lecture 4a - universiti teknologi malaysiajamalt/sme4513/lec4a wireframe.pdfwireframe and curve...
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Lecture 4aLecture 4a
Wireframe and CurveWireframe and Curve
Lecture 3 1
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Type of modelingType of modeling
Wire frameSatah keratan
SurfaceSolid
Satah keratan
Pemodelan Permukaan Pemodelan PepejalPemodelan Kerangkadawai
Lecture 3 2
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What we cover?
T f titiType of entitiesTopologies and geometriesWhy we need certain number of commands to modelParametric entities and its development toward surface
Lecture 3 3
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Types of entityCircleLine
Y
Polyline
Y
Circle
Y
X XX
X X
Curves3D Polyline
Lecture 3 4
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Definition Topology and geometry
TopologyTopologyis the connectivity and associativity of the object entities. Topology states that L1 shares vertex with L2 when L1 and L2 is two connected lineswhen L1 and L2 is two connected lines
Geometryyis metric information which defines the entities of the obejcts. Geometry states that the coordinates of the vertices of a lines.
Geometric modeling requires both topology and geometry as its low-level model definition.
Lecture 3 5
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Importance of topology and geometric in model definition
Make each model unique no two models Make each model unique, no two models has the same low-level definitionDetermine the method to build the entitiesDetermine the method to build the entitiesDetermine the manipulation of the entitiesDetermine the position of the entities in Determine the position of the entities in space
Lecture 3 6
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Understanding topology and geometry
Topology has vertical relationship with Topology has vertical relationship with other topology. Geometry has horizontal relationship with topology.
Line
Topology Geometry
Curve Straight line
Vertex CoordinatesVertex Coordinates
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Understanding topology and geometry – con’tSame geometry but different topologySame geometry but different topology
Same topology but different geometry
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Geometry of curves
The geometry of the curves is defined by The geometry of the curves is defined by their arrays of points or by their mathematical representations.pMathematical representation is preferable description due to its practicality for computational purposes.The mathematical representation can be divided into two types; implicit and divided into two types; implicit and parametric
Lecture 3 9
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Curve representation: implicitImplicit LineImplicit f(x,y) = 0
f(x,y,z) = 0
0)x(xy)y(yx)xy(x)yx(y 1211211212 =−+−−−−−
Circle
Line
f(x,y,z) 00ryx 222 =−+
Ellipse
Circle
1by
ax 22
=⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛
p
xayn
0i
ii=∑
=
Curve
Lecture 3 10
verticescontroltotal1nwithtcoefficienisawhere +
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Curve representation: parametric
Parametric LineParametric x = x(t)y = y(t)
t)((t)t)x(xxx(t) 121121 −+=−+= yyyy
Circle
Line
z = z(t)where
t)πsin(2r(t)t)πcos(2rx(t) == y
Ellipse
Circle
10 ≤≤ t
t)πsin(2by(t)t)πcos(2ax(t) ==
C
Ellipse
illi hffi iih
ta(t)tax(t)n
0i
ii
n
0i
ii == ∑∑
==
y
Curve
Lecture 3 11
verticescontroltotal1nwithtcoefficienisawhere +
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Circle: Implicit and Parametric representation
ParametricImplicitImplicit
Parametric representation generates evenly spaced points and hence generates more smooth curves.
Lecture 3 12
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Synthetic curve representation
I t l ti th dInterpolation methodCubic, 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 1t0taP(t) ≤≤∑
Parametric Cubic Equation
23
0ii 1t0taP(t) ≤≤=∑
=
012
23
3 atatataP(t) +++=
P(t): point on the curveai:algebraic coefficient
Lecture 3 14
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Parametric Cubic: con’tt = 0 P(0) = a0 eq1t = 0 P(0) = a0 eq1
t = 1 P(1) = a3 + a2 + a1 + a0 eq2
Set the t ngent t P(0) nd P(1)Set the tangent at P(0) and P(1)t = 0 P’(0) = a1 eq3
t =1 P’(1) = 3a3 + 2a2 + a1 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) = (2t3 – 3t2 + 1)P(0) + (-2t3 + 3t2)P(1) +
(t3 – 2t2 + t)P’(0) + (t3 – t2)P’(1)
Lecture 3 15
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Parametric Cubic: con’t
x(t) = (2t3 – 3t2 + 1)x(0) + (-2t3 + 3t2)x(1) + (t3 – 2t2 + t)x’(0) + (t3 – t2)x’(1)
y(t) = (2t3 – 3t2 + 1)y(0) + (-2t3 + 3t2)y(1) + (t3 – 2t2 + t)y’(0) + (t3 – t2)y’(1)
Used constraint: slope of the end point.
Matrix representation
[ ] ⎥⎥⎤
⎢⎢⎡
⎥⎥⎤
⎢⎢⎡
−−−−
=]1[]0[
12331122
1)( 23 PP
ttttP [ ]⎥⎥⎥
⎦⎢⎢⎢
⎣⎥⎥⎥
⎦⎢⎢⎢
⎣
=
]1[']0['
00010100
1)(
PP
ttttP
Lecture 3 16
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Cubic SplineWhat is spline? What is spline?
Spline is introduced to replace flexible curve. Flexible curve enables the continuity yof the curve to second derivative (C2)
C bi li i i l f t i bi Cubic spline is a special for parametric cubic (first derivative at each ends of the segment) with ensure continuity at second derivative. Therefore, smoother curve is generated.
Lecture 3 17
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Curve continuity
Lecture 3 18
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Cubic spline: con’t
012
23
3 atatataP(t) +++=Cubic polynomial equation
23 a2ta6(t)P" +=Second derivative
At Pi 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 Pi+1 end point of segment curve i when t =1 i+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: con’tCubic spline equationCubic spline equation
)(34 '' −=++ i PPPPP )(34 1111 −++− −=++ iiiii PPPPP
If second derivatives both end point of curve segment i-1 and start point of curve segment I is equal to 0 the curve is natural cubic I is equal to 0, the curve is natural cubic spline
Lecture 3 20
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Interpolation vs approximationInterpolationInterpolation
It is originated for data-fitting. The curve generated will go through the g g gvertices
ApproximationApproximationThe curve is not necessarily passing through all of the verticesGenerate free-form surface.Suitable to model car body, hull etc.
Lecture 3 21
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Bezier Curve
Basic EquationBasic Equation
0
10
)()(=
≤≤
=∑N
ii
t
BiPtP
:110
+≤≤
verticestotalNt
Bi is blending function1! NiN 1)1(
)!(! −−−
= Nii vv
iNiNB
Lecture 3 22
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Bezier Curve: example4 vertices: (0 0) (1 4) (2 2) and (3 5)4 vertices: (0,0), (1,4), (2,2) and (3,5)
Based from basic eqBased from basic eqP(t)= P(0)(1-t)3 + P(1)3t(1-t)2 + P(2) 3t2(1-v) + P(3)v3
Therefore
x(t)= x0 (1-t)3 + x13t(1-t)2 + x2 3t2(1-v) + x3 v3
y(t)= y0 (1-t)3 + y13t(1-t)2 + y2 3t2(1-v) + y3 v3
Lecture 3 23
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Bezier Curve: example
1,4
3,5t x y
0 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
0 5 0 75 1 375
2,2
0.5 0.75 1.375
0.6 1.032 1.752
0.7 1.386 2.261
0.8 1.824 2.944
0,0
0.9 2.358 3.843
1 3 5
Lecture 3 24
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B-Spline CurvePiecewise collection of nPiecewise collection of Bezier Curve, connected end to end.
Degree function k is 1
)()(
1
0,
tttforN
VtNtP
ii
n
iiki
+
=
⎨⎧ ≤≤
=∑
Degree function k is introduced. This degree function basically pushes the curve away.
)(
0,
andotherwise
N ti⎩⎨=
)()()(
)()()(
1,11
1,1
, tNtttt
tNtt
ttN kiiki
kiki
iki
iki −+
++
+−
−+ −
−+
−−
=
Ni,k : blending function
Lecture 3 25
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B-Spline: k = 2 and k = 3k 2 li
)()()(
)()()(
2,012
22,0
01
03,0 tN
tttt
tNttttN
−
−+
−−
=
k =2 linear 1
2
)()()(
)()()(
)()(
2,123
32,1
12
13,1
1201
tNtttt
tNttttN
tttt
−
−+
−−
=
)()()(
)()()(
103
100
20 tNtt
tNttN−
+−
=
k =3 0
3
)()()(
)()()(
)()(
)()(
1,124
41,1
13
12,1
1,023
1,002
2,0
tNtttt
tNttttN
tttt
−
−+
−−
=
−−
4
Lecture 3 26
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B-Spline: k = 4
)()( tt
k = 4 1
2
)()(
)()(
)()()(
)()()(
51
4,114
44,0
03
04,0
tNtt
tNttN
tNtttt
tNttttN
−+
−
−
−+
−−
=
...
)()(
)()()(
4,125
4,114
14,1 tN
tttN
ttN
−+
−=
03
4
Lecture 3 27
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B-Spline
{0,1.5,2.5,3}
{0,1.8,2.8,3}t varies
k=2
k=3
k=4
k varies
Lecture 3 28
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Rational Curve
Introduce the homogeneous coordinate and Introduce the homogeneous coordinate and space. It is referred as weight (w(t))
Parametric curveP(t) = [ x(t), y(t), z(t)]P(t) [ x(t), y(t), z(t)]
Rational CurveP(t) = [ x(t)/w(t), y(t)/w(t), z(t)/w(t) ]
Lecture 3 29
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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
M lti l i t t tiMultiple interpretation
Lecture 3 31
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Disadvantages of wire frame
Ambiguous modelmodel
Unreal object (silhouette I li(silhouette line)
Isoline
Lecture 3 32(a) (b)