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Rasterization

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youssef ramzy

The document discusses different algorithms for computer graphics rasterization. It covers line drawing algorithms like the straightforward approach, DDA, midpoint, and Bresenham's. It also discusses circle drawing using the parametric polar form equation and triangle rasterization which examines pixels using edge equations and colors them using barycentric coordinates.

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Rasterization
Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
o Line Drawing 1) Algorithms 1) A straightforward line drawing algorithm & The incremental DDA Algorithm . 2) The Midpoint Line Drawing Algorithm - Pitteway (1967) 3) The Bresenham Line Drawing Algorithm 4) The Double Step Midpoint Line Drawing Algorithm 2) Some points worth considering o Circle Drawing . 1) The Symmetry of a Circle 2) A Straightforward Circle Drawing Algorithm 3) Use of parametric polar form equation o Ellipses . o Triangle Rasterization: 1) Targets & Steps . 2) Edge walking idea(steps). 3) Edge equations 4) Which pixels will be examined? 5) How pixel is colored ? 6) Barycentric coordinates 7) Coloring triangle 8) Shared edges Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses • The output of the Interpolation Block is series output , So this block will be slow as it includes four modules (2 Dividers & 2 FIFOs) .
o Architecture of Rasterization ? o Line Drawing 1) Algorithms 1) A straightforward line drawing algorithm & The incremental DDA Algorithm . 2) The Midpoint Line Drawing Algorithm - Pitteway (1967) 3) The Bresenham Line Drawing Algorithm 4) The Double Step Midpoint Line Drawing Algorithm 2) Some points worth considering o Circle Drawing . 1) The Symmetry of a Circle 2) A Straightforward Circle Drawing Algorithm 3) Use of parametric polar form equation o Ellipses . o Triangle Rasterization: 1) Targets & Steps . 2) Edge walking idea(steps). 3) Edge equations 4) Which pixels will be examined? 5) How pixel is colored ? 6) Barycentric coordinates 7) Coloring triangle 8) Shared edges Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
o Architecture of Rasterization ? o Line Drawing 1) Algorithms 1) A straightforward line drawing algorithm & The incremental DDA Algorithm . 2) The Midpoint Line Drawing Algorithm - Pitteway (1967) 3) The Bresenham Line Drawing Algorithm 4) The Double Step Midpoint Line Drawing Algorithm 2) Some points worth considering o Circle Drawing . 1) The Symmetry of a Circle 2) A Straightforward Circle Drawing Algorithm 3) Use of parametric polar form equation o Ellipses . o Triangle Rasterization: 1) Targets & Steps . 2) Edge walking idea(steps). 3) Edge equations 4) Which pixels will be examined? 5) How pixel is colored ? 6) Barycentric coordinates 7) Coloring triangle 8) Shared edges Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
A straightforward line drawing algorithm Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
The incremental DDA Algorithm Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses 𝑥𝑖+1 = 𝑥𝑖 + 𝑦𝑖+1 = 𝑦𝑖 + 𝑚
The incremental DDA Algorithm (Cont.) Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
o Architecture of Rasterization ? o Line Drawing 1) Algorithms 1) A straightforward line drawing algorithm & The incremental DDA Algorithm . 2) The Midpoint Line Drawing Algorithm - Pitteway (1967) 3) The Bresenham Line Drawing Algorithm 4) The Double Step Midpoint Line Drawing Algorithm 2) Some points worth considering o Circle Drawing . 1) The Symmetry of a Circle 2) A Straightforward Circle Drawing Algorithm 3) Use of parametric polar form equation o Ellipses . o Triangle Rasterization: 1) Targets & Steps . 2) Edge walking idea(steps). 3) Edge equations 4) Which pixels will be examined? 5) How pixel is colored ? 6) Barycentric coordinates 7) Coloring triangle 8) Shared edges Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses • Data we have: start point between 𝑥 𝑎 , 𝑦𝑎 and 𝑥 𝑏 , 𝑦 𝑏
Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
o Architecture of Rasterization ? o Line Drawing 1) Algorithms 1) A straightforward line drawing algorithm & The incremental DDA Algorithm . 2) The Midpoint Line Drawing Algorithm - Pitteway (1967) 3) The Bresenham Line Drawing Algorithm 4) The Double Step Midpoint Line Drawing Algorithm 2) Some points worth considering o Circle Drawing . 1) The Symmetry of a Circle 2) A Straightforward Circle Drawing Algorithm 3) Use of parametric polar form equation o Ellipses . o Triangle Rasterization: 1) Targets & Steps . 2) Edge walking idea(steps). 3) Edge equations 4) Which pixels will be examined? 5) How pixel is colored ? 6) Barycentric coordinates 7) Coloring triangle 8) Shared edges Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
o Architecture of Rasterization ? o Line Drawing 1) Algorithms 1) A straightforward line drawing algorithm & The incremental DDA Algorithm . 2) The Midpoint Line Drawing Algorithm - Pitteway (1967) 3) The Bresenham Line Drawing Algorithm 4) The Double Step Midpoint Line Drawing Algorithm 2) Some points worth considering o Circle Drawing . 1) The Symmetry of a Circle 2) A Straightforward Circle Drawing Algorithm 3) Use of parametric polar form equation o Ellipses . o Triangle Rasterization: 1) Targets & Steps . 2) Edge walking idea(steps). 3) Edge equations 4) Which pixels will be examined? 5) How pixel is colored ? 6) Barycentric coordinates 7) Coloring triangle 8) Shared edges Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
o Architecture of Rasterization ? o Line Drawing 1) Algorithms 1) A straightforward line drawing algorithm & The incremental DDA Algorithm . 2) The Midpoint Line Drawing Algorithm - Pitteway (1967) 3) The Bresenham Line Drawing Algorithm 4) The Double Step Midpoint Line Drawing Algorithm 2) Some points worth considering o Circle Drawing . 1) The Symmetry of a Circle 2) A Straightforward Circle Drawing Algorithm 3) Use of parametric polar form equation o Ellipses . o Triangle Rasterization: 1) Targets & Steps . 2) Edge walking idea(steps). 3) Edge equations 4) Which pixels will be examined? 5) How pixel is colored ? 6) Barycentric coordinates 7) Coloring triangle 8) Shared edges Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
𝑥 𝑏 , 𝑦 𝑏 to 𝑥 𝑎 , 𝑦𝑎 Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
o Architecture of Rasterization ? o Line Drawing 1) Algorithms 1) A straightforward line drawing algorithm & The incremental DDA Algorithm . 2) The Midpoint Line Drawing Algorithm - Pitteway (1967) 3) The Bresenham Line Drawing Algorithm 4) The Double Step Midpoint Line Drawing Algorithm 2) Some points worth considering o Circle Drawing . 1) The Symmetry of a Circle 2) A Straightforward Circle Drawing Algorithm 3) Use of parametric polar form equation o Ellipses . o Triangle Rasterization: 1) Targets & Steps . 2) Edge walking idea(steps). 3) Edge equations 4) Which pixels will be examined? 5) How pixel is colored ? 6) Barycentric coordinates 7) Coloring triangle 8) Shared edges Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
o Architecture of Rasterization ? o Line Drawing 1) Algorithms 1) A straightforward line drawing algorithm & The incremental DDA Algorithm . 2) The Midpoint Line Drawing Algorithm - Pitteway (1967) 3) The Bresenham Line Drawing Algorithm 4) The Double Step Midpoint Line Drawing Algorithm 2) Some points worth considering o Circle Drawing . 1) The Symmetry of a Circle 2) A Straightforward Circle Drawing Algorithm 3) Use of parametric polar form equation o Ellipses . o Triangle Rasterization: 1) Targets & Steps . 2) Edge walking idea(steps). 3) Edge equations 4) Which pixels will be examined? 5) How pixel is colored ? 6) Barycentric coordinates 7) Coloring triangle 8) Shared edges Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
o Architecture of Rasterization ? o Line Drawing 1) Algorithms 1) A straightforward line drawing algorithm & The incremental DDA Algorithm . 2) The Midpoint Line Drawing Algorithm - Pitteway (1967) 3) The Bresenham Line Drawing Algorithm 4) The Double Step Midpoint Line Drawing Algorithm 2) Some points worth considering o Circle Drawing . 1) The Symmetry of a Circle 2) A Straightforward Circle Drawing Algorithm 3) Use of parametric polar form equation o Ellipses . o Triangle Rasterization: 1) Targets & Steps . 2) Edge walking idea(steps). 3) Edge equations 4) Which pixels will be examined? 5) How pixel is colored ? 6) Barycentric coordinates 7) Coloring triangle 8) Shared edges Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
• The equation of a circle may be expressed in parametric polar form as follows: 𝑥 = 𝑥 𝑐 + 𝑟 𝑐𝑜𝑠𝜃 𝑦 = 𝑦𝑐 + 𝑟 𝑠𝑖𝑛𝜃 Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
o Architecture of Rasterization ? o Line Drawing 1) Algorithms 1) A straightforward line drawing algorithm & The incremental DDA Algorithm . 2) The Midpoint Line Drawing Algorithm - Pitteway (1967) 3) The Bresenham Line Drawing Algorithm 4) The Double Step Midpoint Line Drawing Algorithm 2) Some points worth considering o Circle Drawing . 1) The Symmetry of a Circle 2) A Straightforward Circle Drawing Algorithm 3) Use of parametric polar form equation o Ellipses . o Triangle Rasterization: 1) Targets & Steps . 2) Edge walking idea(steps). 3) Edge equations 4) Which pixels will be examined? 5) How pixel is colored ? 6) Barycentric coordinates 7) Coloring triangle 8) Shared edges Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
Ellipses ( 𝑥 − 𝑥 𝑐 𝑟1 )2+( 𝑦 − 𝑦𝑐 𝑟2 )2= 1 Or 𝑦2 = 𝑟2 2(1 − 𝑥2 𝑟1 2 ) 𝑥 = 𝑥 𝑐 + 𝑟 𝑐𝑜𝑠𝜃 𝑦 = 𝑦𝑐 + 𝑟 𝑠𝑖𝑛𝜃 Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
o Architecture of Rasterization ? o Line Drawing 1) Algorithms 1) A straightforward line drawing algorithm & The incremental DDA Algorithm . 2) The Midpoint Line Drawing Algorithm - Pitteway (1967) 3) The Bresenham Line Drawing Algorithm 4) The Double Step Midpoint Line Drawing Algorithm 2) Some points worth considering o Circle Drawing . 1) The Symmetry of a Circle 2) A Straightforward Circle Drawing Algorithm 3) Use of parametric polar form equation o Ellipses . o Triangle Rasterization: 1) Targets & Steps . 2) Edge walking idea(steps). 3) Edge equations 4) Which pixels will be examined? 5) How pixel is colored ? 6) Barycentric coordinates 7) Coloring triangle 8) Shared edges Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
Targets Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
Steps Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
o Architecture of Rasterization ? o Line Drawing 1) Algorithms 1) A straightforward line drawing algorithm & The incremental DDA Algorithm . 2) The Midpoint Line Drawing Algorithm - Pitteway (1967) 3) The Bresenham Line Drawing Algorithm 4) The Double Step Midpoint Line Drawing Algorithm 2) Some points worth considering o Circle Drawing . 1) The Symmetry of a Circle 2) A Straightforward Circle Drawing Algorithm 3) Use of parametric polar form equation o Ellipses . o Triangle Rasterization: 1) Targets & Steps . 2) Edge walking idea(steps). 3) Edge equations 4) Which pixels will be examined? 5) How pixel is colored ? 6) Barycentric coordinates 7) Coloring triangle 8) Shared edges Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
o Architecture of Rasterization ? o Line Drawing 1) Algorithms 1) A straightforward line drawing algorithm & The incremental DDA Algorithm . 2) The Midpoint Line Drawing Algorithm - Pitteway (1967) 3) The Bresenham Line Drawing Algorithm 4) The Double Step Midpoint Line Drawing Algorithm 2) Some points worth considering o Circle Drawing . 1) The Symmetry of a Circle 2) A Straightforward Circle Drawing Algorithm 3) Use of parametric polar form equation o Ellipses . o Triangle Rasterization: 1) Targets & Steps . 2) Edge walking idea(steps). 3) Edge equations 4) Which pixels will be examined? 5) How pixel is colored ? 6) Barycentric coordinates 7) Coloring triangle 8) Shared edges Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
o Architecture of Rasterization ? o Line Drawing 1) Algorithms 1) A straightforward line drawing algorithm & The incremental DDA Algorithm . 2) The Midpoint Line Drawing Algorithm - Pitteway (1967) 3) The Bresenham Line Drawing Algorithm 4) The Double Step Midpoint Line Drawing Algorithm 2) Some points worth considering o Circle Drawing . 1) The Symmetry of a Circle 2) A Straightforward Circle Drawing Algorithm 3) Use of parametric polar form equation o Ellipses . o Triangle Rasterization: 1) Targets & Steps . 2) Edge walking idea(steps). 3) Edge equations 4) Which pixels will be examined? 5) How pixel is colored ? 6) Barycentric coordinates 7) Coloring triangle 8) Shared edges Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
o Architecture of Rasterization ? o Line Drawing 1) Algorithms 1) A straightforward line drawing algorithm & The incremental DDA Algorithm . 2) The Midpoint Line Drawing Algorithm - Pitteway (1967) 3) The Bresenham Line Drawing Algorithm 4) The Double Step Midpoint Line Drawing Algorithm 2) Some points worth considering o Circle Drawing . 1) The Symmetry of a Circle 2) A Straightforward Circle Drawing Algorithm 3) Use of parametric polar form equation o Ellipses . o Triangle Rasterization: 1) Targets & Steps . 2) Edge walking idea(steps). 3) Edge equations 4) Which pixels will be examined? 5) How pixel is colored ? 6) Barycentric coordinates 7) Coloring triangle 8) Shared edges Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
o Architecture of Rasterization ? o Line Drawing 1) Algorithms 1) A straightforward line drawing algorithm & The incremental DDA Algorithm . 2) The Midpoint Line Drawing Algorithm - Pitteway (1967) 3) The Bresenham Line Drawing Algorithm 4) The Double Step Midpoint Line Drawing Algorithm 2) Some points worth considering o Circle Drawing . 1) The Symmetry of a Circle 2) A Straightforward Circle Drawing Algorithm 3) Use of parametric polar form equation o Ellipses . o Triangle Rasterization: 1) Targets & Steps . 2) Edge walking idea(steps). 3) Edge equations 4) Which pixels will be examined? 5) How pixel is colored ? 6) Barycentric coordinates 7) Coloring triangle 8) Shared edges Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
o Architecture of Rasterization ? o Line Drawing 1) Algorithms 1) A straightforward line drawing algorithm & The incremental DDA Algorithm . 2) The Midpoint Line Drawing Algorithm - Pitteway (1967) 3) The Bresenham Line Drawing Algorithm 4) The Double Step Midpoint Line Drawing Algorithm 2) Some points worth considering o Circle Drawing . 1) The Symmetry of a Circle 2) A Straightforward Circle Drawing Algorithm 3) Use of parametric polar form equation o Ellipses . o Triangle Rasterization: 1) Targets & Steps . 2) Edge walking idea(steps). 3) Edge equations 4) Which pixels will be examined? 5) How pixel is colored ? 6) Barycentric coordinates 7) Coloring triangle 8) Shared edges Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
o Architecture of Rasterization ? o Line Drawing 1) Algorithms 1) A straightforward line drawing algorithm & The incremental DDA Algorithm . 2) The Midpoint Line Drawing Algorithm - Pitteway (1967) 3) The Bresenham Line Drawing Algorithm 4) The Double Step Midpoint Line Drawing Algorithm 2) Some points worth considering o Circle Drawing . 1) The Symmetry of a Circle 2) A Straightforward Circle Drawing Algorithm 3) Use of parametric polar form equation o Ellipses . o Triangle Rasterization: 1) Targets & Steps . 2) Edge walking idea(steps). 3) Edge equations 4) Which pixels will be examined? 5) How pixel is colored ? 6) Barycentric coordinates 7) Coloring triangle 8) Shared edges Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses

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Rasterization

  • 2. Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 3. o Line Drawing 1) Algorithms 1) A straightforward line drawing algorithm & The incremental DDA Algorithm . 2) The Midpoint Line Drawing Algorithm - Pitteway (1967) 3) The Bresenham Line Drawing Algorithm 4) The Double Step Midpoint Line Drawing Algorithm 2) Some points worth considering o Circle Drawing . 1) The Symmetry of a Circle 2) A Straightforward Circle Drawing Algorithm 3) Use of parametric polar form equation o Ellipses . o Triangle Rasterization: 1) Targets & Steps . 2) Edge walking idea(steps). 3) Edge equations 4) Which pixels will be examined? 5) How pixel is colored ? 6) Barycentric coordinates 7) Coloring triangle 8) Shared edges Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 4. Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 5. Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses • The output of the Interpolation Block is series output , So this block will be slow as it includes four modules (2 Dividers & 2 FIFOs) .
  • 6. o Architecture of Rasterization ? o Line Drawing 1) Algorithms 1) A straightforward line drawing algorithm & The incremental DDA Algorithm . 2) The Midpoint Line Drawing Algorithm - Pitteway (1967) 3) The Bresenham Line Drawing Algorithm 4) The Double Step Midpoint Line Drawing Algorithm 2) Some points worth considering o Circle Drawing . 1) The Symmetry of a Circle 2) A Straightforward Circle Drawing Algorithm 3) Use of parametric polar form equation o Ellipses . o Triangle Rasterization: 1) Targets & Steps . 2) Edge walking idea(steps). 3) Edge equations 4) Which pixels will be examined? 5) How pixel is colored ? 6) Barycentric coordinates 7) Coloring triangle 8) Shared edges Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 7. Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 8. o Architecture of Rasterization ? o Line Drawing 1) Algorithms 1) A straightforward line drawing algorithm & The incremental DDA Algorithm . 2) The Midpoint Line Drawing Algorithm - Pitteway (1967) 3) The Bresenham Line Drawing Algorithm 4) The Double Step Midpoint Line Drawing Algorithm 2) Some points worth considering o Circle Drawing . 1) The Symmetry of a Circle 2) A Straightforward Circle Drawing Algorithm 3) Use of parametric polar form equation o Ellipses . o Triangle Rasterization: 1) Targets & Steps . 2) Edge walking idea(steps). 3) Edge equations 4) Which pixels will be examined? 5) How pixel is colored ? 6) Barycentric coordinates 7) Coloring triangle 8) Shared edges Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 9. A straightforward line drawing algorithm Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 10. The incremental DDA Algorithm Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses 𝑥𝑖+1 = 𝑥𝑖 + 𝑦𝑖+1 = 𝑦𝑖 + 𝑚
  • 11. The incremental DDA Algorithm (Cont.) Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 12. o Architecture of Rasterization ? o Line Drawing 1) Algorithms 1) A straightforward line drawing algorithm & The incremental DDA Algorithm . 2) The Midpoint Line Drawing Algorithm - Pitteway (1967) 3) The Bresenham Line Drawing Algorithm 4) The Double Step Midpoint Line Drawing Algorithm 2) Some points worth considering o Circle Drawing . 1) The Symmetry of a Circle 2) A Straightforward Circle Drawing Algorithm 3) Use of parametric polar form equation o Ellipses . o Triangle Rasterization: 1) Targets & Steps . 2) Edge walking idea(steps). 3) Edge equations 4) Which pixels will be examined? 5) How pixel is colored ? 6) Barycentric coordinates 7) Coloring triangle 8) Shared edges Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 13. Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses • Data we have: start point between 𝑥 𝑎 , 𝑦𝑎 and 𝑥 𝑏 , 𝑦 𝑏
  • 14. Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 15. Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 16. o Architecture of Rasterization ? o Line Drawing 1) Algorithms 1) A straightforward line drawing algorithm & The incremental DDA Algorithm . 2) The Midpoint Line Drawing Algorithm - Pitteway (1967) 3) The Bresenham Line Drawing Algorithm 4) The Double Step Midpoint Line Drawing Algorithm 2) Some points worth considering o Circle Drawing . 1) The Symmetry of a Circle 2) A Straightforward Circle Drawing Algorithm 3) Use of parametric polar form equation o Ellipses . o Triangle Rasterization: 1) Targets & Steps . 2) Edge walking idea(steps). 3) Edge equations 4) Which pixels will be examined? 5) How pixel is colored ? 6) Barycentric coordinates 7) Coloring triangle 8) Shared edges Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 17. Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 18. Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 19. Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 20. o Architecture of Rasterization ? o Line Drawing 1) Algorithms 1) A straightforward line drawing algorithm & The incremental DDA Algorithm . 2) The Midpoint Line Drawing Algorithm - Pitteway (1967) 3) The Bresenham Line Drawing Algorithm 4) The Double Step Midpoint Line Drawing Algorithm 2) Some points worth considering o Circle Drawing . 1) The Symmetry of a Circle 2) A Straightforward Circle Drawing Algorithm 3) Use of parametric polar form equation o Ellipses . o Triangle Rasterization: 1) Targets & Steps . 2) Edge walking idea(steps). 3) Edge equations 4) Which pixels will be examined? 5) How pixel is colored ? 6) Barycentric coordinates 7) Coloring triangle 8) Shared edges Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 21. Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 22. Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 23. o Architecture of Rasterization ? o Line Drawing 1) Algorithms 1) A straightforward line drawing algorithm & The incremental DDA Algorithm . 2) The Midpoint Line Drawing Algorithm - Pitteway (1967) 3) The Bresenham Line Drawing Algorithm 4) The Double Step Midpoint Line Drawing Algorithm 2) Some points worth considering o Circle Drawing . 1) The Symmetry of a Circle 2) A Straightforward Circle Drawing Algorithm 3) Use of parametric polar form equation o Ellipses . o Triangle Rasterization: 1) Targets & Steps . 2) Edge walking idea(steps). 3) Edge equations 4) Which pixels will be examined? 5) How pixel is colored ? 6) Barycentric coordinates 7) Coloring triangle 8) Shared edges Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 24. 𝑥 𝑏 , 𝑦 𝑏 to 𝑥 𝑎 , 𝑦𝑎 Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 25. o Architecture of Rasterization ? o Line Drawing 1) Algorithms 1) A straightforward line drawing algorithm & The incremental DDA Algorithm . 2) The Midpoint Line Drawing Algorithm - Pitteway (1967) 3) The Bresenham Line Drawing Algorithm 4) The Double Step Midpoint Line Drawing Algorithm 2) Some points worth considering o Circle Drawing . 1) The Symmetry of a Circle 2) A Straightforward Circle Drawing Algorithm 3) Use of parametric polar form equation o Ellipses . o Triangle Rasterization: 1) Targets & Steps . 2) Edge walking idea(steps). 3) Edge equations 4) Which pixels will be examined? 5) How pixel is colored ? 6) Barycentric coordinates 7) Coloring triangle 8) Shared edges Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 26. Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 27. o Architecture of Rasterization ? o Line Drawing 1) Algorithms 1) A straightforward line drawing algorithm & The incremental DDA Algorithm . 2) The Midpoint Line Drawing Algorithm - Pitteway (1967) 3) The Bresenham Line Drawing Algorithm 4) The Double Step Midpoint Line Drawing Algorithm 2) Some points worth considering o Circle Drawing . 1) The Symmetry of a Circle 2) A Straightforward Circle Drawing Algorithm 3) Use of parametric polar form equation o Ellipses . o Triangle Rasterization: 1) Targets & Steps . 2) Edge walking idea(steps). 3) Edge equations 4) Which pixels will be examined? 5) How pixel is colored ? 6) Barycentric coordinates 7) Coloring triangle 8) Shared edges Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 28. Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 29. o Architecture of Rasterization ? o Line Drawing 1) Algorithms 1) A straightforward line drawing algorithm & The incremental DDA Algorithm . 2) The Midpoint Line Drawing Algorithm - Pitteway (1967) 3) The Bresenham Line Drawing Algorithm 4) The Double Step Midpoint Line Drawing Algorithm 2) Some points worth considering o Circle Drawing . 1) The Symmetry of a Circle 2) A Straightforward Circle Drawing Algorithm 3) Use of parametric polar form equation o Ellipses . o Triangle Rasterization: 1) Targets & Steps . 2) Edge walking idea(steps). 3) Edge equations 4) Which pixels will be examined? 5) How pixel is colored ? 6) Barycentric coordinates 7) Coloring triangle 8) Shared edges Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 30. • The equation of a circle may be expressed in parametric polar form as follows: 𝑥 = 𝑥 𝑐 + 𝑟 𝑐𝑜𝑠𝜃 𝑦 = 𝑦𝑐 + 𝑟 𝑠𝑖𝑛𝜃 Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 31. o Architecture of Rasterization ? o Line Drawing 1) Algorithms 1) A straightforward line drawing algorithm & The incremental DDA Algorithm . 2) The Midpoint Line Drawing Algorithm - Pitteway (1967) 3) The Bresenham Line Drawing Algorithm 4) The Double Step Midpoint Line Drawing Algorithm 2) Some points worth considering o Circle Drawing . 1) The Symmetry of a Circle 2) A Straightforward Circle Drawing Algorithm 3) Use of parametric polar form equation o Ellipses . o Triangle Rasterization: 1) Targets & Steps . 2) Edge walking idea(steps). 3) Edge equations 4) Which pixels will be examined? 5) How pixel is colored ? 6) Barycentric coordinates 7) Coloring triangle 8) Shared edges Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 32. Ellipses ( 𝑥 − 𝑥 𝑐 𝑟1 )2+( 𝑦 − 𝑦𝑐 𝑟2 )2= 1 Or 𝑦2 = 𝑟2 2(1 − 𝑥2 𝑟1 2 ) 𝑥 = 𝑥 𝑐 + 𝑟 𝑐𝑜𝑠𝜃 𝑦 = 𝑦𝑐 + 𝑟 𝑠𝑖𝑛𝜃 Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 33. o Architecture of Rasterization ? o Line Drawing 1) Algorithms 1) A straightforward line drawing algorithm & The incremental DDA Algorithm . 2) The Midpoint Line Drawing Algorithm - Pitteway (1967) 3) The Bresenham Line Drawing Algorithm 4) The Double Step Midpoint Line Drawing Algorithm 2) Some points worth considering o Circle Drawing . 1) The Symmetry of a Circle 2) A Straightforward Circle Drawing Algorithm 3) Use of parametric polar form equation o Ellipses . o Triangle Rasterization: 1) Targets & Steps . 2) Edge walking idea(steps). 3) Edge equations 4) Which pixels will be examined? 5) How pixel is colored ? 6) Barycentric coordinates 7) Coloring triangle 8) Shared edges Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 34. Targets Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 35. Steps Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 36. o Architecture of Rasterization ? o Line Drawing 1) Algorithms 1) A straightforward line drawing algorithm & The incremental DDA Algorithm . 2) The Midpoint Line Drawing Algorithm - Pitteway (1967) 3) The Bresenham Line Drawing Algorithm 4) The Double Step Midpoint Line Drawing Algorithm 2) Some points worth considering o Circle Drawing . 1) The Symmetry of a Circle 2) A Straightforward Circle Drawing Algorithm 3) Use of parametric polar form equation o Ellipses . o Triangle Rasterization: 1) Targets & Steps . 2) Edge walking idea(steps). 3) Edge equations 4) Which pixels will be examined? 5) How pixel is colored ? 6) Barycentric coordinates 7) Coloring triangle 8) Shared edges Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 37. Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 38. Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 39. Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 40. o Architecture of Rasterization ? o Line Drawing 1) Algorithms 1) A straightforward line drawing algorithm & The incremental DDA Algorithm . 2) The Midpoint Line Drawing Algorithm - Pitteway (1967) 3) The Bresenham Line Drawing Algorithm 4) The Double Step Midpoint Line Drawing Algorithm 2) Some points worth considering o Circle Drawing . 1) The Symmetry of a Circle 2) A Straightforward Circle Drawing Algorithm 3) Use of parametric polar form equation o Ellipses . o Triangle Rasterization: 1) Targets & Steps . 2) Edge walking idea(steps). 3) Edge equations 4) Which pixels will be examined? 5) How pixel is colored ? 6) Barycentric coordinates 7) Coloring triangle 8) Shared edges Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 41. Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 42. Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 43. o Architecture of Rasterization ? o Line Drawing 1) Algorithms 1) A straightforward line drawing algorithm & The incremental DDA Algorithm . 2) The Midpoint Line Drawing Algorithm - Pitteway (1967) 3) The Bresenham Line Drawing Algorithm 4) The Double Step Midpoint Line Drawing Algorithm 2) Some points worth considering o Circle Drawing . 1) The Symmetry of a Circle 2) A Straightforward Circle Drawing Algorithm 3) Use of parametric polar form equation o Ellipses . o Triangle Rasterization: 1) Targets & Steps . 2) Edge walking idea(steps). 3) Edge equations 4) Which pixels will be examined? 5) How pixel is colored ? 6) Barycentric coordinates 7) Coloring triangle 8) Shared edges Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 44. Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 45. Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 46. o Architecture of Rasterization ? o Line Drawing 1) Algorithms 1) A straightforward line drawing algorithm & The incremental DDA Algorithm . 2) The Midpoint Line Drawing Algorithm - Pitteway (1967) 3) The Bresenham Line Drawing Algorithm 4) The Double Step Midpoint Line Drawing Algorithm 2) Some points worth considering o Circle Drawing . 1) The Symmetry of a Circle 2) A Straightforward Circle Drawing Algorithm 3) Use of parametric polar form equation o Ellipses . o Triangle Rasterization: 1) Targets & Steps . 2) Edge walking idea(steps). 3) Edge equations 4) Which pixels will be examined? 5) How pixel is colored ? 6) Barycentric coordinates 7) Coloring triangle 8) Shared edges Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 47. Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 48. o Architecture of Rasterization ? o Line Drawing 1) Algorithms 1) A straightforward line drawing algorithm & The incremental DDA Algorithm . 2) The Midpoint Line Drawing Algorithm - Pitteway (1967) 3) The Bresenham Line Drawing Algorithm 4) The Double Step Midpoint Line Drawing Algorithm 2) Some points worth considering o Circle Drawing . 1) The Symmetry of a Circle 2) A Straightforward Circle Drawing Algorithm 3) Use of parametric polar form equation o Ellipses . o Triangle Rasterization: 1) Targets & Steps . 2) Edge walking idea(steps). 3) Edge equations 4) Which pixels will be examined? 5) How pixel is colored ? 6) Barycentric coordinates 7) Coloring triangle 8) Shared edges Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 49. Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 50. Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 51. Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 52. Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 53. Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 54. o Architecture of Rasterization ? o Line Drawing 1) Algorithms 1) A straightforward line drawing algorithm & The incremental DDA Algorithm . 2) The Midpoint Line Drawing Algorithm - Pitteway (1967) 3) The Bresenham Line Drawing Algorithm 4) The Double Step Midpoint Line Drawing Algorithm 2) Some points worth considering o Circle Drawing . 1) The Symmetry of a Circle 2) A Straightforward Circle Drawing Algorithm 3) Use of parametric polar form equation o Ellipses . o Triangle Rasterization: 1) Targets & Steps . 2) Edge walking idea(steps). 3) Edge equations 4) Which pixels will be examined? 5) How pixel is colored ? 6) Barycentric coordinates 7) Coloring triangle 8) Shared edges Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 55. Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 56. o Architecture of Rasterization ? o Line Drawing 1) Algorithms 1) A straightforward line drawing algorithm & The incremental DDA Algorithm . 2) The Midpoint Line Drawing Algorithm - Pitteway (1967) 3) The Bresenham Line Drawing Algorithm 4) The Double Step Midpoint Line Drawing Algorithm 2) Some points worth considering o Circle Drawing . 1) The Symmetry of a Circle 2) A Straightforward Circle Drawing Algorithm 3) Use of parametric polar form equation o Ellipses . o Triangle Rasterization: 1) Targets & Steps . 2) Edge walking idea(steps). 3) Edge equations 4) Which pixels will be examined? 5) How pixel is colored ? 6) Barycentric coordinates 7) Coloring triangle 8) Shared edges Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 57. Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses
  • 58. Introduction Architecture Line Drawing Triangle RasterizationCircle Drawing Ellipses