Angel: Interactive Computer Graphics, Fourth Edition
Chapter 12 Solutions
12.1 Let’s do the problem in two dimensions. The solution in three
dimensions is essentially the same. Assume that the vertices are used in a
consistent clockwise or counterclockwise manner. Starting at some vertex,
that vertex and the next determine a line of the form
ax
+
by
+
c
= 0
.
If we evaluate
ax
+
by
+
c
for a given point, the result will be positive or
negative depending on which side of the line the point lies. If we are
following the vertices in a clockwise manner, the point is inside the
polygon if and only if it is to the right of each of these lines.
12.3 Consider two identical circles of radius
r
centered at (
a,
0) and
(
−
a,
0). We can describe them through the single implicit equation
((
x
−
a
)
2
+
y
2
−
r
2
)((
x
+
a
)
2
+
y
2
−
r
2
)
,
by simply multiplying together their individual implicit equations. We can
form the torus by rotation these circles about the
y
axis which is
equivalent to replacing
x
2
by
x
2
+
z
2
.
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 Spring '10
 Alxe
 Computer Graphics, Euclidean geometry, single implicit equation, individual implicit equations

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