Motivation:

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Operators in CAD Systems
Motivation (a) Which operators are available
helps us to plan how to complete the design
of a part. (b) Understand the mathematical
basis, to help us to decide when to use it when
not to use it, understand, if it fails,
why decide how to avoid failures.
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1. Transformations
Use (a) Building any 3D model (b) Viewing any 3D
model Types of affine transformations of
interest object transformations coordinate
transformations
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Mathematical Tool matrices
Shape/size of an entities is invariant with
respect to translation and rotation
gt We need only study translation, rotation of a
point (vertex)
x y z
Vertex Point, column vector x, y, zT
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Translation
Coordinate transformation
Object transformation
Move r x, y, zT by t tx, ty, tzT r
r t,
x P in fixed frame oxyz x P in frame
oxyz (translated w.r.t oxyz by t) r x
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Rotation
Rotations in the XY plane Rotation of P by angle
q about the oz matrix A Rot(z,q)
Object transformation Rot(z, q) Coordinate
transformation by Rot( z, - q )
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Matrix formulas
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2.1. Scaling
Use to shrink / expand size of a part
Uniform scaling Multiply each coordinate in
BREP by the scaling factor
Operation Map vertex, v( x, y, z) -gt v(
sx, sy, sz)
Property Uniform scaling does not change the
topology of the part
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2.2. Non-uniform scaling
Non-Uniform scaling scaling each coordinate,
v( x, y, z) by different scaling factors,
(sx, sy, sz)
Transformed coordinates v(sxs, syy, szz).
Question Does the topology of the part change ?
Applications of Non-uniform scaling (a) Mold
design from part (b) Clothing and Footwear
design,
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Composition of Transformations
Point in E3 as x y z 1T 4x4 matrices for
transformations
Translation
Rot( X, q)
Scaling
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Rotation about an arbitrary axis
u u1 u2 u3 unit vector along given
axis Rotation by angle q about u
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Arbitrary rotation of Coordinate frame
r x y z 1T a point in OXYZ Oxyz A new
coordinate frame What are the coordinates of the
point in Oxyz?
(u1, u2, u3), (v1, v2, v3), (w1, w2, w3) DCs of
u, v, w in OXYZ
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Concatenation of Transformations
Successive transformations (i) Translate -5 0
5 (ii) Rotate(Y, 45) (iii) Translate 5v2 0
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Summary
1. affine transformations preserve
collinearity 2. Affine transformations useful in
CAD 3. Matrices are useful to compute
transformations 4. Translations, Rotations,
Scaling 5. 4x4 matrices can represent all three
of these 6. A series of transformation ?
resultant transformation ? multiply
corresponding matrix in reverse order
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Operators in CAD Boolean operators
Boolean operations U, n, - inputs regular
2-manifold solid(s) outputs regular 2-manifold
solid(s)
Problem
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Operators in CAD Sweeping
Linear and Non-linear sweep
Use solid shapes from 2D sketches
INPUTS Profile (2D sketch, one or more
loops) Sweep path (continuous, bounded curve)
OUTPUT regular 2-manifold solid
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Operators in CAD Sweeping..
Geometric Problem all vertices, edges, faces of
output easy to generate (i) fix sketch in
frame OXYZ (ii) move OXYZ along sweep-path
rotation determined by tangency, torsion
A sweep outer loop B sweep inner loop(s) Out
A - B
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Operators in CAD Sweeping...
Other names Extrude, Extrude-cut
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Operators in CAD Topological problems in Sweeping
(i) sweep path not smooth (C0, C1) (ii)
self-intersection of the swept shape
Example 1. How to generate shape?
OR
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Operators in CAD Topological problems in
Sweeping..
Example 2. How to generate shape?
Profile maintains orientation w.r.t. global frame
Profile maintains orientation w.r.t. path
FAILS!
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Operators in CAD Chamfer, Fillet
Special cases of Blending
INPUTS Surfaces S1 and S2 that share an edge, E
OUTPUT Blending surface, B, between S1 and S2 E
will vanish
Interfaces between (B, S1), an (B, S2) are
continuous
chamfer C0
fillet, round C1
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Operators in CAD Chamfer
Typical Uses angled recess at the end of a
hole angled edge to of shaft for ease of assembly
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Operators in CAD Chamfer computations
(a) For each edge, create chamfer cross-section
a triangle (b) sweep the section of each
edge along the corresponding edge (c) fill
gaps at vertices (filler shapes) (d) Boolean
(Part - solid shape) for each solid shape and
filler shape
PROBLEM what if neighboring faces are curved?
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Operators in CAD Chamfer
Geometry, Topological Problems
step (a) cross-section step (c) filling
gaps step (d) poor/changed topology
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Operators in CAD Fillet, Round
Same as Chamfer, except Geometric section used
is Circular arc
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Operators in CAD Blending
General case of Chamfers, Fillets and Rounds
USES Smooth the edge shared between two
(spline) surfaces Merge two surfaces that are
close, but not touching
INPUTS Two surfaces, and shared/blending
edges Blend radius
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Operators in CAD Blending..
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Operators in CAD Blending...
Problems 1. If edge is not C1, then sweep of
blend arc is not C0 2. How to store the surface
equation ?
Blend computation Rolling Ball Method
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Operators in CAD Tapers and drafts...
Main Use manufacturing related features
casting or molding
INPUTS (a) face / faces to be drafted (b) a
neutral plane, that intersects the faces being
drafted cross-section of the drafting faces on
the neutral face unchanged (c) the draft angle
OUTPUT (a) Modified part geometry
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Operators in CAD Tapers and drafts...
Examples
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Operators in CAD Tapers and drafts...
Examples (4th example is special!)
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Operators in CAD Tapers and drafts...
Example 4 INPUTS (a) Face(s) to be drafted (b)
Draft line (c) Draft angle (d) Neutral plane
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Operators in CAD Tapers and drafts...
Draft method (i) Intersect the neutral plane
with the draft faces, to get a series of
edges (or a loop) that must stay fixed (ii)
Each point, p on the fixed loop belongs to a
Draft face, F. (iii) For each point p, form the
drafted edge (a) P plane through p, with
normal NF X NN (b) Intersect P with F to get
drafting edge, ep. (c) Rotate ep by the drafting
angle in plane P, to get the drafted edge
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Operators in CAD Tapers and drafts...
This Draft method works for (some) non-planar
draft-lines
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Operators in CAD Tapers and drafts...
Draft failures
(a) draft feature results in non-manifold geometry
draft angle 44?
draft angle 45?
X
X
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Operators in CAD Tapers and drafts...
Draft failures
(b) draft face intersection with neighboring face
ill-defined
Requires intersection after draft generation
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Operators in CAD Tapers and drafts...
Draft failures
(c) draft face intersection with neighboring face
ill-defined
NON-PLANAR
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Operators in CAD Surface operations Offsets
Inputs Surface S, offset distance, r, offset
direction Computation for each point p on
the base surface Normal Np Offset point of p
p r Np
Planar surface
offset
offset
surface
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Operators in CAD Surface offsets, cylindrical
surface
q a xÛ (q - p).Û 0 (a xÛ p).Û 0 x
(p a).Û Offset point po p r (p - q)/(p
q)
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Operators in CAD Surface operations
Surface offsets BSpline surface
Bspline
Surface equation p p(u, v)
Tangent vectors
Normal vector
For each point, p offset point po p rN
NOT Bspline
Offset surface equation p(u, v) r N(u, v)
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Operators in CAD Surface operations
Find a grid of points on the surface, For each
point of grid, offset point po p rN Fit a
new BSpline surface through grid po
BSpline offset is UNSTABLE !
poor geometry
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Operators in CAD Surface operations
Surface offsets
Inputs Surface S, offset distance, r, offset
direction Computation for each point p on
the base surface Normal Np Offset point of p
p r Np
Planar surface
offset
offset
surface
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Operators in CAD Offset -- interpretations
non-intersecting faces extended
intersecting faces trimmed
non-intersecting faces extended
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Operators in CAD Trimming
Trimming operation is based on intersection

Trim R by B
B
R

Trim B by R
R must be EXTENDED
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Operators in CAD Trimming
BSpline surface trimming



Curve extension
Intersection curve
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Operators in CAD Shelling
Convert a solid to a shell

Examples

dont shell left, front faces
shell all
dont shell top, left faces
dont shell top face

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Operators in CAD Shelling
Shell computation For each face f, compute
offset face, fo Intersect each pair of offset
faces Re-compute BREP of solid topologically
difficult



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Operators in CAD Lofting/ Skinning
Use a series of (2D) guide profiles Put a
skin surface that interpolates each guide
profile


Example

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Operators in CAD Lofting/ Skinning
Lofting computation Complex, involving BSpline
surfaces

self intersection fails
Guide curves possible improvement

Instability
skinned surface not smooth

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Issues in using CAD for PDM/PLM
- Reuse same model (saves time/money) ?
Parametric designs ? Design tables - Automated
design verifications ? Design rules, triggers
(constraints) ? Knowledge-based designs (using
APIs)



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Conclusions
Understanding of CAD operations ? Using the
correct operation to make a given shape Ability
to anticipate analyze why/when an opn may fail