2 D viewing Prepared by Elizabeth Isaac DCS, RSET

2 D viewing

Prepared byElizabeth Isaac

DCS, RSET

The Viewing PipelineViewing transformations in several stepsWindow to Viewport.

Window: A region of the scene selected for viewing (also called clipping window)Viewport: A region on display device for mapping to window

2-D viewing transformation referred to as Window –to-Viewport Transformation or windowing transformation

World Coordinates Viewing Coordinates

Window

Viewport

The Viewing Pipeline

A viewing transformation using standard rectangles for the window and viewport

World Coordinates

xvmin xvmax

ywmax

ywmin

Device Coordinate

xvmin xvmax

yvmax

yvmin

The two-dimensional viewing-transformation pipeline

Construct World CoordinateScene Using

Modeling-CoordinateTransformation

ConvertWorld-Coordinates

toViewing Coordinates

Map Viewing Coordinates toNormalized Viewing

Coordinates usingWindow-Viewport Specifications

Map NormalizedViewport to

DeviceCoordinates

Setting up a rotated world window in viewing coordinates and the corresponding normalized-

coordinate viewport

NormalizedDevice Coordinates

Viewpoint

0 1

1

World Coordinates

y view

x view

y world

y0

x0 x world

Viewing Coordinate Reference Frame

• The steps in this coordinate transformation– A viewing coordinate frame is moved into

coincidence with the world frame in two stepsa) Translate the viewing origin to the world

origin, thenb)Rotate to align the axes of the two

systems



• Matrix for converting world-coordinate positions to viewing coordinate

R: rotation matrixT: translation matrix

ΤRΜ VCWC ,

Window-To-Viewport Coordinate Transformation

• A point at position (xw, yw) in a designated window is mapped to viewport coordinates (xv, yv) so that relative positions in the two areas are the same


• To maintain the same relative placement

minmax

min

minmax

min

minmax

min

minmax

min

ywywywyw

yvyvyvyv

xwxwxwxw

xvxvxvxv


• Solving these expressions

• The scaling factors

syywywyvyv

sxxwxwxvxv

)(

)(

minmin

minmin


• Conversion sequence of transformation1. Perform a scaling transformation using a fixed-

point position of (xwmin, ywmin) that scales the window area to the size of the viewport

2. Translate the scaled window area to the position of the viewport


• Workstation transformation– Opening any number of output devices in a particular

application– Performing another window-to-viewport transformation

for each open output device


Two-Dimensional Viewing Functions

• Setting up the elements of a window-to-viewport mapping matrix

evaluateViewMappingMatrix (xwmin, xwmax, ywmin, ywmax, xvmin, xvmax, yvmin, yvmax, error, viewMappingMatrix)

Two-Dimensional Viewing Functions

Selection of a workstation window-viewport pair – setWorkstationWindow (ws, xwsWindmin,

xwsWindmax, ywsWindmin, ywsWindmax)– setWorkstationViewport (ws, xwsVPortmin,

xwsVPortmax, ywsVPortmin, ywsVPortmax)

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 clipping

Clip window

Clipping

– Point – Line– Area (or Polygons)– Curve– Text

Clipping

• Point Clipping• Assuming that the clip window is a rectangle

in standard position• Saving a point P=(x, y) for display

maxmin

maxmin

ywyyw

xwxxw

Line Clipping• Line clipping against a rectangular clip window

Line Clipping

Cohen-Sutherland Line Clipping

• Four digit binary code -Region Code

• Identifies the location of the point relative to the boundaries of the clipping rectangle

• Bit 1: left• Bit 2: right• Bit 3: below• Bit 4: above

4 3 2 1



– Calculate differences between endpoint coordinates and clipping boundaries

– Use the resultant sign bit of each difference calculation to set the corresponding value in the region code

• Region code divide space into 9 regions• Computation of region code requires at most

4 subtractions

b0 = 1 if y > ymax, 0 otherwiseb1 = 1 if y < ymin, 0 otherwiseb2 = 1 if x > xmax, 0 otherwiseb3 = 1 if x < xmin, 0 otherwise


• Outside Line Removal Test– A method that can be used to test lines total

clipping is to perform the logical and operation with both region codes

– Not 0000• Completely outside the clipping region!!

• Lines that cannot be identified as completely inside or outside a clip window by this test


Intersection Point

• Calculate Intersection Point– Using the slope-intercept form– Vertical Boundary

– Horizontal Boundary

)( 11 xxmyy

m

yyxx 1

1

)/()( 1212 xxyym

• 1. Start• 2. Read two end points of a line P1,P2 (x1,y1) and (x2,y2)• 3. Read two corners of window(wx1,wy1) and(wx2,wy2)• 4. Assign the region codes for two end points using

following steps– a. Set bit 1 if x < wx1– b. Set bit 2 if x>wx2– c. Set bit 3 if y<wy2– d. Set bit 4 if y> wy1

• 5. Check for the visibility of the line– a. If region codes for both end points are 0

then the line is completely visible. Draw the line. Go to 10

– b. If the region codes are not zero and the logical AND ing of them is also non zero then the line is completely invisible. Go to 10

– c. If the region codes for two end points do not satisfy the conditions in 5 a and 5 b the line is partially visible

• 6. Determine the intersecting edge of the clipping window by inspecting the region codes of two end points

– a. If the region codes foe both end points are non zero find the intersection point P1’ and P2’ with boundary edges of clipping window with respect to P1 and P2 respectively

– b. If the region code for any one end point is non zero then find intersection point P1’ or P2’ with the boundary edge of the clipping window with respect to it.

• 7. Divide the line segments considering the intersection points

• 8. Reject the line segment if any one end point of it appears outside the clipping window

• 9. Draw the remaining line segments• 10. Stop.

Polygon Clipping

Polygon Clipping

36

Sutherland-Hodgman Clipping

37


38


39


40


41


42


43


Sutherland-Hodgeman Polygon Clipping


• 4 possible cases


Weiler-Atherton Polygon Clipping• Rules

– For an outside-to-inside pair of vertices, follow the polygon boundary– For an inside-to-outside pair of vertices, follow the window boundary

in clockwise direction

Other Polygon-Clipping Algorithm

• Clipping a polygonby determining theintersection of twopolygon areas

Text clipping

Clip window

A B C

Text clipping

• All-or-none string clipping– No text is retained after clipping

since part of text is outside clip window

Clip window

A B C

Clip window

Before clipping After clipping

Text clipping

• All-or-none character clipping– Only “B” and “C” are retained after clipping

since part of “A” is outside clip window

Clip window

A B C

Clip window


B C

Text clipping

• Character component clipping– All of “B” and “C” are retained, and those parts of

“A” inside clip window are also retained

Clip window

A B C

Clip window


A B C

interior clipping

• interior clipping– what is to be saved is inside the clip window

• exterior clipping – what is to be saved is outside clip window

Interior clipping- keep point P2

P2(x2, y2)

Exterior Clipping

Exterior clipping- keep point P1

Clip window

P1(x1, y1)

Documents

2 D viewing Prepared by Elizabeth Isaac DCS, RSET