to creating solid models :
Primitive Approach
- allows designers to use pre-defined shapes (primitives) as
building blocks to model solid object.
- Boolean operation are used to combine the primitives to
create complex objects.
- is limited by the restricted shapes of primitives
Feature Approach
- Features are more flexible as they allow the construction
of more complex solid than what the primitive offer.
Feature is defined as a geometric shape (i.e. feature profile or 2-D sketch) and feature
operation to build parts.
Feature
Feature Profile (Sketch)
+
Feature Operation
¾ The entities required for Feature profile (Sketch) are wireframe & surface entities
¾ Feature Operation is an activity that converts the Sketch (2-D) into 3-D shape
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Constraints
Geometric Constraints
Dimension Constraints
Fix
Coincidence
Concentricity
Tangency
Parallelism
Perpendicular
Horizontal
Vertical
Mid point
Equidistant point
¾ Geometric constraints is defined as a geometric condition that relates two or
more sketch entities.
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Constraint-based Feature Modeling
Select a Sketch Plane
Sketching the feature profile (2D profile)
Apply Geometric & Dimensional Constraints
Apply feature operations
Create Base Feature
Combine feature to build part
- is considered the best tool to create solids because of its ability to edit & change the
shape of the solid in the future by using its relations & constraints. It has flexibility to
create complex shape.
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Common Feature Operations & Features
Extrude
Protruded Feature
Revolve
Sweep
Loft
Feature
Operations
Rib
Pocket
Hole
Slot
Shell
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Common Feature Operations & Features
Fillet
Chamfer
Rectangular Pattern
Pattern
Feature
Operations
Circular Pattern
Translation
Transformation
Rotation
Scaling
Mirror
Thread
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Common Features
Extruded or Protruded feature
¾ Extrude command is used to create a solid object by
extruding a given closed profile.
¾ It requires a closed profile (sketch of cross-section of
solid object) & an extrusion vector (length & direction).
¾ The extrusion direction is always perpendicular to the
sketch plane of the profile.
Revolved feature
¾ Revolve command is used to create an axi-symmetric
objects by revolving the selected geometry about an axis.
¾ It requires a profile (sketch of cross-section of solid
object) & a revolution vector (axis & angle of revolution).
¾ The axis of revolution is always in the sketch plane of the
profile.
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Common Features
Sweep feature
¾ is a generalization of extrusion.
¾ Creates a feature by moving a sketch (closed or open)
along a linear or non-linear path, which is not
necessarily perpendicular to the sketch plane.
Loft feature
¾ Use to blend multiple cross-section (two or more) along
the linear or non-linear guide curve to create a solid.
Hole feature
¾ is equivalent to subtracting a cylinder from a solid.
Simple and Taper hole
Counter-bored hole
Type of Hole feature
Counter-sunk hole
Threaded hole
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Common Features
Slot feature
¾ Removes material from solid.
Rectangular slot
Type of Slot feature
T- slot
Dovetail slot
Shell feature
¾ Shell operation is used to create hollow or thin-walled
solids by removing material out.
¾ The input to a shell operation is the faces to be removed
& a wall thickness.
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Common Features
Rib feature
¾ is a special type of extruded feature.
¾ Creates an extension of geometric element by creating a
feature connecting the two faces of the part.
¾ The input to create a rib is a contour & a thickness
Pattern feature
Rectangular Pattern
Circular Pattern
¾ Used to create multiple copy of given
feature in rectangular (linear) array.
¾ Used to create multiple copy of given
feature in circular array.
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Common Features
Fillet
¾ are used to smoothen (round) the sharp edges of solid.
¾ input to create a constant radius fillet feature is to select
the edges to be filleted & fillet radius.
Chamfer
¾ use to remove sharp edges (or corners) from parts
¾ input to create a chamfer is angle-distance, distancedistance.
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Boundary representation (B-rep)
Constructive Solid Geometry (CSG)
Sweeping
Half spaces
Solid Representation Schemes
Analytic Solid Modeling (ASM)
Cell decomposition
Spatial enumeration
Octree encoding
Primitive Instancing
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Boundary representation (B-rep)
¾ B-rep is based on the topological notion that an object is bounded
by a set of faces.
¾ A face is a closed, orientable & bounded (by edges) surface. Each face is
bounded by edges & each edge (bounded curve) is bounded by vertices.
¾ B-rep model of an object consists of faces, edges, vertices.
Vertices (v)
Edge (E)
Face (F)
- The database of a boundary
model contains both its
topology & its geometry.
Topology & Geometry are
interrelated & cannot be
separated entirely from each
other
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B - rep
Advantages
9 B-rep is very appropriate to construct solid models of unusual or
complex shapes ( like turbine blades, auto body etc.) that are
difficult to build using primitives.
9 It is relatively simple to convert a B-rep model into a wireframe
model because the model’s boundary definition is similar to the
wireframe definition.
Disadvantages
¾ It requires large amounts of storage because it stores the explicit
definition of model boundaries i.e. faces, edges & vertices which
tend to grow fairly fast for complex model.
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Constructive Solid Geometry (CSG)
¾ CSG model is based on the topological notion that a physical object
can be divided into set of primitives (basic elements & shapes) that
can be combined in a certain order following a set of rules (Boolean
operation) to form the object.
ƒ
Primitives are considered as building block
ƒ
Primitives are simple, basic shapes which can be combined by a mathematical
set of Boolean operations to create the complex solid object.
Primitives
Block
Cylinder
Cone
Sphere
Wedge
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Boolean Operations
Union
(U)
Intersection
(∩)
Difference
(-)
Used to combine or
add two primitives/
objects
Intersecting two
primitives give a
shape equal to their
common value
Used to subtract one
primitive from other &
results in shape equal
to the difference in
their volume
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CSG
Advantages
9 It is easy to construct out of primitives & Boolean operation.
9 It is concise & requires minimum storage.
Disadvantages
¾ Major disadvantage of CSG is in its inability to represent sculptured
surfaces i.e. non-analytical (synthetic surface) cannot be modeled
using CSG scheme.
- Modern CAD systems provide both approaches to increase their modeling domain
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Solid Model-based Down-stream Application
Solid Model-based Assembly Design
Solid Model-based Drafting
Solid Model-based Mechanism Design
Finite Element Modeling from Solid Models
CNC machining based on Solid Modeling
…
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