第六章視景圖形座標轉換

1

第六章視景圖形座標轉換

World → Device Transformation

World Device

Viewport

2

Windows

WCS (World Coordinate System)

NDCS (Normalized Coordinate System)

PDCS (Physical Device Coordinate System)

UCS (User Coordinate System)

WCS ： theoretically infinite

Window ： a finite region in WCS

Viewport ： defines the physical location of the display area within the physical raster

3

Normalization Transformation (N) ：

Workstation Transformation (W) ：

World Coordinate NormalizedDevice Coordinates

map

Normalized DeviceCoordinate

mapPhysical DeviceCoordinate

Viewing Transformation (V) ：V = W N‧

4

maxyw

minyw

minxw maxxw

P

Window

World Coordinate

0 1

1

Normalized DeviceCoordinate (parameter 0~1)

maxyv

minyv

minxv maxxv

WorkstationCoordinate

Viewport

100

0

0

minmin

minmin

yvywSS

xvxwSS

N yy

xx

5

Window – to – Viewport Transformation

Window Viewport

World Normalization Device

6

Mapped to viewport area

maxyw

minyw

minxw maxxw

ww yx ,

maxyv

minyv

minxv maxxv

vv yx ,

7

Point in Window ww yx , vv ,yx in a viewport

minmax

min

minmax

min

xvxv

xvxv

xwxw

xwxw

andminmax

min

minmax

min

yvyv

yvyv

ywyw

ywyw

minminminmax

minmax xvxwxwxwxw

xvxvxv

minminminmax

minmax yvywywywyw

yvyvyv

8

Window – to – Viewport Transformation

minmin

minmin

yvywywSy

xvxwxwSx

yv

xv

Scaling

y

x

S

SMaintain relative proportion( 保持比例 )

如果 yx SS Window size = Viewport size( 大小不變 )

1

1

1 不變放大縮小

Translation平移

min

min

yv

xv

9

Boundary Conflicts( 衝突 )

處理方法

1. Wraparound( 捲折 )

轉折至下方繼續

2. Scissoring( 切斷 )

在 Hardware上切斷

10

3. Saturation( 滲透 )

Boundary Conflicts （ cont. ）處理方法

以為基準向下擠

maxy

4. Clipping( 修剪 )

以為基準切斷

maxy

11

CLIPPING

1. Analytical World

2. Raster World

幾何計算畫素計算

Point Clipping

PA point

is inside the window yxP ,

Visible

12

Both True maxmin xxx

maxmin yyy

Point Clipping (cont.)

Inside the Window

maxy

miny

maxxminx

13

Clipping lines

Analytical Clipping 1P2P

Check In or Out of 2 end points

Both In

Both Out

One In / One Out

Need to Check Parametric Equation

Need to Calculate Intersection points

14

Line Clipping 修剪

Cohen – Sutherland Algorithm

將 Clip rectangle 分為 9 份將 4-bit code assign 給每一點( 四個邊 )

Each end point

一條線的 2 端點，每一端點給一個 Code

0 0 0 0 4 3 2 bit 1

從右至左，視 Window之關係位置給予 Code

15

Cohen – Sutherland Algorithm （ cont. ）

bit 1 ： left bit 2 ： right

bit 3 ： below bit 4 ： above

1001

0001

0101 0100

0000Window

1000 1010

0010

0110

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Cohen – Sutherland Algorithm （ cont. ）bit 1 ： sign bit of minxwx bit 2 ： sign bit of xxw max

bit 3 ： sign bit of minywy bit 4 ： sign bit of yyw max

1001

0001

0101

1000

0000

0100

1010

0010

01102

4

3

51

10

6

78

9

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Logical AND of both codes of end points

Line End point bit codes Logic AND Conclusion

12345

0000 00000100 01001000 00100000 00100001 0100

0000 00100 10000 00000 00000 0

Completely Visible Completely Invisible

No Conclusion Partially VisibleNo Conclusion

1. 0 0 2. 1 0 3. 0

4. 0 0

ANDANDAND

AND

結果，兩端全結果，兩端非結果，

結果，兩端皆非

Completely Visible

Completely Invisible

1 0 兩端有、 Partially Visible

No Conclusion

判斷準則

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Logical AND of both codes of end points (cont.)

ConclusionNo

VisiblePartiallyNeed to check intersection points

Line Codes of 2 end points AND Conclusion

6789

10

1001 10001000 00001001 00001001 00001001 0001

1000 0000 0000

0000 0001

Completely Invisible Partially Partially Partially

Completely Invisible

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Partially Visible or No Conclusion( 求線與 Clipping Windows 之交點 )

1. Find Intersection Points

Line 2211 ,,, yxyx

12

12

xx

yym

2. In , Out Intersection Points

2211 ,,, yxyx

(1) 11 yxxmy

xx

LI

LI

(2) 11 yxxmy

xx

RI

RI

Left

Right

Ty

By

RxLx

TL xx , TR xx ,

BL xx , BR xx ,

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(3)

BI

BI

yym

yyxx

1

1

(4)

TI

TI

yym

yyxx

1

1

Bottom

Top

3. Check Parametric Equations

Each Intersection

Point

Line4 Boundary Line

防止延伸線上的交點

(1) 121

121

,

yyyy

xxxx

yx

I

I

II

Line （線）10

21

(2) LRLI xxxx Boundary（邊界）

10

(3) BTTI yyyx 10

If (1) 、 (2) 、 (3) 皆成立 Intersection True

If 2 intersection points 2211 ,,, IIII yxyx

New Line

If 1 intersection point 2211 ,,, yxyxcheck

the point in window

( code 0000 )

for example 11, yx ( code 0000 ) and II yx ,

定義 New Line

22

Cyrus – Beck Technique

Algorithm ：Clipping a 2-D line against a rectangle or a convex 凸 polygon

PEi

Edge Ei

P1P0

Ni

Normal of Edge Ei

（邊線法向量）

0 ii PEtPN

0 ii PEtPN

0 ii PEtPN

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10PP

EEdge i

被切的線：多邊型的一邊：

Cyrus – Beck Technique (cont.)

（反時針）的：線的參數式：

NormalEEdgeN

tPPPtP

ii

010

在 Edge Ei 上任取一點 P Ei

取 3 個向量：（ 1 ） PEi to P0

（ 2 ） PEi 在 Edge Ei 上（ 3 ） PEi to P1

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根據


0 ii PEtPN 判斷

0 ii PEtPN

0 ii PEtPN

0 ii PEtPN 在外面在 Edge 上在內部

（ Counterclockwise Edge Index ）

Intersection Point ： 10PP with Edge Ei

公式推導如下頁

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0 ii PEtPN tP


代入

0010 ii PEtPPPN

0010 tPPNPEPN iii

DN

PEPNt

i

ii

0

and 01 PPD

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P0

P1

P0

P1

P0

P1

PL

PLPL

PE

PE

PE

PEPL

Line 2Line 1 Line 3

交點分類： PE （ Potentially Energy ）

PL （ Potentially Leaving ）

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0DN i


PE 090angle

0DN i PL 090angle

Choose a ( PE , PL ) pair

PE ： with tE

PL ： with tL

The intersecting line segment is defined by ( tE , tL )

28

Calculations for Parametric Line Clipping

Clip Edge Ei

Normal Ni

PEi P0-PEi

Left

Right

Bottom

Top

minxx

minxx

minxx

minxx

yx ,min

yx ,max

min, yx

max, yx

yyxx 0min0 ,

yyxx 0max0 ,

min00 , yyxx

max00 , yyxx

DN

PEPNt

i

ii

0

01

min0

xx

xx

01

max0

xx

xx

01

min0

yy

yy

01

max0

yy

yy

0,1

0,1

1,0

1,0

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Calculations for Parametric Line Clipping

maxy

miny

minx maxx

11, yx 00 , yx

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Clipping Polygons

2 Polygons Clipping Each Line

before after

31

Sutherland – Hodgman Algorithm

Polygon ： a set of ordered vertices

Compare a polygon to each boundary

Output is a set of vertices defining polygon

依序比較 2 polygons 利用切斷各邊

Original Clip Left Clip Right Clip Bottom Clip Top

32

Polygon – Polygon Clipping

Sutherland – Hodgman Algorithm

Polygon nvvvG ,, 21 clockwise

Traverse vertices clockwise

S ：前一點 P ：目前點

S

P P SI

S

P S

IP

IN→INSave P

IN→OUTSave I

OUT→OUTNo Save

OUT→INSave I , P

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Saved Vertices New Polygon

2

6

1

3

4

5

1‘

5‘

2‘

4‘3‘

1‘

5‘

2‘

4‘3‘

34

View Volume

In 3 – D viewing ：Projection( 投影 ) Window is used to define a View Volume.

View plane is z = 0 plane

Window is defined by maxmaxminmin ,,, ywxwywxw

on the view plane

Only those objects within the view volume areprojected and displayed on the view plane.

只有在 view volume 內的物體才會被投影及顯示

35

View Volume (cont.)

View volome

WindowWindow

View volome

Parallel Projection Perspective Projection

Truncated Pyramid Frustum截斷角錐幾何體

36

View Volume (cont.)

y

x

z

No display on view plane

y

x

zNo display on view plane

37

Parallel Projection

Far Plane

Window

View VolumeNear Plane

dndf

near ： hither此處

far ： yon plane彼處 near ： front far ： back

38

Perspective Projection

dn

df

View Volume

Near Plane

Window

Center ofProjectio

n

39

3 - D

Viewport is a regular parallelepiped( 平行六面體 ) defined innormalized coordinates

3 - D Window - to – viewport mapping

1

000

000

000

zyx

z

y

x

KKK

D

D

D比例

位移

40

3 - D Window - to - viewport mapping (cont.)

dndf

minmax ywyw

minmax xwxw

minmax zvzv

minmax ywyw

minmax xwxw

View Volume 3 - D viewport

y

xz

41

3 - D Window - to - viewport mapping (cont.)

minmax

minmax

xwxw

xvxvDx

minmax

minmax

ywyw

yvyvDy

max min

max minz

zv zvD

zw zw

View Volume ： defined by maxmaxminmin ,,, ywxwywxw

and dndf ,

3 - D viewport ： defined by maxmaxminmin ,,, vwxvyvxv

and maxmin , zvzv

xx DxwxvK minmin yy DywyvK minmin zz DdnzvK min

42

Clipping against a normalized View Volume

Extension of

orithmABeckCyrus

orithmASutherlandCohen

lg

lg

Code of 6 bits right to left( 六個面 )

000000 back front above below right left

Bit6

Bit1

後前上下右左

43

Code of 6 bits right to left (cont.)

bit 1bit 2

bit 3

bit 4

bit 5

bit 6

leftright

below

above

front

back

minxvx

maxxvx

minyvy

maxyvy

minzvz

maxzvz

CodeCode

Logical AND of 2 codes

2 end points of a line

44


判斷結果AND = 0 ，兩端皆 0 Inside

AND = 1 ，兩端有 1 Outside

AND = 0 ，兩端有 1 , 0 Partially Inside

AND = 0 ，兩端都有 1 No Conclusion

Check Intersection Points 6 Intersection Points

45


Line

1111 ,, zyxP 2222 ,, zyxP

uzzzz

uyyyy

uxxxx

121

121

121

其中 1~0u

1. Intersection of a line with a ( face plane ) of view volume

2. with front minzvz

12

1min

zz

zzvu

46

2. with front

(1) If u not in (0 ~ 1)

Line segment does not intersect

(2) If uin (0 ~ 1)

min

121

12

1min121

zvz

uyyyy

zz

zzvxxxx

I

I

I

(3) If neither Ix Iynor is in

beyond boundary( 超出邊界 )

47

y

z

3. Check Intersection with 6 ( faces ) planes

P2

P1

P2

P1

B

A

maxzvminzv

minyv

maxyv

Side View

48

Cyrus - Beck Algorithm 用在 3 - D

Documents

第六章 視景圖形座標轉換

第六章視景圖形座標轉換