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Homework 0: Transformations & OpenGL Warmup
The goal of this warmup assignment is to get comfortable with the
programming environment you will be using for this class and
familiarize yourself with a simple library that we will use for linear
algebra. It's also
an opportunity for a crash course in C++ and OpenGL (if you're not
already familiar with them).
This assignment is only worth 1/4 of the points of a
regular homework assignment. It will not be rigorously graded, but
you are expected to work through and submit the exercises. Please try
to finish this assignment by the end of the first week of classes;
however, there is no firm deadline for this assignment.
Here are a couple tutorials you may want to check out:
The incidental goal is also to have fun with bizarre fractal objects.
IFS are selfsimilar fractals: a subpart of the object is similar to
the whole. The classic example of an IFS is Barnsley's fern, where
each subpart of the fern is exactly the same as the whole fern. IFS
are described by a set of affine transformations (rotations,
translations, scale, skew, etc.) These transformations capture the
selfsimilarity of the object. IFS can be defined in any dimension,
and we will play with both twodimensional and threedimensional ones.
Formally, an IFS is defined by n affine transformations. Each
transformation f_{i} must be contractive: The
distance between points must be reduced. An attractor of the
IFS is the object such that A = U f_{i} (A).
A is unchanged by the set of transformations: It is a fixed
point.
We can render an IFS by iterating the transform on random input points
from the unit square. We approximate the fixed point by applying the
transformation many times. The algorithm is as follows:
for "lots" of random points (x_{0}, y_{0})
for k=0 to num_iters_{ }
pick a random transform f_{i}
(x_{k+1}, y_{k+1}) = f_{i}(x_{k}, y_{k})
display a dot at (x_{k}, y_{k})
To reduce the number of points necessary to make an image of
reasonable quality, probabilities are assigned to each transformation,
instead of choosing a transformation with uniform probability.
Tasks
 Download the provided source code and set up your C++ development
environment. To receive full credit, your homework assignments must
compile and run without errors using gcc/g++. Even if you plan to do
much of your development in another environment, you'll probably still
want to set up gcc so that you can test it before submission. For
interactive display of your IFS, you will use the OpenGL API that uses
graphics hardware for fast rendering of 3D primitives. Note: with some
configurations, software emulation might be used, resulting in slower
rendering. Here are some notes:
A Makefile is provided for use with either Unix or Cygwin. On
Linux or FreeBSD, type: make unix . On Windows/Cygwin, if
you have the files OpenGL32.lib, GLU32.lib, and glut32.lib, type:
make cygwin_lib ; if you have libopengl32.a, libglu32.lib,
and libglut32.a, type: make cygwin_a ; and if you have
libopengl.a, libglu.a, and libglut.a, type: make cygwin_x .
On Mac OSX, type: make osx . You may need to adjust the CC,
INCLUDE_PATH, and LIB_PATH lines of the Makefile for your particular
installation to point to the right files. Please see the TA if you
have trouble setting up your development environment.
All files implementing OpenGL code should include the OpenGL
header files:
#include <GL/gl.h>
#include <GL/glu.h>
#include <GL/glut.h>
When provided with appropriate arguments, the initial executable
should launch an OpenGL window and draw a cube and you should be able
to navigate the scene with the mouse (left button rotates, middle
button translates, right button zooms). Try this at the command
prompt:
./ifs size 500
 Now you're ready to start coding. Write a C++ class
IFS that renders iterated function systems, including the
class declaration (in the header file ifs.h) and the
implementation (ifs.C). The IFS class should include:
 a field to store n, the number of transformations,
 an array/vector of matrices representing the n transformations,
 an array/vector of the corresponding probabilities for choosing a
transformation,
 a function that parses the input file and creates an IFS,
 a paint method which makes appropriate OpenGL calls to draw points
or polygons to the OpenGL window, and
 When you're done, adjust the main function (provided in
main.C) as necessary.
 Use the linear algebra library for the point and transformation
representations.
 Consider the performance of your programming environment.
How many polygons/points can you render interactively? What
improvements could you make to your code?
Possible Extensions
 design a new IFS  figure out the transformations and probabilities,
 automatically determine the bounding box of the fractal object,
 change the color scheme,
 implement antialiasing,
 experiment with depthfirst vs. breadthfirst, etc.
Hints
 Random numbers can be obtained using the drand48() or
rand() and RAND_MAX. See
stdlib.h.
 To debug your code, set the number of iterations to one. This
will allow you to check that you got the transformations right.
 Be careful, arrays are indexed from 0 to n1 in
C++. Reading beyond the bounds of the array will probably result in a
segmentation fault.
 Use assert() to check function preconditions, array
indices, etc. See assert.h.
 To perform transformations in OpenGL, read about the Modelview
matrix stack and the OpenGL commands glMatrixMode(),
glPushMatrix(), glPopMatrix(), and
glMultMatrix() in the
OpenGL Programming Guide, Chapter 3.
 Linear Algebra Library (vectors.h & matrix.h & matrix.cpp)
Linear algebra support for floating point vectors with 2, 3, and 4
elements (Vec2f, Vec3f, and Vec4f) and 4x4
floating point matrices (Matrix). For this assignment, the
void Matrix::Transform(Vec3f &v) function will be handy.
 Parsing code for commandline arguments and input files (argparser.h & parse.cpp)
Your program should take a number of command line arguments to specify
the input file (input), number of points (points),
number of iterations (iters), and output size
(size). Your program should render points by default, or
cubes if cubes is specified. Examples are shown below.
Code to parse input files and command line arguments is provided.
 OpenGL and main code (main.cpp, ifs.h, glCanvas.h, glCanvas.cpp, camera.h, camera.cpp)
OpenGL programs can be tricky to set up from scratch. This base code
should do all that work for you.
 Makefile for g++ on UNIX or Windows/Cygwin
On the command line simply type make unix (or make
cygwin_lib or make cygwin_a or make cygwin_x or
make osx). To force a rebuild of all object files, type
make clean first.
 Data files (fern.txt, dragon.txt, sierpinski_triangle.txt, and giant_x.txt)
The input data for an IFS is a file which contains n,
the number of transforms, followed by the probability of choosing each
transform and a 4x4 floating point matrix representation of the
transform.
Sample Results
./ifs input sierpinski_triangle.txt points 10000 iters 0 size 200
./ifs input sierpinski_triangle.txt points 10000 iters 1 size 200
./ifs input sierpinski_triangle.txt points 10000 iters 2 size 200
./ifs input sierpinski_triangle.txt points 10000 iters 3 size 200
./ifs input sierpinski_triangle.txt points 10000 iters 4 size 200
./ifs input sierpinski_triangle.txt points 10000 iters 30 size 200
./ifs input fern.txt points 50000 iters 30 size 400
./ifs input giant_x.txt points 10000 size 400 iters 0
./ifs input giant_x.txt points 10000 size 400 iters 1
./ifs input giant_x.txt points 10000 size 400 iters 2
./ifs input giant_x.txt points 10000 size 400 iters 3
./ifs input giant_x.txt points 10000 size 400 iters 4
./ifs input giant_x.txt size 400 iters 0 cubes
./ifs input giant_x.txt size 400 iters 1 cubes
./ifs input giant_x.txt size 400 iters 2 cubes
./ifs input giant_x.txt size 400 iters 3 cubes
./ifs input giant_x.txt size 400 iters 4 cubes
Please read the Homework information page again before submitting.
