Graphics Reference
In-Depth Information
correspondence between
L
and [a,b,c] sets up a one-to-one and onto correspondence
between lines
L
in
P
2
and points [a,b,c] in
P
2
.
in
R
2
defined by the equation
Notation.
Given a line
ax
++=0
by
c
or a line
L
in
P
2
defined by the equation
aX
++=0,
bY
cZ
] and [
L
] denote the point [a,b,c] in
P
2
.
we shall let [
With this notation, consider a line
L
and assume that [
L
] = [
a
], where
a
Œ
R
3
. It
follows that
L
is just the set of all points [
p
] in
P
2
, so that
ap
•
= 0
.
(3.16)
This gives us a complete duality between points on a line and lines on a point in
P
2
.
This observation was already implicit in our discussion earlier of equation (3.15). We
make this duality a little more explicit.
Definition.
Given a statement concerning the projective plane, the
dual
of that state-
ment is the statement where every occurrence of the words “point” or “collinear” is
replaced by “line” or “concurrent,” respectively, and vice versa.
The principle of duality in the projective plane:
Given any theorem for the
projective plane, its dual is also a theorem.
The duality principle is very useful in the study of projective geometry. Even
though it will only get an occasional mention in this topic, it is worth being aware of
it.
All lines are alike in
P
2
(topologically they are circles), but with our imbedding of
R
2
in
P
2
, we can distinguish between two types of lines.
An
ordinary
line
L
in
P
2
Definition.
is a line that consists of an ordinary line
in
•
}.
(See Exercise 3.4.1.) The line defined by the equation Z = 0, which consists of the set
of ideal points, is called the
ideal
line.
the Euclidean plane with an ideal point adjoined, that is, a line of the form
L
=
»{
Note that there is no such thing as “parallel” lines in the projective plane.
Every pair of distinct lines in
P
2
intersects in a point.
3.4.2. Theorem.
Proof.
Since every line has an ideal point, every line intersects the ideal line. It
suffices to show that two ordinary lines intersect. The only possible problem could
come from two lines whose real parts in
R
2
are parallel, but those will intersect in
their common ideal point.
