1. World → Device Transformation
第六章 視景圖形座標轉換
World → Device Transformation
Viewport
World
Device

2. Windows WCS (World Coordinate System)
PDCS (Physical Device Coordinate System)
NDCS (Normalized 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)：
World Coordinate
Normalized
Device Coordinates
map
Workstation Transformation (W)：
Normalized Device
Coordinate
Physical Device
Coordinate
map
Viewing Transformation (V)：
V = W‧N

4. Coordinate (parameter 0~1)
Viewport
Window
Normalized Device
Coordinate (parameter 0~1)
Workstation
Coordinate
World Coordinate

5. Window – to – Viewport Transformation
World
Normalization
Device

6. Mapped to viewport area

7. Point in Window
in a viewport
and

8. Window – to – Viewport Transformation
Scaling
Maintain relative proportion
( 保持比例 )
Window size = Viewport size 如果
( 大小不變 ) 不變 放大 縮小
Translation 平移

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. Boundary Conflicts（cont.）
處理方法
3. Saturation(滲透) 以
為基準
向下擠
4. Clipping(修剪) 以
為基準 切斷

11. CLIPPING Point Clipping
幾何計算
1. Analytical World
畫素計算
2. Raster World
Point Clipping
A point
is inside the window
Visible

12. Point Clipping (cont.)
Both True
Inside the Window

13. Clipping lines Analytical Clipping 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. Cohen – Sutherland Algorithm
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
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
1000
1010
0001
0000
Window
0010
0101
0100
0110

16. Cohen – Sutherland Algorithm （cont.）
bit 1：sign bit of
bit 2：sign bit of
bit 3：sign bit of
bit 4：sign bit of 6 8
1001
1000 3
1010 9 7 10 1
0000
0001
0010 5 4 2
0101
0110
0100

17. Logical AND of both codes of end points
Line
End point bit codes
Logic AND
Conclusion 1 2 3 4 5
Completely Visible
Completely Invisible
No Conclusion
Partially Visible
Completely Visible
Completely Invisible 判 斷 準 則
Partially Visible
No Conclusion

18. Logical AND of both codes of end points (cont.)
Need to check intersection points
Line
Codes of 2 end points
AND
Conclusion
Completely Invisible
Partially
1000
0000
0001 6 7 8 9 10

19. Partially Visible or No Conclusion (求線與Clipping Windows之交點)
1. Find Intersection Points
Line
In , Out
Intersection Points
(1)
Left
(2)
Right

20. 3. Check Parametric Equations
(3)
Bottom
(4)
Top
防止延伸線上的交點
3. Check Parametric Equations
Line
Each
Intersection
Point
4 Boundary Line
(1)
Line（線）

21. If 2 intersection points
(2)
Boundary
（邊界）
(3)
If (1)、(2)、(3)皆成立
Intersection True
If 2 intersection points
New Line
If 1 intersection point
check
( code 0000 )
the point in window
for example
( code 0000 )
and
定義 New Line

22. Cyrus – Beck Technique Algorithm： Clipping a 2-D line against a
rectangle or a convex凸 polygon
Edge Ei
PEi P1 P0 Ni
Normal of Edge Ei
（邊線法向量）

23. Cyrus – Beck Technique (cont.)
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

24. Cyrus – Beck Technique (cont.)根據 判斷
在外面
在Edge上
在內部
（Counterclockwise Edge Index）
Intersection Point：
with Edge Ei
公式推導如下頁

25. Cyrus – Beck Technique (cont.)代入
and

26. Cyrus – Beck Technique (cont.)
Line 1 P1
Line 3 PE P1
Line 2 PL P1 PL PL P0 PE P0 PL PE PE P0
交點分類：PE（Potentially Energy）
PL（Potentially Leaving）

27. Cyrus – Beck Technique (cont.)PE PL
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

29. Calculations for Parametric Line Clipping

30. 2 Polygons Clipping Each Line
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
clockwise
Traverse vertices clockwise
S：前一點
P：目前點 S P P S I I P P S S
IN→IN
Save P
IN→OUT
Save I
OUT→OUT
No Save
OUT→IN
Save I , P

33. Saved Vertices
New Polygon 2 3 1‘ 2‘ 1‘ 2‘ 4 1 3‘ 3‘ 5‘ 4‘ 5‘ 4‘ 6 5

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
on the view plane
Only those objects within the view volume are
projected 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.) Parallel Projection Perspective Projection
Truncated Pyramid Frustum
截斷角錐幾何體

36. View Volume (cont.)

37. Parallel Projection near：hither 此處 far：yon plane 彼處 near：front
far：back

38. Perspective Projection
Center of
Projection

39. 3 - D Window - to – viewport mapping
Viewport is a regular parallelepiped(平行六面體)
defined innormalized coordinates
3 - D Window - to – viewport mapping 比例 位移

40. 3 - D Window - to - viewport mapping (cont.)y z x
View Volume
3 - D viewport

41. 3 - D Window - to - viewport mapping (cont.)
View Volume：defined by
and
3 - D viewport ：defined by
and

42. Clipping against a normalized View Volume
Extension of
Code of 6 bits right to left(六個面)
back front above below right left 後 前 上 下 右 左
Bit 6
Bit 1

43. Code of 6 bits right to left (cont.)
below
bit 4
above
bit 5
front
bit 6
back
Code
Code
Logical AND of 2 codes
2 end points of a line

44. Code of 6 bits right to left (cont.)
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. Code of 6 bits right to left (cont.)
Line 其中
1. Intersection of a line with a ( face plane ) of view volume
2. with front

46. Line segment does not intersect
2. with front
not in (0 ~ 1)
(1) If
Line segment does not intersect
(2) If
in (0 ~ 1)
(3) If neither
nor
is in
beyond boundary(超出邊界)

47. 3. Check Intersection with 6 ( faces ) planes
Side View P2 A P1 P2 B P1
3. Check Intersection with 6 ( faces ) planes

48. Cyrus - Beck Algorithm 用在 3 - D