Graphics Reference
In-Depth Information
×
v
)
that arose in studying line intersections
occurs quite often (as do its generalizations to higher dimensions). Show that it
equals the determinant of a matrix whose first column is
v
and whose second
column is
u
. As a point of information, this works more generally: In 3-space,
u
Exercise 7.3:
The expression
u
·
(
w
)
is the same as the determinant of a matrix whose columns are
v
,
w
, and
u
and similar formulas hold in higher dimensions.
Exercise 7.4:
Find the parametric form of the line between the points
(
1, 1
)
and
(
2, 2
)
. Then find the parametric form of the line between the points
(
3, 3
)
and
(
5, 5
)
. Are the two functions identical? Are the lines they describe identical?
Explain why the mapping defined by the parametric line formula is actually a
map from “two distinct points in the plane” to “a parametric representation of
the line between these points,” rather than from the line itself to a parametric
representation.
Exercise 7.5:
The same reasoning as in Exercise 7.4 applies to the implicit
form of a line: Depending on the points chosen, we get different “standard implicit
forms.” Fortunately, all define the same line. Furthermore, any two standard
implicit forms are proportional. Write down the implicit form of the line between
the points
(
1, 1
)
and
(
2, 2
)
. Then find the implicit form of the line between the
points
(
3, 3
)
and
(
5, 5
)
. Are the two implicit functions identical? Are they propor-
tional? Are the points where they are zero identical?
Exercise 7.6:
Since any two standard implicit forms of a line are proportional,
is there a way to choose a “standard representative” once and for all? Suppose
that
Ax
+
By
+
C
and
A
x
+
B
y
+
C
=
0 describe the same line. Then we
know that the triples
(
A
,
B
,
C
)
and
(
A
,
B
,
C
)
are proportional, and that any other
triple proportional to these (except (0,0,0)) will determine the same line. Can we
choose just
one
and call it the “normal form of the line”? For instance, we might
say, “Take your triple
(
A
,
B
,
C
)
and convert to a normal form where
B
=
1by
dividing through by
B
to get
(
A
·
(
v
×
B
)
.” Unfortunately, this doesn't work if
B
=
0; the same g
oes for
A
and
C
.We
can
take the triple
(
A
,
B
,
C
)
and divide
/
B
,1,
C
/
by
√
A
2
+
B
2
+
C
2
to get a “normal form”; if we do this, the only ambiguity is
a sign: The standard form for
(
A
,
B
,
C
)
and the one for
(
−
A
,
−
B
,
−
C
)
, which
determine the same line, end up being opposites. Explain why (a) if
λ
=
0, the
lines determined by
(
A
,
B
,
C
)
and
(
λ
A
,
λ
B
,
λ
C
)
are identical;
(b) if
λ>
0, they
have the same normal form; and (c) the factor
√
A
2
+
B
2
+
C
2
is never zero when
Ax
+
By
+
C
=
0 determines a line.
Exercise 7.7:
Barycentric coordinates can be used to describe lines within a
triangle
PQR
. For instance, all points on the line
PQ
satisfy the equation
γ
=
0
(where
are the barycentric coordinates). Let
S
be a point that's one-
third of the way from
P
to
Q
, and consider the line passing through
S
and
R
. What
equation, in barycentric coordinates, determines this line? (Hint: Draw a picture,
and find the barycentric coordinates of at least two points on the line.)
Exercise 7.8:
The inequality 4
x
+
2
y
α
,
β
, and
γ
−
6
≥
0 defines a half-plane; its boundary
is the line
consisting of solutions to 4
x
+
2
y
−
6
=
0. Find the point of
that's
on the
x
-axis, and the point of
that's on the
y
-axis. Is the origin in the half-space
defined by the inequality? Draw the normal ray for the equation of
, and verify
that it points toward the positive half-space, as described in Section 7.9.3.
Exercise 7.9:
Generalize to 3-space the result about normal vectors and the
defining equation for half-planes: that for the half-plane defined by
Ax
+
By
+
C
≥
0, the vector
A
B
points from the edge of the half-plane into the positive half-
plane.
