Computação Gráfica 2D
2D Computer Graphics

Este é um curso introdutório de computação gráfica 2D. O tema central é a renderização de ilustrações vetoriais. Ao final do curso, os alunos compreenderão completamente, tanto na teoria como na prática, todas as etapas necessárias para produzir imagens de alta qualidade a partir de ilustrações vetoriais. Para melhor aproveitar o curso, os alunos devem estar previamente familiarizados com álgebra linear, cálculo, e ter conhecimentos de programação em Lua ou C++.

This is an introductory course in 2D Computer Graphics. The main topic of study is vector graphics rendering. By the end of the course, students will thoroughly understand, both in theory and in practice, all the steps required to produce high-quality renderings of vector graphics illustrations. Students are expected to be familiar with linear algebra, calculus, and have basic computer programming skills in Lua or C++.


Class dates and locations:
Mon (Room 347), Wed (Room 347), Fri (Room 236)

Thu (Room 232, exceptionally Room 333 on 30/01)

Discussion list:

Prof. Diego Nehab
Office: 410
E-mail: diego@...

Teaching assistant:
Pedro Arthur dos Santos Souza
Office: Visgraf
E-mail: pasouza@...


Practical aspects will be tested by programming assignments distributed throughout the course. Assignemts will be based on software infrastructure we will provide. Students without a solid programming experience can use the Lua Programming Language. More advanced students can use C++ instead.

Theory will evaluated with a final exam at the end of the course.

The final grade will be formed by the assignments (60%), the exam (30%), and participation in class (10%).


Date Topic Details
1. M 06/01 Introduction Reading list:
class slides,
Lua [PIL],
line rasterization [Bresenham1965],
polygon rasterization (edge flag) [AcklandWeste1981],
polygon rasterization (active-edge list) [WylieEtAl1967],
error diffusion dithering [FloydSteinberg1976],
general books covering computer graphics [FoleyVanDam1990] [GomesVelho2008]
2. W 08/01 Geometry and transformations Warmup assignment posted: run examples from class
Reading list:
class slides,
Conics, affine geometry, and projective geometry [BrannanEtAl2012, chapters 0—3],
transformations [Mathematica: xforms]
3. F 10/01 Vector graphics Warmup due
Assignment 1 posted: triangles, circles, and polygons
Reading list:
class slides,
PostScript tutorial [PSBlueBook],
PostScript reference [PSRedBook],
PDF reference [PDF],
OpenXPS reference [OpenXPS],
SVG reference [SVG],
OpenVG API [OpenVG],
NV Path Rendering API [NVPR],
inside-outside test for triangles and polygons [Mathematica: polygons],
inside-outside test, center, and bounding-box for ellipses [Mathematica: ellipses]
4. M 13/01 Bézier curves Reading list:
class slides,
integral Bézier curves [Mathematica: Bézier],
general books covering these topics [Sederberg2014, chapters 1—3] [Farin2002, chapters 1—6]
5. W 15/01 More on Bézier curves Reading list:
class slides,
blossoms (multiaffine maps) [Mathematica: Multiaffine maps],
rational quadratic Bézier curves [Mathematica: Rational Bézier],
general books covering these topics [Sederberg2014, chapters 1—3] [Farin2002, chapters 13]
6. F 17/01 Floating-point and root-finding Assignment 1 due
Assignment 2 posted: paths
Reading list:
class slides
floating-point [MullerEtAl2010],
finding roots [Mathematica: roots],
solving quadratics [Blinn2005],
solving cubics [Blinn2007],
root finding [Press2007, chapter 9],
root finding in Bernstein form [Sederberg2014, chapter 9],
integration [Press2007, chapter 4]
7. M 20/01 Color and compositing Reading list:
class slides,
great webpage covering color [Hardprint],
sRGB color model [sRGB],
seminal works on compositing [Wallace1981] [PorterDuff1984],
FAQs on color and gamma [ColorFAQ] [GammaFAQ],
general book covering these topics [GomesVelho2008, chapters 5 and 17]
8. W 22/01 Gradient paints Reading list:
class slides,
EPS/PDF gradient samples,
[PSRedBook, section 4.9.3],
PDF reference [PDF, section 4.6],
SVG reference [SVG, chapter 13]
9. F 24/01 Differential geometry Assignment 2 due
Assignment 3 posted: transparency and gradients
List 1 posted
Reading list:
class slides,
integral Bézier curves [Mathematica: Bézier],
rational quadratic Bézier curves [Mathematica: Rational Bézier],
general books covering these topics [Kreyszig1991, chapters 1–2], [ONeil2006, chapters 1–3],
offset curves [Sederberg2014, chapter 8], [Hoschek1988],
arc-length approximation [Juttler1997]
10. M 27/01 Resultants and implicitization Reading list:
class slides,
resultants for CAGD [MontaudouinTiller1984] [GoldmanEtAl1984],
resultants [Mathematica: resultants],
implicitization by affine functionals [LoopBlinn2005] [LoopBlinn2007],
implicitizing integral quadratics [Mathematica: implicit quadratics],
implicitizing rational quadratics [Mathematica: implicit rational quadratics],
implicitizing integral cubics [Mathematica: implicit cubics]
11. W 29/01 Inflection points and double points Reading list:
class slides,
computing inflection points and double-points [Mathematica: inflection-double],
tensor notation [Blinn1992a] [Blinn1992b],
cubic inflection points [Blinn1999]
12. F 31/01 Abstract segments Assignment 3 due
Reading list:
class slides,
abstract segments [LoopBlinn2005] [LoopBlinn2007] [GanacimEtAl2014]
13. M 03/02 Digital images and anti-aliasing Assignment 4 posted: implicit intersection tests
Assignment 4 addition: add supersampling
Reading list:
class slides,
analytic anti-aliasing [Duff1989] [MansonSchaefer2013],
anti-aliasing and gamma, analytic anti-aliasing approximation [NehabHoppe2008],
Fourier transforms [Teixeira2001],
perceptual image metric [ZhangWandell1996]
14. W 05/02 Texture mapping Assignment 4 addition: add texture mapping
Reading list:
class slides
Monte Carlo methods [Anderson1999],
sampling and reconstruction [NehabHoppe2014]
15. F 07/02 Acceleration datastructures List 2 posted
Reading list:
polygon clipping [SutherlandHodgman1974],
Quad trees [Warnock1969],
k-d trees [Bentley1975] [FriedmanEtAl1977],
BSP trees [FuchsEtAl1980],
R-trees (BVH) [Guttman1984]
16. M 10/02 The shortcut tree and regular grid Reading list:
lattice clipping [NehabHoppe2008],
“tripod” line rasterization [Cohen1994],
shortcut tree [GanacimEtAl2014]
17. W 12/02 Typesetting Assignment 4 due
Group assignment 5 posted: accelerated rendering
Readling list:
caligraphy video,
Making manuscripts video,
Jaquet Droz's writer automaton video,
KUKA scribe video,
METAFONT tutorial [Grandsire2004],
Gutenberg's printing press video,
Linotype machine video,
OpenType specification [OpenType],
FreeType library [FreeType],
HarfBuzz library [HarfBuzz],
Unicode standard [Unicode],
hyphenation and justification used in TeX [KnuthPlass1981],
microtypography in TeX [Thanh2000],
general book covering these topics [Felici2012]
18. F 14/02 Computational geometry Reading list:
reentrant polygon clipping [SutherlandHodgman1974],
general polygon clipping [Vatti1992],
triangulation by ear-clipping [Meisters1975],
plane-sweep convex hull [Andrew1979],
plane-sweep all segment intersections [BentleyOttmann1979],
plane-sweep closest pair of points [HinrichsEtAl1988],
plane-sweep Voronoi diagram [Fortune1986],
general books covering these topics [deFigueiredoCarvalho1991] [deBergEtAl2008]
19. M 17/02 Stroked primitives Reading list:
SVG stroke properties [SVG, chapter 11.4],
OpenXPS stroke rendering [OpenXPS, chapter 18.6],
arc-length approximation [Juttler1997],
offset (and evolute) approximation [Hoschek1988],
GPU-accelerated path rendering [KilgardBolz2012],
differential geometry of stroking [Mathematica: stroke],
joining flattened offsets [Mathematica: join],
20. W 19/02 Screen space effects (blur, clipping)
21. T 20/02 Other rendering algorithms
22. F 21/02 Test Group assignment 5 due


May require Inkscape or Wolfram CDF Player.

Lua and C++ materials C++ source-code for Lua modules and the C++ framework. Sample vector graphics inputs in the RVG format. Ground-truth renderings of RVG inputs. Corresponding SVG files. A collection of freely avaialable fonts.
docker image. A Docker image configured for the course (sources). Win64 libraries and Lua executable (Requires the Visual Studio Runtime.)


xforms-1.0.cdf Linear, affine, and projective transformations.
polygons-1.0.cdf Inside-outside tests for triangles and polygons, winding rules.
ellipses-1.0.cdf Inside-outside test, transformation, center, and bounding-box for implicit ellipses.
bezier-1.0.cdf Bézier curves, polynomial basis (Bernstein and Power), De Casteljau, tangents,
affine reparametrization, subdivision, monotonization, flattening,
silly bisection, curvature, oscullating circle.
rational-bezier-1.1.cdf Same and more for rational quadratic Bézier curves.
multiaffine-1.0.cdf Blossoms (Multiaffine maps).
roots-1.0.cdf Bisection search, Newton-Raphson, Safe Newton-Raphson, polynomial root-finding.
resultants-1.0.cdf Resultants via the Sylvester and Cayley-Bézout matrices.
implicit-quadratics-1.0.cdf Implicitization of integral quadratic Bézier curves.
implicit-rational-quadratics-1.0.cdf Implicitization of rational quadratic Bézier curves.
implicit-cubics-1.0.cdf Implicitization of integral cubic Bézier curves.
inflection-double-2.1.cdf Computing inflection points and double points of rational parametric cubics.

Exercices and programming assignments Due 10/01/2020 (before class). Due 17/01/2020 (before class). Due 24/01/2020 (before class).
list-1.pdf Try to solve the exercices. No need to hand in your solutions. Due 31/01/2020 (before class). Due 12/02/2020 (before class).
list-2.pdf Try to solve the exercices. No need to hand in your solutions. Due 21/02/2020 (before class).


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