Lect2 scan convertinglines

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Computer Graphics:Line Drawing Algorithms

Sudipta Mondal

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Contents

Graphics hardwareThe problem of scan conversionConsiderationsLine equationsScan converting algorithms

– A very simple solution– The DDA algorithm

Conclusion

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Graphics Hardware

It’s worth taking a little look at how graphicshardware works before we go any furtherHow do things end up on the screen?

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Architecture Of A Graphics System

System Bus

CPU DisplayProcessor

SystemMemory

DisplayProcessorMemory

FrameBuffer

VideoController MonitorMonitor

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Output Devices

There are a range of output devicescurrently available:

– Printers/plotters– Cathode ray tube displays– Plasma displays– LCD displays– 3 dimensional viewers– Virtual/augmented reality headsets

We will look briefly at some of the morecommon display devices

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Basic Cathode Ray Tube (CRT)

Fire an electron beam at a phosphor coatedscreen

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Raster Scan Systems

Draw one line at a time

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Colour CRT

An electron gun for each colour – red, greenand blue

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Applying voltages tocrossing pairs ofconductors causesthe gas (usually amixture includingneon) to breakdown into a glowingplasma of electronsand ions

Plasma-Panel DisplaysIm

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Liquid Crystal Displays

Light passingthrough the liquidcrystal is twistedso it gets throughthe polarizerA voltage isapplied using thecrisscrossingconductors to stopthe twisting andturn pixels offIm

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The Problem Of Scan Conversion

A line segment in a scene is defined by thecoordinate positions of the line end-points

x

y

(2, 2)

(7, 5)

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The Problem (cont…)

But what happens when we try to draw thison a pixel based display?

How do we choose which pixels to turn on?

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Considerations

Considerations to keep in mind:– The line has to look good

• Avoid jaggies– It has to be lightening fast!

• How many lines need to be drawn in a typicalscene?

• This is going to come back to bite us again andagain

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Line Equations

Let’s quickly review the equations involvedin drawing lines

x

y

y0

yend

xendx0

Slope-intercept lineequation:

bxmy +×=where:

0

0

xxyym

end

end

--

=

00 xmyb ×-=

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Lines & Slopes

The slope of a line (m) is defined by its startand end coordinatesThe diagram below shows some examplesof lines and their slopes

m = 0

m = -1/3

m = -1/2

m = -1

m = -2m = -4

m = ∞

m = 1/3

m = 1/2

m = 1

m = 2m = 4

m = 0

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A Very Simple Solution

We could simply work out the correspondingy coordinate for each unit x coordinateLet’s consider the following example:

x

y

(2, 2)

(7, 5)

2 7

2

5

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A Very Simple Solution (cont…)

1

2

3

4

5

0

1 2 3 4 5 60 7

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A Very Simple Solution (cont…)

x

y

(2, 2)

(7, 5)

2 3 4 5 6 7

2

5

53

2725=

--

=m

542

532 =*-=b

First work out m and b:

Now for each x value work out the y value:

532

543

53)3( =+×=y

513

544

53)4( =+×=y

543

545

53)5( =+×=y 5

24546

53)6( =+×=y

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A Very Simple Solution (cont…)

Now just round off the results and turn onthese pixels to draw our line

3532)3( »=y

3513)4( »=y

4543)5( »=y

4524)6( »=y

0 1 2 3 4 5 6 7 8

0

1

23

4

56

7

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A Very Simple Solution (cont…)

However, this approach is just way too slowIn particular look out for:

– The equation y = mx + b requires themultiplication of m by x

– Rounding off the resulting y coordinatesWe need a faster solution

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A Quick Note About Slopes

In the previous example we chose to solvethe parametric line equation to give us the ycoordinate for each unit x coordinateWhat if we had done it the other wayaround?So this gives us:

where: and

mbyx -

=

0

0

xxyym

end

end

--

= 00 xmyb ×-=

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A Quick Note About Slopes (cont…)

Leaving out the details this gives us:

We can see easily thatthis line doesn’t lookvery good!We choose which wayto work out the linepixels based on theslope of the line

0 1 2 3 4 5 6 7 8

0

1

23

4

56

7

4323)3( »=x 5

315)4( »=x

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A Quick Note About Slopes (cont…)

If the slope of a line is between -1 and 1 thenwe work out the y coordinates for a line basedon it’s unit x coordinatesOtherwise we do the opposite – x coordinatesare computed based on unit y coordinates

m = 0

m = -1/3m = -1/2

m = -1

m = -2m = -4

m = ∞

m = 1/3m = 1/2

m = 1

m = 2m = 4

m = 0

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A Quick Note About Slopes (cont…)

1

2

3

4

5

0

1 2 3 4 5 60 7

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The DDA Algorithm

The digital differentialanalyzer (DDA) algorithmtakes an incrementalapproach in order tospeed up scan conversionSimply calculate yk+1based on yk

The or ig ina l d i f fe ren t ia lanalyzer was a physicalm a c h i n e d e ve l o p e d b yVannevar Bush at MIT in the1930’s in order to solveordinary differential equations.

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The DDA Algorithm (cont…)

Consider the list of points that wedetermined for the line in our previousexample:

(2, 2), (3, 23/5), (4, 31/5), (5, 34/5), (6, 42/5), (7, 5)Notice that as the x coordinates go up byone, the y coordinates simply go up by theslope of the lineThis is the key insight in the DDA algorithm

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The DDA Algorithm (cont…)

When the slope of the line is between -1 and 1begin at the first point in the line and, byincrementing the x coordinate by 1, calculatethe corresponding y coordinates as follows:

When the slope is outside these limits,increment the y coordinate by 1 and calculatethe corresponding x coordinates as follows:

myy kk +=+1

mxx kk

11 +=+

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The DDA Algorithm (cont…)

Again the values calculated by the equationsused by the DDA algorithm must be roundedto match pixel values

(xk, yk) (xk+1, yk+m)

(xk, round(yk))

(xk+1, round(yk+m))

(xk, yk) (xk+ 1/m, yk+1)

(round(xk), yk)

(round(xk+ 1/m), yk+1)

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DDA Algorithm Example

Let’s try out the following examples:

x

y

(2, 2)

(7, 5)

2 7

2

5

x

y (2, 7)

(3, 2)

2 3

2

7

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DDA Algorithm Example (cont…)

7

2

3

4

5

6

1 2 3 4 5 60 7

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The DDA Algorithm Summary

The DDA algorithm is much faster than ourprevious attempt

– In particular, there are no longer anymultiplications involved

However, there are still two big issues:– Accumulation of round-off errors can make

the pixelated line drift away from what wasintended

– The rounding operations and floating pointarithmetic involved are time consuming

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Conclusion

In this lecture we took a very brief look athow graphics hardware worksDrawing lines to pixel based displays is timeconsuming so we need good ways to do itThe DDA algorithm is pretty good – but wecan do betterNext time we’ll like at the Bresenham linealgorithm and how to draw circles, fillpolygons and anti-aliasing