Computer Graphics Inf4/MSc

1

Computer Graphics

Lecture 7

Scanline algorithm and polygon clipping

Taku Komura

Computer Graphics Inf4/MSc

Today’s topics

• Scanline algorithm

• Clipping

Computer Graphics Inf4/MSc

Scanline algorithm

• Computing the barycentric coordinates for all the pixels inside the bounding box can

be costly

• We can try to scan only the pixels inside the polygon

Computer Graphics Inf4/MSc

Scanline algorithm

For each scan line:

1. Find the intersections of the scan 　 line with 　 all edges of the

polygon.

2 Sort the intersections by increasing　 x coordinate.

3 Fill in all pixels between pairs of 　　 intersections.

Can also deal with concave polygons

Computer Graphics Inf4/MSc Span extrema

• Only turn on pixels whose centers are interior to the polygon:

– Otherwise will intrude other adjacent polygons

• round up values on the left edge of a span, round down on the right edge

midpoint algorithm interior

Computer Graphics Inf4/MSc Edge Coherence

• Computing the intersections between scan lines and edges can be costly

• Use a method similar to the midpoint algorithm

• y = mx + b, x = (y – b) / m• At y = s, xs = (s – b ) / m

At y = s + 1, xs+1 = (s+1 - b) / m = xs + 1 / m• Incremental calculation: xs+1 = xs + 1 / m

Computer Graphics Inf4/MSc

Pseudo code of computing the left span extrema (m > 1)

}

}

/* Overflow*/

{ )( if

;

);( ePixel Writ

){;;(for

int

int

int

1

maxmin

minmax

minmax

minmax

minmax

or; denomenatnt - increme

;x

or denomenatincrement

numeratorincrement

x,y

yyyyy

rdenomenatoincrement

yyrdenomenato

xxnumerator

yy

xx

m

Computer Graphics Inf4/MSc Active Edge Table • A table of edges that are currently used to fill the polygon

• Scan lines are processed in increasing Y order.

• Polygon edges are sorted according to their minimum Y.

• When the current scan line reaches the lower endpoint of an edge it becomes active.

• When the current scan line moves above the upper endpoint, the edge becomes inactive.

Computer Graphics Inf4/MSc

•Active edges are sorted according to increasing X.

•Filling in pixels between left most edge intersection and stops at the second.

•Restarts at the third intersection and stops at the fourth.

Active Edge Table

Computer Graphics Inf4/MSc

Polygon fill rules(to ensure consistency)

• Horizontal edges: Do not include in edge table

• Vertices: If local max or min, then count twice, else count once.

• If pixel is on edge, only draw left / bottom edges

Computer Graphics Inf4/MSc

Today’s topics

• Scanline algorithm

• Clipping

Computer Graphics Inf4/MSc Clipping• We need to clip objects outside the canonical view volume

• Clipping lines (Cohen-Sutherland algorithm)

• Clipping polygons (Sutherland-Hodgman algorithm)

Computer Graphics Inf4/MScCohen-Sutherland algorithm

While (true) {

1. Check if the line segment is trivial accept/reject

2. Clip the edge and shorten }

Computer Graphics Inf4/MSc

What is a trivial accept / reject?

• Trivial acceptanceAll line vertices lie inside box accept.

Computer Graphics Inf4/MSc

All line vertices lie outside and on same side reject.

What is a trivial accept / reject?

Computer Graphics Inf4/MSc

Cohen-Sutherland 2D outcodes

– 4-bit code called: Outcode

– First bit : above top of window, y > ymax

– Second bit : below bottom, y < ymin

– Third bit : to right of right edge, x > xmax

– Fourth bit : to left of left edge, x < xmin

Computer Graphics Inf4/MSc

Cohen-Sutherland 2D outcodes

0000

0100

0001

1001 1000 1010

0010

01100101

Computer Graphics Inf4/MSc

Cohen-Sutherland 2D outcodes

0000

0100

0001

1001 1000 1010

0010

01100101

Both endpoint codes 0000, trivial acceptance, else:

Do logical AND of outcodes

Computer Graphics Inf4/MSc

0000

0100

0001

1001 1000 1010

0010

01100101

1000

00000000

Logical AND between codes for 2 endpoints,Reject line if non-zero – trivial rejection.

0001

Cohen-Sutherland 2D outcodes

Computer Graphics Inf4/MSc

0000

0100

0001

1001 1000 1010

0010

01100101

Logical AND between codes for 2 endpoints,Reject line if non-zero – trivial rejection.

What about this one?

Computer Graphics Inf4/MSc

Line Intersection.

• Clip the line by edges of the rectangle• Select a clip edge based on the outcode, split

and feed the new segment on the side of the rectangle back into algorithm

• Need to perform 4 intersection checks for each line.

Computer Graphics Inf4/MSc

Polygon Clipping

• Sutherland-Hodgman’s algorithm

• Polygons are clipped at each edge of the window while traversing the polygon

• Output : a list of vertices of the clipped polygon

Computer Graphics Inf4/MScSutherland-Hodgman’s algorithm

• The edges of the polygon are traversed

• The edges can be divided into four types

Inside Outside Inside Outside Inside Outside Inside Outside

Case 3No

output.Case 1

OutputVertex

Case 2.

OutputIntersection

Case 4

SecondOutput

FirstOutput

Computer Graphics Inf4/MScSutherland-Hodgman’s algorithm

• For each of the edges of the clipping rectangle

– For each edge of the polygon (connecting pi, pi+1)

• If case 1 add p+1 to the output

• If case 2 add interaction to output

• If case 4 add intersection and p+1 to outputInside Outside Inside Outside Inside Outside Inside Outside

Case 3No

output.

Case 1

OutputVertex

Case 2.

OutputIntersection

Case 4

SecondOutput

FirstOutput

Computer Graphics Inf4/MSc

Example

http://www.sunshine2k.de/stuff/Java/SutherlandHodgman/SutherlandHodgmanApplet.html

Computer Graphics Inf4/MSc

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References

• Scanline algorithm

Foley et al., Chapter 3.5, 3.6• Clipping lines, polygons

– Foley et al. Chapter 3.12, 3.14

– http://www.cc.gatech.edu/grads/h/Hao-wei.Hsieh/Haowei.Hsieh/mm.html