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Cell["Coordinates systems and transformations", "Title",
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Cell["\<\
Comp160 Course Module 5
\
\>", "Subtitle",
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 "Organizing the graphical components of a game requires that each element be \
correctly positioned.  In this model, we review the basics of coordinate \
systems and affine transformation.  We begin by discussing the math behind \
these concepts and then describe ",
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Cell["\<\
Observe this picture doesn't make to much geometric sense since the point is \
drawn with no frame of reference.  To give the point geometric location, \
let's redo the plot with an origin and axes drawn.\
\>", "Text",
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Cell["\<\
produces the same vector shifted down and to the left by a unit.\
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Cell[TextData[{
 "As you have probably learned in physics, vectors form a ",
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operation can simply be computed on a coordinate by coordinate basis.  For \
example, the sum of the vectors ",
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Cell["\<\
Taking combinations of points is a trickier task.  The obvious approach would \
be to add and scale the underlying coordinates of the points.  However, as we \
shall see when we consider transformations, this approach has some peril.\
\>", "Text",
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Cell["\<\
In 2D games, graphical objects typically move and change shape in response to \
the user's action.  For now, we will focus on transformations applied to \
points.  Mathematically, we will model a transformation as a function that \
takes a point as input and returns a new point as output.\
\>", "Text",
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Cell[TextData[{
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Note that translating both endpoints of a vector does not change the vector \
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Clearly, translations, rotations and scaling can be written in this form.  \
One nice property (that you will show in the exercises) is that the \
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Another interesting property of affine transformation is that they can always \
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scaling.  Due to these properties, affine transformation are a standard class \
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