06 2D Viewing

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1 / NVortragender Computergraphik @ TU Wien


Viewing Pipeline:Viewing Pipeline:WindowWindow--ViewportViewport TransformationTransformation

clipping window: what to displayclipping window: what to display

viewportviewport: where to be viewed: where to be viewed

translation, rotation, scaling, clipping,...translation, rotation, scaling, clipping,...

world coordinates viewport coordinates

Clipping WindowViewport


constructworld-coordinate

scene usingmodeling-coordinate

transformations

22--dim. Viewingdim. Viewing--TransfTransf. Pipeline. Pipeline

constructworld-coordinate

scene usingmodeling-coordinate

transformations

convertworld-coordinates

to viewingcoordinates

WCMC

DCVCVCtransform viewing

coordinates tonormalizedcoordinates

map normalizedcoordinates to

devicecoordinates

NC

convertworld-coordinates

to viewing

coordinates


WCWC VC TransformationVC Transformationsetting up a rotated world window in

viewing coordinates and the corresponding

normalized-coordinate viewport

World Coordinates Viewing Coordinates

Viewport

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ViewingViewing CoordCoord. Reference Frame. Reference Frame

TRM VCWC ,

A viewing-

coordinate

frame is

moved into

coincidence

with the world

frame in two

steps:

(a) translate the viewing origin to the world origin

(b) rotate to align the axes of the two systems.


WC

convert

world-coordinatesto viewingcoordinates


construct

world-coordinatescene usingmodeling-coordinate

transformations

convert

world-coordinatesto viewing

coordinates

transform viewingcoordinates to

normalizedcoordinates


devicecoordinates

MC

VC DC

WC

NCNC




WindowWindow -- ViewportViewport Transform (1)Transform (1)

point (xw,yw) in a designated window is mapped

to viewport coordinates (xv,yv) so that relative

positions in the two areas are the same.


WindowWindow -- ViewportViewport Transform (2)Transform (2)linear in x and y

(xwmin/ywmin) (xvmin/yvmin)(xwmax/ywmax) (xvmax/yvmax)

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xw = xwmin + (xwmaxxwmin) where 01xv = xvmin + (xvmaxxvmin)

= xvmin + xwxwmaxxwmin

xwmin(xvmaxxvmin)

xwmaxxwmin

xvmaxxvmin

xwmaxxwmin

xwxwmin =

xv = xvmin + (xvmaxxvmin)xwmaxxwmin

xwxwmin

tx sxwindow-viewport

transformation: xv = sxxw+tx yv = sy.yw+ty

10/ NVortragender Computergraphik @ TU Wien




construct

world-coordinatescene usingmodeling-coordinate

transformations

convert

world-coordinatesto viewingcoordinates



map normalizedviewport to

devicecoordinates

MC WC

NC DCVCVC


devicecoordinates

DC


Workstation TransformationWorkstation Transformation

Mapping

selected parts

of a scene in

normalizedcoordinates to

different video

monitors with

workstation

transformations


ClippingClipping

partly visible or completely invisible partspartly visible or completely invisible parts

must not be ignored and must not be drawnmust not be ignored and must not be drawn

must be cut offmust be cut off(if possible in world coordinates)(if possible in world coordinates)

ignored

vertices edges

scrolled

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Clipping OperationsClipping Operations

remove objects outside a clip windowremove objects outside a clip window

clip window: rectangle, polygon, curvedclip window: rectangle, polygon, curvedboundariesboundaries

applied in world or viewing coordinatesapplied in world or viewing coordinates

combined with scan conversioncombined with scan conversion

objects to clip: points, lines, polygons,objects to clip: points, lines, polygons,curves, text, ...curves, text, ...


3 Possibilities for Clipping3 Possibilities for Clipping

analyticallyanalytically= in world coordinates= in world coordinates

reduces WCreduces WC

DC transformationsDC transformations

during raster conversationduring raster conversation

= as part of the= as part of the rasterizationrasterization algorithmalgorithm

efficient for complex primitivesefficient for complex primitives

pixel by pixelpixel by pixeltesttest biggest effort, very primitive algorithmbiggest effort, very primitive algorithm


Line Clipping (1)Line Clipping (1)

line clipping against a rectangular clip window

before clipping after clipping


Line Clipping (2)Line Clipping (2)

goalsgoals

eliminate simple cases fasteliminate simple cases fast

avoid intersection calculationsavoid intersection calculations

for endpoints (x0,y0), (xend,yend)intersect parametric representation

x = x0 + u(xend x0)y = y0 + u(yend y0)

with window borders:

intersection 0 < u < 1

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CohenCohen--Sutherland Line ClippingSutherland Line Clipping

assignment of region codes to lineassignment of region codes to line

endpointsendpoints

bit1: leftbit1: left

bit2: rightbit2: right

bit3: belowbit3: below

bit4: abovebit4: above

binary region codes assigned to line endpoints

according to relative position with respect to the

clipping rectangle18/ NVortragender Computergraphik @ TU Wien

CohenCohen--Sutherland Line ClippingSutherland Line Clipping oror of codes of bothof codes of both

points 0000points 0000

line entirelyline entirely visiblevisible andand of codes of bothof codes of both

pointspoints 00000000 line entirelyline entirely invisibleinvisible

all othersall others intersect!intersect!



lines extending

from one

coordinate region

to another maypass through the

clip window, or

they may intersect

clipping

boundaries

without entering

the window



remaining linesremaining lines

intersection test with bounding lines ofintersection test with bounding lines ofclipping windowclipping window

left, right, bottom, topleft, right, bottom, top discard an outside partdiscard an outside part

repeat intersection test up to four timesrepeat intersection test up to four times

}yw,{ywy,/)(

}xw,{xwx),(

maxmin00

maxmin00

+=

+=

myyxx

xxmyyvertical

horiz.

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CohenCohen--

SutherlandSutherland

Line ClippingLine Clipping passes throughclipping window

intersects boundaries

without entering

clipping window

}yw,{ywy,/)(

}xw,{xwx),(

maxmin00

maxmin00

+=

+=

myyxx

xxmyyvertical

horiz.


Polygon ClippingPolygon Clipping

modification of line clippingmodification of line clipping

goal: one or more closed areasgoal: one or more closed areas

display of a correctly

clipped polygon

display of a polygon

processed by a line-

clipping algorithm


processing polygon boundary as aprocessing polygon boundary as awhole against each window edgewhole against each window edge

output: list of verticesoutput: list of vertices

clipping a polygon against successive

window boundaries

original

polygonclip left clip right

clip

bottomclip top

SutherlandSutherland--HodgmanHodgman Polygon ClippingPolygon Clipping



four possible edge casesfour possible edge cases

successive processing of pairs of polygon

vertices against the left window boundary

out inoutput: V1, V2

in inV2

in outV1

out outno output

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V2:= 1st vertexfor1 edge:

Sutherland-

Hodgman Alg.

V1:=V2V2:=next vertex V2 result list

V2 visible?no

V1 visible?

yes

yesV1 visible?

no

V1 = clip edge V1V2 result list

yes no


clipping a polygon

against the leftboundary of a

window, starting with

vertex 1. Primed

numbers are used to

label the points in

the output vertex listfor this window

boundary




finished!


SutherlandSutherland--HodgmanHodgman Algorithm:Algorithm:

Combination of the 4 PassesCombination of the 4 Passes

the polygon is clipped against each of the 4the polygon is clipped against each of the 4

borders separately,borders separately,that would produce 3 intermediate results.that would produce 3 intermediate results.

by calling the 4 testsby calling the 4 tests recursivelyrecursively,,

(or by using a clipping pipeline)(or by using a clipping pipeline)

every result point is immediately processedevery result point is immediately processed

on, so that onlyon, so that only oneone result list is producedresult list is produced

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2

2

3

3

1

1

2 pipeline ofpipeline of

boundaryboundary

clippers toclippers toavoidavoid

intermediateintermediate

vertex listsvertex lists

Processing the vertices of the polygon through

a boundary-clipping pipeline.

SutherlanSutherlandd--HodgmanHodgman Clipping ExampleClipping Example

1st clip:

left


2

2

3

3

1

1

2

After all vertices are processed through the

pipeline, the vertex list for the clipped polygon is[1, 2, 2, 2]

SutherlanSutherlandd--HodgmanHodgman Clipping ExampleClipping Example

2nd clip:

bottom


extraneous lines for concave polygons:extraneous lines for concave polygons:

split into separate convex parts orsplit into separate convex parts or

final check of output vertex listfinal check of output vertex list

clipping the

concave polygon

with the

Sutherland-

Hodgeman clipper

produces twoconnected areas



Text Clipping (1)Text Clipping (1)

1. text clipping using a bounding

rectangle about the entire string

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2. text clipping using a bounding

rectangle about individual characters



3. text clipping performed on the

components of individual characters


SummarySummary:: ClippingClipping

CohenCohen--SutherlandSutherland lineline clippingclipping

HodgmanHodgman--SutherlandSutherland polygonpolygon clippingclipping

texttext clippingclipping

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06 2D Viewing