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Lecture 3 1
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Wire frameSurface
Solid
Pemodelan Permukaan Pemodelan PepejalPemodelan Kerangkadawai
Lecture 3 2
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What we cover?
ype o en esTopologies and geometriesWhy we need certain number of
commands to modelParametric entities and itsdevelopment toward surface
Lecture 3 3
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Types of entityLine
Y
Polyline
Y
Y
X
Curves3D Polyline
Lecture 3 4
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Definition Topology and geometry
is the connectivity and associativity of the objectentities. Topology states that L1 shares vertex with L2
Geometris metric information which defines the entities of theobejcts. Geometry states that the coordinates of thevertices of a lines.
Geometric modeling requires both topology and geometry asits low-level model definition.
Lecture 3 5
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Importance of topology andgeometric in model definition
,has the same low-level definition
Determine the manipulation of the entities
space
Lecture 3 6
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Understanding topology andgeometry
other topology. Geometry has horizontalrelationship with topology.
Line
Topology GeometryCurve Straight line
Lecture 3 7
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Understanding topology andgeometry cont
Same topology but different geometry
Lecture 3 8
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Geometry of curves
their arrays of points or by theirmathematical re resentations.Mathematical representation is preferable
description due to its practicality forcomputational purposes.The mathematical representation can be
parametric
Lecture 3 9
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Curve representation: implicit Line
f(x,y) = 0 =
0)x(xy)y(yx)xy(x)yx(y 1211211212 =+
Circle 0r yx 222 =+
Elli se
1 by
ax
22
=
+
xayn
0i
ii
= =
Curve
Lecture 3 10
verticescontroltotal1nwithtcoefficienisawhere +
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Curve representation: parametric
Line x = x(t)y = y(t)
t)((t)t)x(xxx(t) 121121 +=+= y y y y
z = z(t)where
t)sin(2r (t)t)cos(2r x(t) == y
10 t
t)sin(2 by(t)t)cos(2ax(t) ==
ta(t)tax(t)n
0i
ii
n
0i
ii
== ==
y
urve
Lecture 3 11
verticescontroltotal1nwithtcoefficienisawhere +
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Circle: Implicit and ParametricrepresentationParametric
Parametric representation generates evenly spaced points andhence generates more smooth curves.
Lecture 3 12
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Synthetic curve representation
n erpo a on me oCubic, cubic spline (piecewise polynomial)
Approximation methodBezier, B-Spline, Non-rational & rational curve,uniform and non-uniform
Lecture 3 13
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Parametric Cubic
3i
Parametric Cubic Equation
0iia=
=
0123 atatataP(t) +++=
P(t): point on the curve
a i:algebraic coefficient
Lecture 3 14
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Parametric Cubic: cont
t = 1 P(1) = a 3 + a 2 + a 1 + a 0 eq2
e e angen a ant = 0 P(0) = a 1 eq3
t =1 P(1) = 3a 3 + 2a 2 + a 1 eq4
Set the equation a0, a1, a3 and a4 in terms of P(0), P(1),P(0) and P(1) and insert into the parametric cubic equation.
P(t) = (2t 3 3t 2 + 1)P(0) + (-2t 3 + 3t 2 )P(1) +
(t 3 2t 2 + t)P(0) + (t 3 t 2 )P(1)
Lecture 3 15
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Parametric Cubic: cont
x(t) = (2t 3 3t 2 + 1)x(0) + (-2t 3 + 3t 2 )x(1) +
(t 3 2t 2 + t)x(0) + (t 3 t 2 )x(1)
y(t) = (2t 3 3t 2 + 1)y(0) + (-2t 3 + 3t 2 )y(1) +(t 3 2t 2 + t)y(0) + (t 3 t 2 )y(1)
Used constraint: slope of the end point.
Matrix representation
]1[
]0[
1233
1122
23 P
P
]1[']0['
00010100
PP
Lecture 3 16
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Cubic Spline
Spline is introduced to replace flexiblecurve. Flexible curve enables the continuitof the curve to second derivative (C 2)
u c sp ne s a spec a or parame r c cu c(first derivative at each ends of thesegment) with ensure continuity at secondderivative. Therefore, smoother curve isgenerated.
Lecture 3 17
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Curve continuity
Lecture 3 18
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Cubic spline: cont
01
2
2
3
3 atatataP(t) +++=Cubic polynomial equation
23 a2ta6(t)P" +=Second derivative
At P i end point of segment curve i-1 when t =1
start point of segment curve I when t = 0
P i-1 (1) = P i(0)
At P i+1 end point of segment curve i when t =1
start point of segment curve i+1 when t = 0
P i(1) = P i+1 (0)
Lecture 3 19
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Cubic spline: cont
'' i1111 ++ iiiii
If second derivatives both end point of curvesegment i-1 and start point of curve segment
,spline
Lecture 3 20
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Interpolation vs approximation
It is originated for data-fitting.The curve enerated will o throu h thevertices
The curve is not necessarily passingthrough all of the vertices
Generate free-form surface.Suitable to model car body, hull etc.
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Bezier Curve
0)()( ==
N
ii BiPt P
:1+ verticestotal N
Bi is blending function
)1()!(
=i vvi N i B
Lecture 3 22
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Bezier Curve: example , , , , , ,
P(t)= P(0)(1-t) 3 + P(1)3t(1-t) 2 + P(2) 3t 2 (1-v) + P(3)v 3
Therefore
x(t)= x 0 (1-t) 3 + x 13t(1-t) 2 + x 2 3t 2(1-v) + x 3 v 3
y(t)= y 0 (1-t) 3 + y 13t(1-t) 2 + y 2 3t 2(1-v) + y 3 v 3
Lecture 3 23
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Bezier Curve: example
1,4
3,5t x y0 0 0
0.1 0.102 0.347
0.2 0.216 0.616
0.3 0.354 0.849
0.4 0.528 1.088
2,2
. . .
0.6 1.032 1.752
0.7 1.386 2.261
0.8 1.824 2.944
0,0
0.9 2.358 3.8431 3 5
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B-Spline Curve
Bezier Curve,connected end to end.
1
)()(
1
0 ,
t t t for
V t N t P
ii
i ik i
+
=
=
introduced. Thisdegree functionbasically pushes thecurve away.
0,
and
otherwiset i =
)()(
)()(
)(1,1
11,
1, t N t t
t t t N
t t t t
N k iik i
k ik i
ik i
ik i +
++
+
+
+
=
Ni,k : blending function
Lecture 3 25
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B-Spline: k = 2 and k = 3
)()(
)()(
2,02
2,00
3,0 t N t t
t t t N
t t t t
N
+=
= near 1
2
)()(
)()(
)()(
2,1
23
32,1
12
13,1 t N t t
t t t N
t t t t
N
+=
)()(
)()(
103
100
20 t N t t
t N t t
N
+=
k =30
3
)()(
)()(
)()(
1,124
41,1
13
12,1
,
23
,
02
,
t N t t
t t t N
t t t t
N
t t t t
+=
4
Lecture 3 26
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B-Spline: k = 4
k = 4 1
2
)()(
)()(
)()()(
5
4,114
44,0
03
04,0
t t t t
t N t t
t t t N
t t t t
N
+=
...
)()( 4,1254,1
144,1 t t t t
=0
3
4
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Rational Curve
space. It is referred as weight (w(t))
Parametric curve = , ,
Rational CurveP(t) = [ x(t)/w(t), y(t)/w(t), z(t)/w(t) ]
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Homogenous CoordinateCoordinate declaration
(x, y, z)
Homogenous coordinate(x*, y*, z*, h)
W
h: scalar vector
Homogenous coordinate
( x*/h, y*/h, z*/h, 1)Ph (x,y,h)
P2d (x/h,y/h,1)1
X
Lecture 3 30
Y
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Disadvantages of wire frame
u p e n erpre a on
Lecture 3 31
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Disadvantages of wire frame
Ambiguousm l
Unreal object
line)so ne
Lecture 3 32
(a) (b)