Graphics Reference
In-Depth Information
-
s
(
u
,
v
)
= signed angle between vectors
u
and
v
B
n
(
p
,r)
= {
q
Œ
R
n
| |
pq
| < r}
B
n
(r)
=
B
n
(
0
,r)
B
n
=
B
n
(
0
,1)
= the open (n-dimensional) unit disk in
R
n
D
n
(
p
,r)
= {
q
Œ
R
n
| |
pq
| £ r}
= an n-dimensional closed disk
D
n
=
D
n
(
0
,1)
= the closed (n-dimensional) unit disk in
R
n
S
n-1
= {
q
Œ
R
n
| |
q
| = 1}
= the (n - 1)-dimensional unit sphere in
R
n
S
n-1
=
S
n-1
«
R
+
= the upper hemisphere
S
n-1
=
S
n-1
«
R
-
= the lower hemisphere
P
n
= n-dimensional projective space
P
n
(k)
= n-dimensional projective space over a field k
= [a,b,c], where
L
is a line in
P
2
defined, in homogeneous coor-
dinates, by the equation
[
L
]
aX
++=0.
bY
cZ
There are natural inclusions: 0 =
R
0
Ã
R
1
Ã
R
2
à ...
Similarly for the other spaces above.
The map f:
S
n
Æ
S
n
, f(
p
) = -
p
, is called the
antipodal map
of
S
n
and
p
and -
p
are
called
antipodal points
.
X
k
= the k-fold Cartesian product
XX
¥
¥◊◊◊¥
k
X
of the set
X
12
44
44
3
X
D
Y
= (
X
-
Y
) » (
Y
-
X
)
(symmetric difference)
inf
X
= infimum or greatest lower bound of the set
X
of real numbers
sup
X
= supremum or least upper bound of the set
X
of real numbers
cl(
X
)
= closure of
X
int(
X
)
= interior of
X
aff(
X
)
= affine hull of
X
conv(
X
)
= convex hull of
X
f(a
+
)
= right-handed limit of f at a
f(a
-
)
= left-handed limit of f at a
f
(d)
(x)
= the dth derivative of f
I = I
n
= n ¥ n
identity matrix
that consists of 1s along the diagonal and
0s elsewhere
E
ij
(c)
= n ¥ n elementary matrix that consists of 1s on the diagonal,
the value c in the ijth position, and 0s elsewhere (if i = j, then
the ith element on the diagonal is c, not 1)
D(c
1
,c
2
,...,c
n
)
= n ¥ n diagonal matrix whose ith diagonal entry is c
i
and which
has 0s elsewhere
A
T
= transpose of the matrix A
det(A)
= determinant of matrix A
tr(A)
= trace of matrix A
