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\section*{Chapter 2 Solutions}

2.3 Each iteration removes the central triangle of the four created
by subdivision. If we start with an equalateral triangle, the area
is reduced by 3/4 by each iteration. Hence, in the limit we have
no area left. If we consider the perimeter, again starting with an equaliateral
triangle, we find that if the original triangle had a side length of one unit
and a perimeter of three units, the three remaining triangles created by the
subdivision have a combined perimeter of 4 1/2 units. Thus the perimeter increases
at each iteration and in the limit we have an object with no area and an infinite
perimeter.

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2.9 The subtractive colors are often called the complementary colors.
Suppose each color in the RGB system can be written as the triplet
(r,g,b) where each value is between 0 and 1. Then, the complement
of the color (r,g,b) is the color in CMY space (c,m,y) = (1-r, 1-g, 1-b).
If we want a color C in RGB space, we specify its complement in CMY space.
For example, if we want a full red, in RGB it is (1,0,0) and its complement
is (0,1,1). In CMY, (0,1,1) is magenta and yellow, which have red in common.

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2.11 The biggest advantage of relative positioning is that it corresponds
to the way we think. We use terms lke ``in front of,'' ``next to,'', ``to the
right,'' ``forward,'' and ``back'' in speech to describe the location of
objects. For structures that we will consider in Chapter~8, we shall see
that when objects have parts such as a model of the human figure, we can
describne their motion much easier in terms of the relative positioning of
the components. In implementation, there are additional advantages to
relative positioning. For example, if we store the postions of the vertices
that determine and object in terms of their relative positions from a 
particular vertex, we can move the whole object by simpling moving the
base vertex.

The major disadvantage of relative positioning is that if an error is
made, all furthor positions are incorrect.

To add relative positioning we must add some sort of graphics cursor 
which contains the position we measure from.

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2.13 One strategy is to work a scanline at a time. Thus, we can find all
the points along a scanline that are inside the polygon in a single
calculation. Another strategy is to work with the bounding box of the 
polygon, the smallest box with sides aligned with the sides of the window that
contains the polygon. We can obtain the bounding box by a single scan
of the vertices. Then we check if a point is inside the bounding box before 
we check if it is inside the polygon, thus enabling us to eliminate
many points without detailed consideration of the polygon.

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2.15 A conceptually simple test can be based on the fact that if we 
circle a convex polygon in a clockwise direction, a point inside 
the polygon is to the right of every line determined by an edge of
the polygon. We can compute the equation of each of these lines from
successive pairs of vertices. If we insert the location of the
interior point to be tested into the equation of the line, the resulting
sign of the right side determines if the point is on the right side, left
side or on the line.

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