Unit – IV 2D Viewing. 2 of 30 Contents Windowing Concepts The viewing pipeline viewing coordinate...

Unit – IV 2D Viewing

.

2of30

Contents

Windowing Concepts• The viewing pipeline• viewing coordinate reference frame,• window to view-port coordinate transformation, • viewing functions Line Clipping

– Introduction – Cohen-Sutherland Clipping Algorithm– Cyrus-beck line clipping algorithms

Polygon Clipping– Sutherland-Hodgman Area Clipping Algorithm

3of30

Where we require clipping?

An architect may have a graphics program to draw an entire building but be interested in only the swimming pool.

4of30

Viewing in 2D - Viewport

Window in world coordinates.

45

250

Viewport inDevice coords

250 x 250Pixels.

Display Device

5of30

The computer is used in design applications because it can easily and accurately create, store and modify very complex drawings. When drawings are too complex, however, they become difficult to read. In such situations it is useful to display only those portions of the drawing that are of immediate interest. Sometimes, it is desirable to enlarge these portions to take full advantage of the available display surface.

The method for selecting and enlarging portions of a drawing is called windowing.

The technique for not showing the part of the drawing which one is not interested is called a clipping.

6of30

Windowing I

A scene is made up of a collection of objects specified in world coordinates

World Coordinates

7of30

Window

A world coordinate area selected for display is called a window.

wymax

wymin

wxmin wxmax

Window

World Coordinates

8of30

View port

An area on a display device to which a window is mapped is called a view port.

Vymax

Vymin

Vxmin Vxmax

Device Coordinates

View port

9of30

VIDEOCON

Display Screen

View Port

10of30

The window defines what is to be viewed. And the view port defines where it is to be displayed.

Window and view ports are rectangles in standard positions, with rectangle edges parallel to the coordinate axes.

11of30

Window to viewport mappingor

windowing transformation.

The mapping of a part of a world coordinate scene to device coordinates is referred to as a viewing transformation. 2-d viewing transformation is also referred as window to view port or windowing transformation.

Y wmax

Y wmin

X wminX wmax

window

Y vmax

Y vmin

X vmin X vmax

View port

12of30

The Viewing Pipeline

Window

• A world-coordinate area selected for display.

defines what is to be viewed

Viewport

• An area on a display device to which a window is mapped.

defines where it is to be displayed

Viewing transformation

• The mapping of a part of a world-coordinate scene to device coordinates.

A window could be a rectangle to have any orientation.

13of30

14of30

15of30

2D Viewing Pipeline

To make the viewing process independent of the requirements of any output device, graphics systems convert object descriptions to normalized coordinates range from 0 to 1

16of30

Window to viewing coordinate transformation

First, we construct the scene in world coordinates using the output primitives and attributes. Next, to obtain a particular orientation for the window, we can set up a 2-dimenstional viewing coordinate system in the world coordinate plane and define a window in the viewing coordinate system. The viewing coordinate reference frame is used to provide a method for setting up arbitrary orientation of rectangular windows. Once the viewing reference frame is established, we can transform descriptions in world coordinates to viewing coordinates( in the range from 0 to 1) and map viewing coordinate description of the scene to normalized coordinates. At the final step, all parts of the picture that lie outside the view port are clipped and the contents of the view port are transferred to device coordinates.

17of30

Cntd..

By changing the position of the view port, we can view objects at different positions on the display area of an output device.

Also, by varying the size of view port, we can change the size and proportions of displayed objects.

We achieve zooming effects by successively mapping different sized windows on a fixed size view port. As the windows are made smaller, we zoom in on some part of a scene to view details that are not shown with larger windows. Similarly, more overview is obtained by zooming out from a section of a scene with successively larger windows.

18of30

The Viewing Pipeline

Mwc,vc = R. T

19of30

20of30

21of30

(0,0)

(100,50)

(0,25)

(50,50)

So Sx = 50/100 = ½And Sy = 25/50 = ½

XWmin,YWmin

XWmax,YWmax

XVmin,YVmin

XVmax,YVmax

XVmax-XVmin

XWmax-XWmin

Sx=

YVmax-YVmin

YWmax-YWmin

Sy=

22of30

Window to View port coordinate Transformation

23of30

24of30

Window-to-Viewport Coordinate Transformation

minmax

min

minmax

min

xwxw

xwxw

xvxv

xvxv

minmax

min

minmax

min

ywyw

ywyw

yvyv

yvyv

sxxwxwxvxv )( minmin

syywywyvyv )( minmin minmax

minmax

ywyw

yvyvsy

minmax

minmax

xwxw

xvxvsx

To map a point in the window at world position (xw, yw ) in the associated view port (xv, yv ) we use the concept of relative distance.

25of30

Workstation transformation

Mapping selected parts of a scene in normalized coordinates to different video monitors with workstation transformations

26of30

Clipping Operations

Clipping • Identify those portions of a picture that are either

inside or outside of a specified region of space.

Clip window • The region against which an object is to be

clipped.

• The shape of clip window

Applications of clipping

World-coordinate clipping

27of30

Clipping Operations

Viewport clipping • It can reduce calculations by allowing

concatenation of viewing and geometric transformation matrices.

Types of clipping • Point clipping • Line clipping • Area (Polygon) clipping • Curve clipping • Text clipping

Point clipping (Rectangular clip window)

28of30

Point clipping

Point clipping: Remove points outside window. – A point is either entirely inside the region or notAssuming that the clip window is a rectangle in standard

position, we save a point p=(x,y) for display if the following inequalities are satisfied:

Where the edges of the clip window (xwmin, xwmax, ywmin, ywmax) can be either world coordinate window boundaries or viewport boundaries.

If any one of these four inequalities is not satisfied, the point is clipped (not saved for display)

maxmin

maxmin

yyy

xxx

29of30

(xmin , ymin )

(xmax , ymax )

x = xmin x = xmax

y = ymin

y = ymax

(x1, y1)

cliprectangle

maxmin

maxmin

yyy

xxx

For a point For a point ((x,yx,y) ) to be inside the clip rectangleto be inside the clip rectangle::

Point Clipping

30of30

Line Clipping

Basic calculations:– Is an endpoint inside or outside the clipping window?– Find the point of intersection, if any, between a line

segment and an edge of the clipping window.

Both endpoints inside:

trivial accept

One inside: find

intersection and clip

Both outside: either

clip or reject

31of30

Line Clipping

For the image below consider which lines and points should be kept and which ones should be clipped

wymax

wymin

wxmin wxmax

Window

P1

P2

P3

P6

P5P7

P10

P9

P4

P8

32of30

Point Clipping

Easy - a point (x,y) is not clipped if:

wxmin ≤ x ≤ wxmax AND wymin ≤ y ≤ wymax

otherwise it is clipped

wymax

wymin

wxmin wxmax

Window

P1

P2

P5

P7

P10

P9

P4

P8

Clipped

Points Within the Window are Not Clipped

Clipped

Clipped

Clipped

33of30

Line Clipping

Harder - examine the end-points of each line to see if they are in the window or not

Situation Solution Example

Both end-points inside the window

Don’t clip

One end-point inside the window, one outside

Must clip

Both end-points outside the window

Must clip

34of30

Cohen-Sutherland Clipping Algorithm

An efficient line clipping algorithm

The key advantage of the algorithm is that it vastly reduces the number of line intersections that must be calculated

Dr. Ivan E. Sutherland co-developed the Cohen-Sutherland clipping algorithm. Sutherland is a graphics giant and includes amongst his achievements the invention of the head mounted display.

35of30

cliprectangle

Cases for clipping linesCases for clipping lines

Line Clipping

36of30

Cases for clipping linesCases for clipping lines

A

B

A

B

cliprectangle

Line Clipping

37of30

Cases for clipping linesCases for clipping lines

D

A

BC

D'

A

BC

D'

cliprectangle

Line Clipping

38of30

Cases for clipping linesCases for clipping lines

D

E

F

A

BC

D'

A

BC

D'

cliprectangle

Line Clipping

39of30

Cases for clipping linesCases for clipping lines

D

E

F

A

BC

D'

G

H

G'

H'A

BC

D'

G'

H'

cliprectangle

Line Clipping

40of30

Cases for clipping linesCases for clipping lines

D

E

F

A

BC

D'

G

H

G'

H'

I

J

I'

J'

A

BC

D'

G'

H'

cliprectangle

Line Clipping

41of30

Cohen-Sutherland: World Division

World space is divided into regions based on the window boundaries

– Each region has a unique four bit region code– Region codes indicate the position of the

regions with respect to the window

1001 1000 1010

00010000

Window0010

0101 0100 0110

42of30

1. End-points pairs are checked for trivial acceptance or rejection using outcode;

2. If not trivially accepted or rejected, divide the line segment into two at a clip edge;

3. Iteratively clipped by test trivial-acceptance or trivial-rejection, and divided into two segments until completely inside or trivial-rejection.

A

B

C

D

E

F

G

H

I1001

0001

0101 0100

0000

1000 1010

0010

0110

Steps for Cohen-Sutherland alg.

43of30

Cohen-Sutherland: Labelling

Every end-point is labelled with the appropriate region code

wymax

wymin

wxmin wxmax

Window

P3 [0001]P6 [0000]

P5 [0000]

P7 [0001]

P10 [0100]

P9 [0000]

P4 [1000]

P8 [0010]

P12 [0010]

P11 [1010]

P13 [0101] P14 [0110]

44of30

Cohen-Sutherland: Lines In The Window

Lines completely contained within the window boundaries have region code [0000] for both end-points so are not clipped

wymax

wymin

wxmin wxmax

Window

P3 [0001]P6 [0000]

P5 [0000]

P7 [0001]

P10 [0100]

P9 [0000]

P4 [1000]

P8 [0010]

P12 [0010]

P11 [1010]

P13 [0101] P14 [0110]

45of30

Cohen-Sutherland: Lines Outside The Window

Any lines with a common set bit in the region codes of both end-points can be clipped

– The AND operation can efficiently check this

wymax

wymin

wxmin wxmax

Window

P3 [0001]P6 [0000]

P5 [0000]

P7 [0001]

P10 [0100]

P9 [0000]

P4 [1000]

P8 [0010]

P12 [0010]

P11 [1010]

P13 [0101] P14 [0110]

46of30

Cohen-Sutherland: Other Lines

Lines that cannot be identified as completely inside or outside the window may or may not cross the window interior

These lines are processed as follows:– Compare an end-point outside the window to a

boundary (choose any order in which to consider boundaries e.g. left, right, bottom, top) and determine how much can be discarded

– If the remainder of the line is entirely inside or outside the window, retain it or clip it respectively

47of30

Cohen-Sutherland: Other Lines (cont…)

– Otherwise, compare the remainder of the line against the other window boundaries

– Continue until the line is either discarded or a segment inside the window is found

We can use the region codes to determine which window boundaries should be considered for intersection

– To check if a line crosses a particular boundary we compare the appropriate bits in the region codes of its end-points

– If one of these is a 1 and the other is a 0 then the line crosses the boundary

48of30

Cohen-Sutherland Examples

Consider the line P9 to P10 below

– Start at P10

– From the region codes of the two end-points we know the line doesn’t cross the left or right boundary

– Calculate the intersection of the line with the bottom boundary to generate point P10’

– The line P9 to P10’ is completely inside the window so is retained

wymax

wymin

wxmin wxmax

Window

P10 [0100]

P9 [0000]

P10’ [0000]

P9 [0000]

49of30

Cohen-Sutherland Examples (cont…)

Consider the line P3 to P4 below

– Start at P4

– From the region codes of the two end-points we know the line crosses the left boundary so calculate the intersection point to generate P4’

– The line P3 to P4’ is completely outside the window so is clipped

wymax

wymin

wxmin wxmax

WindowP4’ [1001]

P3 [0001]

P4 [1000]

P3 [0001]

50of30

Cohen-Sutherland Examples (cont…)

Consider the line P7 to P8 below

– Start at P7

– From the two region codes of the two end-points we know the line crosses the left boundary so calculate the intersection point to generate P7’

wymax

wymin

wxmin wxmax

Window

P7’ [0001]

P7 [0001] P8 [0010]P8’ [0010]

51of30

Cohen-Sutherland Examples (cont…)

Consider the line P7’ to P8

– Start at P8

– Calculate the intersection with the right boundary to generate P8’

– P7’ to P8’ is inside the window so is retained

wymax

wymin

wxmin wxmax

Window

P7’ [0001]

P7 [0001] P8 [0010]P8’ [0010]

52of30

Cohen-Sutherland Worked Example

wymax

wymin

wxmin wxmax

Window

53of30

Calculating Line Intersections

Intersection points with the window boundaries are calculated using the line-equation parameters

– Consider a line with the end-points (x1, y1) and (x2, y2)

– The y-coordinate of an intersection with a vertical window boundary can be calculated using:

y = y1 + m (xboundary - x1)

where xboundary can be set to either wxmin or wxmax

54of30

Calculating Line Intersections (cont…)

– The x-coordinate of an intersection with a horizontal window boundary can be calculated using:

x = x1 + (yboundary - y1) / m

where yboundary can be set to either wymin or wymax

– m is the slope of the line in question and can be calculated as m = (y2 - y1) / (x2 - x1)

55of30

Initial ConditionInitial Condition

Sutherland-Hodgeman Algo.

Clip Against Right Clipping Clip Against Right Clipping BoundaryBoundary

Clip Against Top Clipping Clip Against Top Clipping BoundaryBoundary

The Clipped PolygonThe Clipped PolygonClip Against Bottom Clipping Clip Against Bottom Clipping BoundaryBoundary

Clip Against Left Clipping Clip Against Left Clipping BoundaryBoundary

56of30

Polygon ClippingPolygon Clipping

v1 v2v1’

out in

Output: v1’ and v2

v1

v2

in in

Output: v2

v1v2 v1’

in out

Output: v1’

v1

v2

out out

Output: none

outside inside

Sutherland-Hodgeman Polygon Clipping against a rectangle (or a window)

For each stage, the boundary clipper divides the plane into an inside half-planeAnd outside half-plane

Inside half-plane contains the clipping window; Outside half-plane doesn’t.

Given an edge going from vertex V[i] to vertex V[i+1], during the transversal, there are four cases (assuming the V[i] has been checked in the previous edge):

57of30

Case 1Case 1

Inside Outside

Polygonbeingclipped

Clipboundary

s

p:output

4 Cases of Polygon Clipping

Case 2Case 2

Inside Outside

s

p

i:output

Case 3Case 3

Inside Outside

s

p

(no output)

Case 4Case 4

Inside Outside

s

i:first output

p:second output

58of30

Cohen-Sutherland – polygons

s

p

Inside Outside

s

p

Inside Outside

s

pInside Outside

s

p

Inside Outside

1st Output

i

Output No Output

i

2nd Output

Output

59of30

Area Clipping

Similarly to lines, areas must be clipped to a window boundary

Consideration must be taken as to which portions of the area must be clipped

60of30

Sutherland-Hodgman Area Clipping Algorithm

A technique for clipping areas developed by Sutherland & Hodgman

Put simply the polygon is clipped by comparing it against each boundary in turn

Original Area Clip Left Clip Right Clip Bottom Clip Top

Sutherland turns up again. This time with Gary Hodgman with whom he worked at the first ever graphics company Evans & Sutherland

61of30

Sutherland-Hodgman Area Clipping Algorithm (cont…)

To clip an area against an individual boundary:– Consider each vertex in turn against the

boundary– Vertices inside the boundary are saved for

clipping against the next boundary– Vertices outside the boundary are clipped– If we proceed from a point inside the boundary

to one outside, the intersection of the line with the boundary is saved

– If we cross from the outside to the inside intersection point and the vertex are saved

62of30

Sutherland-Hodgman Example

Each example shows the point being processed (P) and the previous point (S)

Saved points define area clipped to the boundary in question

S

P

Save Point P

S

P

Save Point I

I

P

S

No Points Saved

S

P

Save Points I & P

I

63of30

Other Area Clipping Concerns

Clipping concave areas can be a little more tricky as often superfluous lines must be removed

Clipping curves requires more work– For circles we must find the two intersection

points on the window boundary

Window WindowWindow Window

64of30

Summary

Objects within a scene must be clipped to display the scene in a window

Because there are can be so many objects clipping must be extremely efficient

The Cohen-Sutherland algorithm can be used for line clipping

The Sutherland-Hodgman algorithm can be used for area clipping

65of30

66of30

Cohen-Sutherland Clipping Algorithm VI

Let’s consider the lines remaining below

wymax

wymin

wxmin wxmax

Window

P3 [0001]P6 [0000]

P5 [0000]

P7 [0001]

P10 [0100]

P9 [0000]

P4 [1000]

P8 [0010]

P12 [0010]

P11 [1010]

P13 [0101] P14 [0110]

67of30

Cohen-Sutherland Clipping Algorithm

Easy - a point (x,y) is not clipped if:

wxmin ≤ x ≤ wxmax AND wymin ≤ y ≤ wymax

otherwise it is clipped

wymax

wymin

wxmin wxmax

Window

P1

P2

P5

P7

P10

P9

P4

P8

Clipped

Points Within the Window are Not Clipped

Clipped

Clipped

Clipped

68of30

Clipping

Point clipping is easy:– For point (x,y) the point is not clipped if

wxmin ≤ x ≤ wxmax AND wymin ≤ y ≤ wymax

wymax

wymin

wxmin wxmax

Before Clipping

Window

P1

P2

P3

P6

P5P7

P10

P9

P4

P8

69of30

Cohen-Sutherland – polygons

s

p

Inside Outside

s

p

Inside Outside

s

pInside Outside

s

p

Inside Outside

1st Output

i

Output No Output

i

2nd Output

Output

Polygon Clipping

.

71of30 Ex: Simple Convex

72of30

Ex: Multiple Components

73of30 Nonconvex with Many Edges

74of30

Polygons : Before Clipping

75of30

Polygons: Clip on Top

Top ClipBoundary

76of30

Polygons: Clip on Right

Right ClipBoundary

77of30

Polygons: Clip on Bottom

Bottom ClipBoundary

78of30

Polygons: Clip on Left

Left ClipBoundary

79of30

Polygons: Clipped Polygon

80of30

Begin Animated Recap

81of30

Polygons: Before Clipping

82of30

Polygons: Clip on Top

Top ClipBoundary

83of30

Polygons: Clip on Top

Top ClipBoundary

84of30

Polygons: Clip on Right

Right ClipBoundary

85of30

Polygons: Clip on Right

Right ClipBoundary

86of30

Polygons: Clip on Bottom

Bottom ClipBoundary

87of30

Polygons: Clip on Bottom

Bottom ClipBoundary

88of30

Polygons: Clip on left

Left ClipBoundary

89of30

Polygons: Clip on left

Left ClipBoundary

90of30

Polygons: Fully Clipped

91of30

Polygons: Clip on Left

Left ClipBoundary

92of30

Polygons: Clipped Polygon

93of30

End Animated Recap

Hodgeman-Sutherland

95of30 Polygons: Case 1Polygons: Case 1

INOUT

Si output

P

Case 1: S out; P in

Action: Output i and P

97of30 Polygons: Case 2Polygons: Case 2

(no output)

P

S

INOUT

Case 2: S & P both in

Action: output p

99of30 Polygons: Case 3Polygons: Case 3

P: second output

i: first output

S

INOUT

Case 3: S IN; P OUT

Action: Output i

101of30 Polygons: Case 4Polygons: Case 4

Polygonbeing clipped

P: output Clip boundary

S

INOUT

Case 4: S & P both out

Action: None

103of30

Case 2: S & P both IN

Action: Output P

Case 3: S IN; P OUT

Action: Output i

(S already output

under Case 1)

Case 4: S & P both OUT

Action: None

Case 1: S OUT; P IN

Action: Output i and P

104of30 Hodgeman-Sutherland

v1 v2

v3

v4

v5 v6

v7v8

105of30

Hodgeman-Sutherland

v1 v2

v3

v4

v5 v6

v7v8

106of30

Hodgeman-Sutherland: Top

2

v3

v5 v6

v7v8

34

1

107of30

Hodgeman-Sutherland: Top

)( 87654321 vvvvvvvv

)( 875 643321 vvvvv

108of30 Hodgeman-Sutherland: Bottom

2

v3

v5 v6

v7v8

34

1

109of30 Hodgeman-Sutherland: Bottom

2

v3

v5 v6

65

34

1

110of30 Hodgeman-Sutherland: Bottom

)( 875 643321 vvvvv

)( 665433215 vvv

111of30

Hodgeman-Sutherland: Right

2

v3

v5 v6

65

34

1

112of30

)( 665433215 vvv

)( 665433215 vvv

This case is trivial: do nothing

Hodgeman-Sutherland: Right

113of30

Hodgeman-Sutherland: Left

2

v3

v5 v6

65

34

1

114of30

Hodgeman-Sutherland: Left

2

v3

7

v6

39

8

6

10

115of30

Hodgeman-Sutherland: Left

)( 665433215 vvv

)( 5610933287 vv

116of30

Hodgeman-Sutherland: Done

117of30

Begin Animated Recap

118of30

v1 v2

v3

v4

v5 v6

v7v8

1 2 3 4 5 6 7 8( )v v v v v v v v

119of30 ( )1)(

v1 v2

v3

v5 v6

v7v8

v4

2 3)v

341

3) 4 5)v 6)v 7)v 8)v Done

Top21

121of30 ( ) 15( ) 2) 3)v 3) 4) 5)v 6)v 6)

v3

v5 v6

v7v8

2341

5 6

Done

Bottom

122of30 7( )

v3

v5 v6

327 8( )v ( )

2341

37 28 3( )v 32 37 8 9 10( )v 3 62 37 8 9 10( )v v 3 5 62 3 45 61( )v v v

5

7 8 2( )

6

37 8 92 3( )v

7

89

10

Nothing needs to be clipped against Right

123of30

v3

v5 v6

234

67

89

10

3 5 62 3 45 61( )v v v

124of30

End Animated Recap

125of30

Hodgeman-Sutherland: Done

126of30 Hodgeman-Sutherland Problem

Extra Lines

127of30

Correct Result

How to Fix Up?

128of30 Distinguish Between Case 2Distinguish Between Case 2

Si output

P

IN OUT

129of30

And, Case 4And, Case 4

P: second output

i: first output

S

IN OUT

130of30

Distingish Between Case 2Distingish Between Case 2

SP

is output

IN OUT

131of30

And, Case 4And, Case 4

P: second output S

is output first

IN OUT

132of30

Extended H-S Algorithm

Tag the different new vertices

Call the one generated on transition from IN to OUT

Call the one generated on transition from OUT to IN

Starting from OUT, connect as soon as is generated

and

133of30

Example of Extended Algo

IN OUT

134of30

Result

IN OUT

135of30

Extended H-S Algorithm

Connect and as soon as the . is generated

Generate next polygon beginning with next , i.e.,

( ,....)

Another Polygon Clipping Algorithm

Weiler-Atherton Clipping

137of30

Another approach to polygon clipping

No extra clipping outside window

Works for arbitrary shapes

Avoids degenerate polygons

Weiler-Atherton Polygon Clipping

138of30

Weiler-Atherton Clipping

13

11

12 6

5

414

3

2

79

810

1

Clip Polygon (ep)

Subject Polygon (sp)

139of30 Gives “Right” Answer

140of30

Start at first (inside) vertexTraverse polygon until hitting a window boundary

Output intersection point iTurn rightFollow window boundary until next intersection

Weiler-Atherton Clipping

141of30

Output second intersection

Turn right, again, and follow subject polygon until closed

Continue on subject polygon from first intersection point.

Repeat processing until complete

Weiler-Atherton Clipping

142of30

Approximate description

Can be improved

Weiler-Atherton Clipping

143of30 Generalizations of W-A

Can be extended to complex situations,

arbitrary windows

Stability issues can arise for such cases

General Boolean Operations

via Boundary Representations

(B-reps)

General Clipping

with abitrary polygons

146of30 General Boolean Operations

A

B

147of30

A

B

BA in

148of30

B

BA in

149of30 BA in

A

B

150of30 BA in

151of30

A

B

A

AB in

152of30

A

AB in

153of30 AB in

154of30

BA out

A

B

155of30

A

B

BA out

156of30

BA out

157of30

A

B

AB out

158of30

AB out

159of30 BA on

B

on

160of30

BA on

161of30

AB on

A

B

on

162of30

AB on

163of30 ((anti-)shared )

anti-sharedshared

B A

164of30

A

B

BA

165of30

A B

n BAo

BAin

n AB i

166of30

A

B

BA

167of30 BA

n BAo BAout

t AB ou

168of30

A

B

BA

169of30 BAout n ABi BA

The End

Polygon Clipping

Lect

ure

Set

5