Line Drawing and Clipping - Drexel CCIdavid/Classes/CS430/Lectures/L-02_Lines... · 2 7 Bresenham’s Algorithm •Suppose we just finished –(assume 0

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CS 430Computer Graphics I

Line Drawing and ClippingWeek 1, Lecture 2

David Breen, William Regli and Maxim Peysakhov

Geometric and Intelligent Computing Laboratory

Department of Computer Science

Drexel Universityhttp://gicl.mcs.drexel.edu

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Outline

• Line drawing

• Digital differential analyzer

• Bresenham’s algorithm

• Cohen-Sutherland line clipping• Parametric line clipping

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Scan-Conversion Algorithms

• Scan-Conversion:Computing pixelcoordinates forideal line on2D raster grid

• Pixels bestvisualized ascircles/dots– Why? Monitor

hardware

1994 Foley/VanDam/Finer/Huges/Phillips ICG

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Digital Differential Analyzer(DDA)

• If slope is less then 1 _x= 1 (otherwize _y = 1)

• Compute corresponding_y (_x)

• xk+1 = xk + _x• yk+1 = yk + _y• Issues:

– Avoiding floatingpoint ops,multiplications, andreal numbers


xk+1

_xxk

yk+1

yk

_y

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Bresenham’s Algorithm

• 1965 @ IBM

• Basic Idea:– Only integer

arithmetic

– Incremental

• Consider the implicitequation for a line:


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Given:

implicit line equation:

Let:where r and q are points on the line

Then:

Observe that all of these are integers

and: for points above the line for points below the line

Now…..Pics/Math courtesy of Dave Mount @ UMD-CP

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• Suppose we justfinished– (assume 0<slope<1)

other cases symmetric

• Which pixel next?– E or NE

Pics/Math courtesy of Dave Mount @ UMD-CP

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Assume:

• = exact y value at

• y midway between E and NE:

Observe:

If , then pick E

Else pick NE

If ,it doesn’t matter


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• Create “modified” implicit function (2x)

• Create a decision variable D to select,where D is the value of f at the midpoint:


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• If then m is below the line– NE is the closest pixel

• If then m is above the line– E is the closest pixel

• Note: because we multiplied by 2x, D isnow an integer---which is very good news

• How do we make this incremental??

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Case I: When E is next

• What increment for computing an new D?• Next midpoint is:

• Hence, increment by:Pics/Math courtesy of Dave Mount @ UMD-CP

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Case II: When NE is next

• What increment for computing an new D?• Next midpoint is:

• Hence, increment by:Pics/Math courtesy of Dave Mount @ UMD-CP

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How to get an initial value for D?

• Suppose we start at:

• Initial midpoint is:

Then:


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The Algorithm

Assumptions:

Pre-computed:Pics/Math courtesy of Dave Mount @ UMD-CP

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Bresenham’s Algorithm:Example

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Some issues withBresenham’s Algorithms

• Pixel ‘density’ variesbased on slope– straight lines look

darker, more pixels perCM

• Endpoint order– Line from P1 to P2

should match P2 to P1

– Always choose E whenhitting M, regardless ofdirection


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Some issues withBresenham’s Algorithms

• How to handle the line when it hits the clipwindow?

• Vertical intersections– Could change line slope

– Need to change init cond.

• Horizontal intersections– Again, changes in the

boundary conditions

– Can’t just intersectthe line w/ the box


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Scissoring Clipping


Performed during scan conversion of theline (circle, polygon)

Compute the next point (x,y)If xmin £ x £ xmax and ymin £ y £ ymax

Then output the point.

Else do nothing

• Issues with scissoring:– Too slow

– Does more work then necessary

– We are not going to use it.

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The Cohen-Sutherland LineClipping Algorithm

• How to clip lines to fit inwindows?– easy to tell if whole line

falls w/in window

– harder to tell what partfalls inside

• Consider a straight line

• And window:


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Cohen-Sutherland

Basic Idea:

• First, do easy test– completely inside or

outside the box?

• If no, we need amore complex test

• Note: we will alsoneed to figure outhow scan line meetsthe box


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Cohen-Sutherland

• The Easy Test:

• Compute 4-bit codebased on endpointsP1 and P2


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Cohen-Sutherland

• Line is completelyvisible iff both codevalues of endpointsare 0, i.e.

• If line segments arecompletely outsidethe window, then


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Cohen-Sutherland

Otherwise,we clip the lines:

• We know that thereis a bit flip, w.o.l.g.assume its (x0 ,x1 )

• Which bit? Try `em all!– suppose its bit 4

– Then x0 < WL and we know that x1 WL

– We need to find the point: (xc ,yc )≥


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Cohen-Sutherland

• Clearly:

• Using similar triangles

• Solving for yc gives


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Cohen-Sutherland

• Replace (x0 ,y0 ) with (xc ,yc )

• Re-compute values

• Continue until allpoints are inside theclip window

QED


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Parametric Line Clipping

• Developed by Cyrus and Beck in 1978• Used to clip 2D/3D lines against convex

polygon/polyhedron• Liang and Barsky (1984) algorithm efficient

in clipping upright 2D/3D clipping regions• First we will follow original Cyrus-Beck

development to introduce parametric clipping• Then we will reduce Cyrus-Beck to more

efficient Liang-Barsky case

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The Cyrus-Beck Technique

• Cohen-Sutherland algorithm computes (x,y)intersections of the line and clipping edge

• Cyrus-Beck finds a value of parameter t forintersections of the line and clipping edges

• Simple comparisons used to find actualintersection points

• Liang-Barsky optimizes it by examining tvalues as they generated to reject some linesegments immediately

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Finding the Intersection Points

Line P(t) = P0 +t(P1-P0)Point on the edge PEi

Make sure1. D ≠ 0, or P1 ≠ P0

2. Ni• D ≠ 0 lines are not parallel


DN

]P[PNt

)-P (PLet D

]-Pt[PN ]-P[PN

])-P-Pt(P[PN

][P(t)-PN

i

Eii

iEii

Eii

Eii

•-

-•=

=

=•+•

=+•

=•

0

01

010

010

0

0

0

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Finding the Line Segment

• Classify point as potentially entering (PE) or leaving (PL)

• PE if crosses edge into inside half plane => angle P0 P1and Ni greater 90° => Ni• D < 0

• PL otherwise.

• Find Te = max(te)

• Find Tl = min(tl)

• Discard if Te > Tl

• If Te < 0 Te = 0

• If Tl > 1 Tl = 1

• Use Te, Tl to compute intersection coordinates (xe,ye), (xl,yl)


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Line Drawing and Clipping - Drexel CCIdavid/Classes/CS430/Lectures/L-02_Lines... · 2 7 Bresenham’s Algorithm •Suppose we just finished –(assume 0