lec4a wireframe

8/13/2019 Lec4a Wireframe

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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)

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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.

Lecture 3 21


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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

Lecture 3 24


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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

Lecture 3 27


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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) ]

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

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)

lec4a wireframe

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