Two-Dimensional Viewing

Viewing Pipeline

Two-Dimensional Viewing

• Two dimensional viewing transformation– From world coordinate scene description to device

(screen) coordinates

Normalization and Viewport Transformation

• World coordinate clipping window• Normalization square: usually [-1,1]x[-1,1]• Device coordinate viewport

OpenGL 2D Viewing

• Projection Mode– glMatrixMode(GL_PROJECTION);

• GLU clipping-window function– gluOrtho2D(xwmin,xwmax,ywmin,ywmax);– Normalized to [-1,1]x[-1,1]

• OpenGL viewport function– glViewport(xvmin,xvmax,yvmin,yvmax);– Rarely used because of GLUT device independent

library

Clipping

• Remove portion of output primitives outside clipping window

• Two approaches– Clip during scan conversion: Per-pixel bounds check– Clip analytically, then scan-convert the

modified primitives

Two-Dimensional Clipping

• Point clipping – trivial• Line clipping

– Cohen-Sutherland– Cyrus-beck– Liang-Barsky

• Fill-area clipping– Sutherland-Hodgeman– Weiler-Atherton

• Curve clipping• Text clipping

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

Cohen-Sutherland Line Clipping

• One of the earliest algorithms for fast line clipping• Identify trivial accepts and rejects by bit operations

< Region code for each endpoint >

above below right leftBit 4 3 2 1

0000

10001001

0001

0101 0100 0110

0010

1010Clipping window

Cohen-Sutherland Line Clipping

• Compute region codes for two endpoints• If (both codes = 0000 ) trivially accepted• If (bitwise AND of both codes 0000) trivially rejected• Otherwise, divide line into two segments

– test intersection edges in a fixed order.(e.g., top-to-bottom, right-to-left)

0000

10001001

0001

0101 0100 0110

0010

1010Clipping window

Cohen-Sutherland Line Clipping

• Fixed order testing and clipping cause needless clipping (external intersection)

Cohen-Sutherland Line Clipping

• This algorithm can be very efficient if it can accept and reject primitives trivially– Clip window is much larger than scene data

• Most primitives are accepted trivially– Clip window is much smaller than scene data

• Most primitives are rejected trivially

• Good for hardware implementation

Cyrus-Beck Line Clipping

• Use a parametric line equation

• Reduce the number of calculating intersections by exploiting the parametric form

• Notations– Ei : edge of the clipping window

– Ni : outward normal of Ei

– An arbitrary point PEi on edge Ei

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Cyrus-Beck Line Clipping

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halfplane inside in thepoint a0))((

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PtPN

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Cyrus-Beck Line Clipping

• Solve for the value of t at the intersection of P0P1 with the edge– Ni · [P(t) - PEi] = 0 and P(t) = P0 + t(P1 - P0)– letting D = (P1 - P0),

– Where• Ni 0• D 0 (that is, P0 P1)• Ni · D 0 (if not, no intersection)

DNPPNt

i

Eii

][ 0

Cyrus-Beck Line Clipping• Given a line segment P0P1, find intersection

points against four edges– Discard an intersection point if t [0,1] – Label each intersection point either PE

(potentially entering) or PL (potentially leaving)– Choose the smallest (PE, PL) pair that defines the

clipped line

PL0PE0

10

10

PPNPPN

i

i

Cyrus-Beck Line Clipping

• Cyrus-Beck is efficient when many line segments need to be clipped

• Can be extended easily to convex polygon (rather than upright rectangle) clip windows

Liang-Barsky Line Clipping

• Liang-Barsky optimized Cyrus-Beck for upright rectangular clip windows

),(1),(0

22

11

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tdyyyytyytdxxxxtxx

1121

1121

)()(

P(x1,y1)

Q(x2.y2)

tB

tT

tR

tL

Liang-Barsky Line Clipping

T

B

RLtT

tB

General Clipping Window• Line clipping using nonrectangular polygon clip

windows– Convex polygon

• Cyrus-Beck algorithm can be readily extended– Concave polygon

• Split the concave polygon into convex polygons

Vector Method for Concave Splitting

• Calculate edge-vector cross products in a counterclockwise order

• If any z component turns out to be negative, the polygon is concave

Polygon Fill-Area Clipping• Polyline vs polygon fill-area

• Early rejection is useful

Clipping Window

Bounding box of polygon fill area

Sutherland-Hodgman Polygon Clipping

• Clip against 4 infinite clip edges in succession

Sutherland-Hodgman Polygon Clipping

• Accept a series of vertices (polygon) and outputs another series of vertices

• Four possible outputs

Sutherland-Hodgman Polygon Clipping

• The algorithm correctly clips convex polygons, but may display extraneous lines for concave polygons

Weiler-Atherton Polygon Clipping

• 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 a clockwise direction

Weiler-Atherton Polygon Clipping

• Polygon clipping using nonrectangular polygon clip windows

Text Clipping• All-or-none text clipping

– Using boundary box for the entire text

• All-or-non character clipping– Using boundary box for each individual character

• Character clipping– Vector font: Clip boundary polygons or curves– Bitmap font: Clip individual pixels