Lec5 Editing and Transformation

8/13/2019 Lec5 Editing and Transformation

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Editing and Transformation


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Modeling

combination of entities that haven i .

2D: circle, arc, line, ellipse

3D: primitive bodies, extrusion and revolved of a profile

Beforeediting

Afterediting

Beforeediting

Afterediting

(a)


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Editing: surface and solid

Surface 1Surface 2 Surface 1

Surface 2

Part to be edited

Editing on surfaces

Before After

A BB - AA BCutting of

Solid modell


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Differences: 2D and 3D editing(X cos - Y sin , x sin + Y cos)A

(X,Y)A

Axis

Plane

Center ofrotation

(0,0) 2D: axis3D: lane

2D: point3D: axis Z

YPlane XY

Plane YZ

Center of

rotation

Rotation axis

X

ane


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Editing

Fillet TranslationChamfer Rotation

Trim Copy

Scale Array

3D: Boolean Operation

Some editing preserves it original shape

suc eng an ang e an some on . 2D editing is similar to 3D editing.

Understandin CS is ver im ortant to findthe relationship.


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Example editing

Circle SplineLine

A

Y

Surface 3D ine

Y

Z XA

Array 2D and 3D

Paksi Z

Z

X

Y

Paksi X Paksi Y


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Geometric Transformation

a cu a es e new coor na es w entransformation is applied to the

.model can be wireframe, surface and

.

Editing commands are in fact

. , ,move.


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Editing and transformation:relationship

OBA

(XA+X, YA+Y)

Line is extended from OA 2x

x A

Scaling points ABCD on point O

C D

(XB+X, YB+Y)

(XA, YA)

X

(XC, Y

C)

C , C

Y

Y

(a)

O

(b)

O

A

A B

(XB, Y

B)

X

(c)

DC

CD

O


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Model Representation

geometrical properties of the entities must berepresented as matrices in its homogeneouscoordinates in 2D or 3D.

1

1

22

11

yx

yxRectangle

0,0 10,0 10,10 0,10

1010

100

1nn yx

1100

1

1

222

111

zyx

zyx

10010

1000Cube

(0,0, 0) (10,0,0)

1nnn zyx

1101010

(10,10, 10)


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Convert to their homogeneousrepresentation

30 30,

(0,1,1)

1 1 1

(10,20) (20,20)

(A)

Y

(0,10,0)

(0,0,1)

0 1 0XZ

(10,0,0)(0,0,10)

(1,1,0)

(0,0,0)

ZY

(C)(1,0,0)


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Applying transformation matrices

''

[ ]Dyx

yx

yx2

22

11

'' 1122

11

=

nn yxyx

nn

''11

zyxzyx 111'''

'''11

111

[ ]D

x

zyx

x

zyx3

222

'''11

222

=

nnnnnn


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Translation

[ ]

= 010

001

TR

2D Translation

yx

0001

Component-wise addition ofvectors

[ ]

=

1

0100

0010

zx

TR

x = x + dxy = y + dyz = z + dz

To move polygons: translate

vertices (vectors) and redrawlines between themPreserves lengths and anglesDoes not require referencepoint


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Rotation

[ ]

=

100

0cossin

s ncos

RRotation 2D

Rotation of vectors through anangle

x=x cos y sin =

(X,Y)A

(X cos - Y sin , x sin + Y cos)A

Preserves lengths and anglesPoint of rotation point (0,0)

Center of

rotation

(0,0)


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Rotation at other point than originpoint

e say a e cen er o ro a on s x , y .

Translation has to be carried before and after rotation

Procedure1. Convert the (x1, y1) to (0,0) by applying translation

x = -x1 and y= -y1.

2. Apply the rotation matrix. , ,applying translation x = x1 and y = y1.

11''

xx

[ ]

=

010

00101000111 22

''

22

11

yxyxR

11

''yxyx nnnn


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3D Rotation 0001 0sin0cos

[ ]

=

1000

0cossin0

0sincos0

XR [ ]

= 0cos0sin

0010

YR

00sincos

Axis of rotation: x Axis of rotation: y

[ ]

=

1000

0100

00cossin ZR

Axis of rotation: z

If axis of rotation other than specified axis: apply translationbefore and after 3D rotation is applied


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Scaling 2D and 3D

[ ]

=000000

Z

y

x

SS

S[ ]

=100

00 y

y

R S

Line is extended from OA 2x

Scaling points ABCD on point

O

OBA

C D

Component-wise scalarmulti lication of vectors

(a)

OA

(b)

O

x AA

x = Sxxy = Syy

Does not preserve lengthsDoes not reserve an les exce t

(c)

A

D

B

CC

D

O

when scaling is uniform)Point of reference for scaling is(0,0) 2D and (0,0,0) 3D

Scaling at other than (0,0) or (0,0,0), apply translation beforeand after applying scaling transformation


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Foundation of transformations

other transformation such as shear,

Important, if the transformation

re uires oint 2D axis 2D & 3Dand plane (3D) at the origin CS.Translation must be applied before

r y r vtransformation.


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Applying transformation compositematrices

1. Draw rectangle any size

2. Translate dx = 6

3. Rotate it by 450

1. The same rectangle2. Rotate it 450

3. Translate dx = 6

Q: Do both rectangles overlapping each other? Why?


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Y

ompos e rans orma ons

2

3

4

5

X0

1

1

2 3 4 5 6 7 8 9 10

TranslationRotation

Y

4

5

6

X0

1

1

2

2

3 4 5 6 7 8 9 10

RotationTranslation


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Composite Transformation

[ ] [ ][ ][ ] [ ]nC = 321 ...

[ ] [ ][ ]CXX ='

Composite Transformation:

om na on o mu p e rans orma onsmust be in sequence.


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Mapping

= 0

0

322212

312111

aaa

aaa

=

02111 aa

1

0332313

zyx

aaaMP

1yx

YY

X

P

Mapping combines the rotation andtranslation.

Y

X

(X1,Y1)

X

mapp ng:a11 to a22: TR

3D mapping - , - -


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Mapping: example

Write the composite transformation for mirroring square ABCD

using axis ( 300 slope line through (0,2))

Axis

CD

(0,2)

A B


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Solution 1Y

Y

Y

X

CD(0,0)

Y

X

CD(0,0)

1. Translation2. Rotation3. Mirror

X

A B

X

A B

(a) Translation (b) RotationB

. o a on5. Translation

A

B

C

A

B

CA

D

C

Y

XC

(0,0)

D

D

Y

X

C(0,0)

D

D

A

D

B

C

Y

A B

(d) Rotation

A B

(c) Mirror

X(0,0)

(e) Translation


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Solution 1 Matrices

=

0)30sin()30cos(001 00

2211 TRRTR

[ ] [ ]

=

=

120

0)30cos()30sin(,

120

010 0011 RTR

[ ] [ ]

=

= ,

120

0)30cos()30sin(

0)30sin()30cos(

,

100

010

00100

2RMR

[ ]

= 010

001

2TR

120


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Solution 2

B

. app ng

2. Mirror 3. Mapping 2

A

B

C

A

D

C

D

D

Y

X

CD

Y

X

20

A B

CD

YY

A B

Y

(0,0)X

XA B

Y

X

X

(0,0)

X(0,0)

(c) Mapping 2

(a) Mapping 1 (b) Mirror


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Solution 2 Matrices

=

0010)30sin()30cos( 00

21 MPMRMP

[ ] [ ]

=

=

100

010,

1)60sin(2)60cos(2

0)30cos()30sin(00

00

1 MRMP

[ ]

=

160sin260cos2

0)30cos()30sin(

0)30sin()30cos(

00

00

2MP


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Discuss the following problem

, , , , ,

D(10,20)

without mathematically solving them to

establish the following transformation. X axis ( Y = 0)

Y = -10

-

Y = -X -5


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Application of transformation

Modeling Head

composition decomposition

Shaft

Point

Aids realism in computer graphic: projection

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Lec5 Editing and Transformation