2-D Clipping

8/7/2019 2-D Clipping

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Topics to be Covered

Two-Dimensional Viewing The 2-D Viewing transformation Pipeline

Window-to-Viewport Coordinate Transformation

Clipping Operations Line Clipping

Polygon Clipping

Text Clipping

Shielding-Definition


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Two-Dimensional Viewing

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 ismapped

Defines where it is to be displayed

NDCS

Normalized device coordinate system is deviceindependent tool, in which a unit (1 x 1) square whoselower left corner is at the origin of the coordinatesystem, defines the display are of a virtual device.


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The process that converts objectcoordinates in WCS to normalized device

coordinates is called Normalization

Mapping of a part of a world coordinate

scene to device coordinates is referred to

as viewing transformation or window to

viewport transformation or windowing

transformation

Two-Dimensional Viewing


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The Two-Dimensional Viewing Transformation Pipeline

Construct

WCS

Convert

WCS to

Viewing

Coordinates

MC WC Map VC to

NVC

VC NVC Map NVC

to DC

DC


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A point at position (xw,yw) in the window is mapped into

position (xv,yv) in the associated viewport.

To maintain the same relative placement in the viewport as in

the window, we require that

minmax

min

minmax

min

minmax

min

minmax

min

ywyw

ywyw

yvyv

yvyv

xwxw

xwxw

xvxv

xvxv

!

!

Window to Viewport Transformation


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Solving the above expression, we have

minmax

minmax

minmax

minmax

minmin

minmin

*)(

*)(

yy

yvyv

sy

xx

xvxvsx

syyyyvyv

sxxxxvxv

!

!

!

!

Window to Viewport Transformation


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Clipping

Any procedure that identifies those

portions of a picture that are either inside

or outside of a specified region of space is

referred to as a clipping algorithm, or

simply clipping.

The region against which an object is to be

clipped is called a clip. window


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Applications of Clipping

1. Extracting part of a defined scene for viewing

2. Identifying visible surfaces in 3D views

3. Antialiasing line segments or object

boundaries

4. Displaying a multiwindow environment

5. Drawing and painting operations that allow

parts of a picture to be selected for copying,

moving, erasing or duplicating


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POINT CLIPPING

Point Clipping is essentially the evaluation of the

following inequalities

xmin x xmax and ymin y ymax

where xmin, xmax, ymin and ymax define theclipping window. A point (x, y) is considered

inside the window when the inequalities all

evaluate to true.


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The Cohen-Sutherland Algorithm

One of the Oldest and Most popular procedure

The method speeds up the processing o line segments by

performing initial tests that reduce the number of intersectionsthat must be calculated

Can be divided into two phases

Step 1: Identify those lines which intersect the clipping window and

so need to be clipped

Step 2: Perform the Clipping

LINE CLIPPING


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LINE CLIPPING


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Cohen-Sutherland Line Clipping Algorithm

Step 1: Identify those lines which need to beclipped

The lines fall into following four categories

1. Visible Both endpoints of the line within the window

2. Not Visible The line definitely lie outside the window.This will occur if the line from (x1,y1) to (x2,y2) satisfies

any of the following inequalities

x1,x2 > xmax y1,y2 > ymax

X1,x2 < xmin y1,y2 < ymin3. Clipping candidate - The line in neither category 1 nor 2


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1. Assign a 4-bit region code to each endpoint of the line.

2. Line is Visible - Both region codes are 0000

3. Line is Invisible Logical AND of endpoints is not0000

4. Clipping Candidate - Logical AND of endpoints is

0000

0000

10001001

0001

0101 0100 0110

0010

1010ymax

ymin

xmax xmin

Above, Below , left, right

Step 1: Identify the Category


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C(0100)

D(1010)

C

D

D

xmin

ymax

ymin


Clipping can be performed in two ways

Method 1: Clipping process begins by comparing outside point to a clipping

boundary to determine how much of the line can be discarded

Then remaining part of the line is checked against the other boundaries

and the process is continued until either the line is totally discarded or asection is found inside the window.


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Consider the line CD as it is clipping candidate

The coordinates of the intersection point are

xi = x in or xmax for vertical boundary line

yi = y1+m(xi-x1)

Or

xi = x1 + (yi-y1)/m for horizontal boundary line

Yi = ymin or ymax

where m =(y2-y1)/x2-x1) is the slope of the line.



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Replace endpoint (x1,y1) with the

intersection point (xi, yi)

Assign region code to new endpoint (xi, yi) Re-categorize

Iterate the process till the line segment is

either in the category 1 or category 2



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Based on Binary Search

Divide the line segment into two at its midpoint

Determine the Clipping categories

Each segment in category 3 is divided again intoshorter segments and categorized

This process continues until Each line segment that spans across a window boundary

reaches a threshold for line size

and all other segments are either in Category 1 or in Category2

The midpoint coordinates (xm, ym ) of a line joining(x1,y1) and (x2,y2) are given by

2

21 xxxm

!

2

21 yyym

!

Method 2: Midpoint Subdivision


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POLYGON CLIPPINGSutherland Hodgeman Algorithm

L

Pi

Pi-1

Output Pi

R

Pi-1

Pi

Output i

I

L R

Nooutput

Pi-1

Pi

L R

Output I,Pi

Pi-1Pi

L R

I


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Let P1,,Pn be the vertex list of Polygons to be clipped

Let Edge E be any edge of the positively oriented convex clippingpolygon.

We clip each edge of the polygon in turn against the edge E of the

clipping polygon , forming a new polygon whose vertices are

determined as follows

Consider the edge Pi-1Pi1. If both Pi-1 and Pi are to the left of the edge, vertex Pi is placed on the

vertex output list of the clipped polygon

2. If both Pi-1 and Pi are to the right of the edge, nothing is placed on the

vertex output list

3. If Pi-1 is to the left and Pi to the right of E, the intersection point I of

line segment Pi-1Pi with the extended edge E is calculated andplaced on the vertex output list

4. If Pi-1 is to the right and Pi to the left of E, the intersection point I of

line segment Pi-1Pi with the extended edge E is calculated and Both

are placed on the vertex output list

POLYGON CLIPPING


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Convex polygons are correctly clipped by

the Sutherland-Hodgeman algorithm, butconcave polygons may be displayed with

extraneous lines

POLYGON CLIPPING


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TEXT CLIPPING

Several techniques are used All or none-string Clipping

Simplest method for processing strings relative to awindow boundary

STRING 1

STRING 1

Before Clipping

STRING 1

After Clipping


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all- or-none character-clipping

Discard only those characters that are not

completely inside the window

STRING 1

STRING 3

Before Clipping

STRING2

STRING 3

STRING 1

RING 3

Before Clipping

STRING 3

TEXT CLIPPING


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A final method for handling text clipping is

to clip the components of individual

characters

STRING 1

Before Clipping

STRING 1

After Clipping

STRING 1

TEXT CLIPPING


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Exterior Clipping

Clipping a picture to the interior of a region by

eliminating everything outside the clipping

region.

Exterior clipping is saving the picture partsthat are outside the region.

Applications

Multiple Windowing System

Where pictures overlap

TEXT CLIPPING

2-D Clipping

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