Drawing Lines

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# Drawing Lines - PowerPoint PPT Presentation

Drawing Lines. The Bresenham Algorithm for drawing lines and filling polygons. Plotting a line-segment. Bresenham published algorithm in 1965 It was originally to be used with a plotter It adapts well to raster “scan conversion” It uses only integer arithmetic operations

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### Drawing Lines

The Bresenham Algorithm for drawing lines and filling polygons

Plotting a line-segment
• Bresenham published algorithm in 1965
• It was originally to be used with a plotter
• It adapts well to raster “scan conversion”
• It uses only integer arithmetic operations
• It is an “iterative” algorithm: each step is based on results from the previous step
• The sign of an “error term” governs the choice among two alternative actions
Scan conversion

The actual line is comprised of points drawn from a continuum,

but it must be “approximated” using pixels from a discrete grid.

The various cases
• Horizontal or vertical lines are easy cases
• Lines that have slope 1 or -1 are easy, too
• Symmetries leave us one remaining case: 0 < slope < 1
• As x-coodinate is incremented, there are just two possibilities for the y-coordinate: (1) y-coordinate is increased by one; or (2) y-coordinate remains unchanged
0 < slope < 1

Y-axis

X-axis

y increases by 1

y does not change

Integer endpoints

ΔY = Y1 – Y0

ΔX = X1 – X0

(X1,Y1)

0 < ΔY < ΔX

ΔY

(X0,Y0)

ΔX

slope = ΔY/ΔX

Which point is closer?

y = mx + b

A

yi -1+1

yi -1

B

ideal line

xi

xi -1

error(A) = (yi -1 + 1) – y*

error(B) = y* - (yi -1)

The Decision Variable
• Choose B if and only if error(B)<error(A)
• Or equivalently: error(B) – error(A) < 0
• Formula: error(B) – error(A) = 2m(xi – x0) + 2(y0 – yi -1) -1
• Remember: m = Δy/Δx (slope of line)
• Multiply through by Δx (to avoid fractions)
• Let di = Δx( error(B) – error(A) )
• Rule is: choose B if and only if di < 0
Computing di+1 from di

di+1 = 2(Δy)(xi+1 – x0) +2(Δx)(y0 – yi) – Δx

di = 2(Δy)(xi – x0) + 2(Δx)(y0 – yi-1) – Δx

The difference can be expressed as:

di+1 = di + 2(Δy)(xi+1 – xi) – 2(Δy)(yi – yi-1)

Recognize that xi+1 – xi = 1 at every step

And also: yi – yi-1 will be either 0 or 1

(depending on the sign of the previous d)

How does algorithm start?
• At the outset we start from point (x0,y0)
• Thus, at step i =1, our formula for di is: d1 = 2(Δy) - Δx
• And, at each step thereafter:

if ( d i < 0 ) { di+1 = di + 2(Δy); yi+1 = yi; }

else { di+1 = di + 2(Δy-Δx); yi+1 = yi + 1; }

xi+1 = xi + 1;

‘bresdemo.cpp’
• The example-program is on class website:

http://nexus.cs.usfca.edu/~cruse/cs686/

• It draws line-segments with various slopes
• The Michener algorithm (for a circle-fill) is also included, for comparative purposes
• Extreme slopes (close to zero or infinity) are not displayed in this demo program
• They can be added by you as an exercise
Filling a triangle or polygon
• The Bresenham’s method can be adapted
• But an efficient data-structure is needed
• All the sides need to be handled together
• We let the y-coordinate steadily increment
• For sides which are “nearly horizontal” the x-coordinates can change by more than 1
Bucket-Sort

Y

0

XLO

XHI

1

2

8

8

3

7

9

4

6

10

5

5

11

6

7

12

7

9

13

8

11

14

10

13

15

11

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12

17

17

13

In-class exercises
• For the ‘bresdemo.cpp’ program:
• Supply a function that tests the capability of the Breshenham line-drawing algorithm to draw lines having the full range of slopes
• For the ‘fillpoly.cpp’ program:
• Modify the program code so that it will work with polygons having more than three sides