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Notebook[{
Cell["\<\
Affine Geometry:  A Connect the Dots Approach to Computer Graphics\
\>", "Title"],

Cell["Comp 360 Course Notes", "Subtitle",
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Cell[TextData[{
 "A line is determined by a point and a direction vector.  In turtle geometry \
we applied this observation to generate line segments using the FORWARD \
command.  Generating line segments from a point (turtle position) and a \
direction vector (turtle direction) is the basic drawing operation in turtle \
graphics.  But a line is also determined by two points.  In affine geometry, \
we apply this fact to generate line segments by connecting pairs of points.  \
That is, the primitive drawing operation in affine geometry is to connect two \
points with a straight line.  This choice of the primitive drawing operation \
is how computer graphics based on affine geometry differs fundamentally from \
turtle graphics.\nThe turtle is a metaphor for the human condition.  Turtles \
know only their current location and heading;  turtles do not know their \
relation to any other point in the universe.  In affine geometry, we will \
have simultaneous access to many different points in the plane, and we will \
know how these different locations are related to each other.  This god-like \
perspective makes affine methods very powerful, but also somewhat artificial. \
 We shall need to develop some new techniques to deal effectively with affine \
geometry.  Our main tools will be affine coordinates and affine \
transformations.\nIn these notes, we will develop affine geometry for \
computer graphics inside of ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 ".  We shall then use affine geometry, just as we used turtle geometry, to \
build and to draw a variety of shapes from simple polygons to complex \
fractals.  As you will see, there are both advantages and disadvantages to \
affine methods, depending on the problem at hand.  In the lab, you will \
implement these affine methods using C++ and OpenGL.  Beware that these two \
implementations of computer graphics based on affine geometry are quite \
different because the underlying programming environments are very different. \
 As usual, you can learn a lot from studying this ",
 StyleBox["Mathematica",
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 " implementation, but you cannot just copy these ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " programs and use them in your lab assignment."
}], "Text"],

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Cell["Global Variables", "Section"],

Cell["\<\
For the purpose of display, we shall maintain three global lists:
\ti.   ptlist = list of points
\tii.  linelist = list of line segments
\tiii.  polygonlist = list of (filled in) polygons
Whenever we want to display some geometry, we shall display the objects in \
one of these lists.  Thus we shall display either a collection of points, a \
collection of line segments (determined by pairs of points), or a collections \
of filled in polygons.\
\>", "Text"]
}, Open  ]],

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Cell["Display Programs", "Section"],

Cell["Below are the programs for displaying these three lists.", "Text"],

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Cell["Affine Coordinates and Affine Transformations", "Section"],

Cell[TextData[{
 "In computer graphics, we manipulate geometry by performing transformations \
both to relocate objects in space and to change their shape.  Transformations \
are so fundamental in computer graphics that our underlying coordinate \
representation is chosen in large part to help simplify how we represent and \
compute simple transformations.\nThe simplest and most useful transformations \
in computer graphics are affine transformations.  Affine transformations in \
2-dimensions are maps from the plane to the plane that map points to points \
according to the rules:\n\t",
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 " + f .\nAdding constants (+e or +f) is equivalent to translation.  But \
vectors are invariant under translation.  Therefore affine transformations \
map vectors to vectors according to the rules:\n\t",
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 "= a ",
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 "= c",
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 "\nthat is, affine transformation on vectors are actually just linear \
transformations on vectors.\nThe most common transformations of computer \
graphics -- translation, rotation, and scaling -- are affine transformations, \
but there are many other affine transformations such as mirror image and \
shear.  Nevertheless, for simplicity, we shall restrict ourselves here to \
rotation, translation, and uniform scaling.  \nWe already know the formulas \
for these transformations from our study of turtle geometry:\n\t",
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 "\nRotation and uniform scaling can be represented by 2\[Times]2 matrices, \
but translation cannot, since translation is not a linear transformation!  \
(Remember that translation does not affect vectors.)  This anomaly is very \
annoying, since we would like to compose transformations by matrix \
multiplication.  There is, however, a standard trick in computer graphics to \
overcome this difficulty:  we shall use three coordinates instead of just two \
to represent points and vectors, and we shall use 3\[Times]3 matrices instead \
of 2\[Times]2 matrices to represent affine transformations.  That is, we use \
the following representations:\n\tPoints:  (",
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and always 0 for vectors.  Now the matrix equation\n\t",
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transformation as ",
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Cell["\<\
We can compose these affine transformations by matrix multiplication so that \
we can perform all three operations simultaneously.  That is, we can rotate, \
scale, and translate using a single matrix.  Note that the order in which we \
perform these operations is critical, since rotation and translation do not \
commute.  The convention here is that we rotate and scale first (rotation and \
uniform scaling do commute), and then we translate.\
\>", "Text"],

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Cell["\<\
These matrices work fine as long as we rotate and scale only about the \
origin.  But we shall have occassion to rotate and scale about arbitrary \
points as well.  One way to perform these transformations is first to \
translate the center of rotation or scaling to the origin, then rotate or \
scale about the origin, and then translate the origin back to the original \
center of the transformation.  We shall also want to perform non-unjiform \
scaling and arbitrary affine transformations defined by the image of three \
non-collinear points or the image of one point and twolinearly independent \
vectors.  These operations can be simplified to the following code:   \
\>", "Text"],

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The Star of the Alhambra consists of two overlapping rectangles.  We can \
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Cell["\<\
Rosettes are also very easy to generate using affine geometry.  Again we \
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Spirals are just as easy to generate as polygons and stars.  Instead of \
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To apply an affine transformation to a list of lines or a list of polygons, \
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Similarly we can shift, spin, and scale a list of points, lines, or polygons, \
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