Rasterization: Bresenham s Midpoint Algorithm
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1 Rasterization: Bresenham s Mioint Algorithm CS4600 Comuter Grahics aate from Rich Riesenfel s slies Fall 05 Rasterization General metho: Use equation for geometr Insert equation for rimitive Derive rasterization metho CS4600
2 Line Characterizations licit: m B Imlicit: F(, abc 0 Constant sloe: k Constant erivative: f ( k Line Characterizations - Parametric: where, ( t ( t P0 t P P P( 0 P0 ; P( P Intersection of lanes Shortest ath between oints Conve hull of iscrete oints CS4600
3 Two Line quations licit: m B Imlicit: F(, abc 0 Define: 0 Hence, 0 B From revious We have, B Hence, B 0 CS4600 3
4 Relating licit to Imlicit q s Recall, Multil through b, B 0 ( ( ( B 0 F(, ( ( ( B 0 where, a ( ; b ( ; c B( Discrete Lines Lines vs. Line Segments What is a iscrete line segment? This is a comuter grahics roblem How to generate a iscrete line? CS4600 4
5 Goo Discrete Line No gas in ajacent iels Piels close to ieal line Consistent choices; same iels in same situations Smooth looking ven brightness in all orientations Same line for P0 P as for P P0 Double iels stacke u? How to Draw a Line? Plug into the equation irectl. Comute sloe. Start at on oint ( o, o 3. Increment X an raw How to figure this out? CS4600 5
6 Derive from Line quation Y = mx + b Y i = mx i + b X i+ = X i + Y i+ = mx i+ + b Y i+ = ( / (X i+ + b CS4600 6
7 What about sloe? CS4600 7
8 Derive from Line quation Y = mx + b Y i = mx i + b Y i+ = mx i+ + b mx i+ = Y i+ -b X i+ = (Y i+ - b / m X i+ = (Y i+ - b / (/ X i+ = (Y i+ - b * (/ CS4600 8
9 What about sloe? If 0 > 0 increment Y else increment X Restricte Form Line segment in first octant with 0 < m < After we erive this, we ll look at the other cases (other octants CS4600 9
10 Incremental Function val Recall f ( i Characteristics Fast Cumulative rror Nee to efine f ( i ( i f ( o Investigate Sign of F Verif that F(, 0 below line on line above line Look at etreme values of CS4600 0
11 F (, 0 The Picture above line F(, 0 below line Ke to Bresenham Algorithm Reasonable assumtions have reuce the roblem to making a binar choice at each iel: (net (Previous (net CS4600
12 Decision Variable (logical Define a logical ecision variable linear in form incrementall uate (with aition tells us whether to go or The Picture Q M mioint revious CS4600
13 The Picture (again (, Q M mioint (, (, revious (, Observe the relationshis Suose Q is above M, as before. Then F( M 0 F( M 0, M is below the line So, means line is above M, Nee to move, increase value CS4600 3
14 The Picture (again (, mioint (, Q (, revious (, Observe the relationshis Suose Q is below M, as before. Then F(M < 0, imlies M is above the line So, F(M < 0, means line is below M, Nee to move to ; on t increase CS4600 4
15 M = Mioint = (, : Want to evaluate at M Will use an incremental ecision variable Let, F(, a( b( c Let, Therefore, How will be use? a( b( c (mioint below ieal line (mioint above ieal line (arbitrar CS4600 5
16 CS Case : Suose is chosen Recall... c b a ol ( ( c b a F new ( ( (, : ;,... a c b a c b a ol new ol new ( ( ( ( Case : Suose is chosen
17 CS a c b a c b a ol new ol new ( ( ( ( Case : Suose is chosen... a c b a c b a ol new ol new ( ( ( ( Case : Suose is chosen
18 Case : Suose is chosen new ol a( b( c... a( b( c new ol a Review of licit to Imlicit Recall, Or, B 0 ( ( ( B 0 F(, ( ( ( B 0 where, a ( ; b ( ; c B( CS4600 8
19 Case :. a new ol increment we a if is chosen. So, a. But remember that a (from line equations. Hence, FM ( is not evaluate elicitl. We siml a a to uate for Case : Suose chosen Recall ol a( b( c an, : ;,... 3 new F(, 3 a ( b ( c CS4600 9
20 Case : Suose new ol... new a( a( ol b( a b b( 3 c c So, ab Case :. new ol increment that we a if is chosen. ab. But remember that a, an b (from line equations. Hence, FM ( is not evaluate elicitl. We siml a ab to uate for CS4600 0
21 Case :. new ol ab a b, where a, an b means, we siml a a b, i.e., to uate for. Summar At each ste of the roceure, we must choose between moving or base on the sign of the ecision variable Then uate accoring to, where, where, or CS4600
22 What is initial value of? First oint is First mioint is ( 0, 0 ( 0, 0 What is initial mioint value? ( 0, 0 F( 0, 0 What is initial value of? F( 0, 0 a( 0 b( 0 c b a0 b0 ca b F( 0, 0 a CS4600
23 What is initial value of? Note, F(, 0, since (, is on line Hence, F( 0, 0 0 a b ( What Does Factor of Do? Has the same 0-set F(, ( a b c 0 Changes the sloe of the lane Rotates lane about the 0-set line Gets ri of the enominator CS4600 3
24 What is initial value of? Note, we can clear enominator an not change line, F( 0, 0 ( What is initial value of? F(, ( abc 0 So, first value of ( ( CS4600 4
25 More Summar Initial value ( ( Case :, where ( Case : where {( ( } Note, all eltas are constants, More Summar if 0 Choose otherwise CS4600 5
26 amle Line en oints: ( (5,8;, (9, 0, 0 ( Deltas: = 4; = 3 Grah CS4600 6
27 amle ( = 4; = 3 Initial value of (5, 8 ( ( 640 = ( ( if 0 otherwise Grah CS4600 7
28 amle (=4; =3 Uate value of where Last move was, so ( - (3 4 0, where (, {( ( } if 0 otherwise Grah CS4600 8
29 amle (=4; =3 Previous move was where, where (, {( ( } ( ( if 0 otherwise Grah CS4600 9
30 amle (=4; =3, Previous move was where ( - ( , where ( {( ( } if 0 otherwise Grah CS
31 Grah Meeting Bresenham Criteria Case 0: m0; m trivial cases Case : Case : 0 m fli about -ais m fli about CS4600 3
32 Case 0: Trivial Situations m 0 horizontal line m line Do not nee Bresenham Case : Fli about -ais ( 0, 0 ( 0, 0 (, (, CS4600 3
33 Case : Fli about -ais Suose, 0 m, Fli about -ais ( : (, (, ; (, (, How o sloes relate? m m i ; b efinition Since, m i ( 0 0 CS
34 How o sloes relate? i.e., m m m ( m 0 m Case 3: Fli about line = (, ( 0, 0 CS
35 Case 3: Fli about line = m B, swa an rime them, mb, mb Case 3: m =? B, m m an, m m 0 m CS
36 More Raster Line Issues Fat lines with multile iel with Smmetric lines How shoul en t geometr look? Generating curves, e.g., circles, etc. Jaggies, staircase effect, aliasing... Piel Sace CS
37 amle amle CS
38 Bresenham Circles The n Bresenham s Algorithm CS
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