Graphics Reference
In-Depth Information
}
k
Ê
ˆ
10
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
Á
˜
˜
˜
˜
˜
˜
˜
˜
˜
˜
˜
O
00
01
0
a
M
=
a
nk
}
--
1
00
0 0
00
0001
O
Ë
¯
The general case p is reduced to this special case using a motion like before. Choose
an orthonormal basis for
B
and let G = (
u
1
,
u
2
,...,
u
n
,
p
) be the corresponding aug-
mented frame for F. The map G
-1
maps world coordinates into the coordinates of the
frame G. Let T be the translation
q
Æ
q
- d
e
k+1
. Then p=C
1/d
TG
-1
. The affine version
of the map is
(
)
(
)
qp u
qp u
-
•
•
qp u
pp u
-
•
•
Ê
Ë
ˆ
¯
1
k
k
()
=
p
q
d
,...,
d
, ,...,
00
.
(
)
(
)
-
-
k
+
1
+
1
3.9
The Theorems of Pascal and Brianchon
It did not seem appropriate to leave the subject of projective geometry without men-
tioning two well-known and beautiful theorems.
3.9.1.
Theorem.
(1) (Pascal's Theorem) If the vertices
A
,
B
,
C
,
D
,
E
, and
F
of a hexagon in
P
2
, no
three of which are collinear, lie on a nondegenerate conic, then the pairs of
opposite sides intersect in collinear points, that is, the intersection points
XL
=
«
L
,
YL
=
«
L
,
and
ZL
=
«
L
AB
DE
BC
FE
AF
CD
are collinear, where
L
PQ
is the line through points
P
and
Q
. See Figure 3.32.
(2) Conversely, if the pairs of opposite sides of a hexagon in
P
2
, no three of whose
vertices are collinear, intersect in collinear points, then the vertices of the
hexagon lie on a nondegenerate conic.
Proof.
See [Gans69] or [Fari95].
Pascal proved his theorem in 1640. There are affine versions of these theorems
but they are not as elegant because one has to add assumptions that intersections
