Graphics Reference
In-Depth Information
APPENDIX A
Notation
N
= the natural numbers {0,1,2, . . .}
Z
= the ring of integers
R
= the field of real numbers
R
*
= the extended real numbers, that is,
R
» {•}
I
= the unit interval [0,1]
C
= the field of complex numbers
H
= the non-commutative division ring of quaternions
In the context of an n-tuple
p
, p
i
will always refer to the ith component of
p
. The same
holds for functions. If f:
R
n
Æ
R
m
, then f
i
is the ith component function of f, that is,
()
=
(
() ()
()
)
f
p
f
p
,
f
p
,...,
f
m
p
.
1
2
N
n
= {
z
= (z
1
,z
2
,...,z
n
) | z
i
Œ
N
}
Z
n
= {
z
= (z
1
,z
2
,...,z
n
) | z
i
Œ
Z
}
R
n
= {
p
= (p
1
,p
2
,...,p
n
) | p
i
Œ
R
}
= n-dimensional Euclidean space
R
+
= {
p
Œ
R
n
|
p
n
≥ 0}
= the upper halfplane of
R
n
R
-
= {
p
Œ
R
n
|
p
n
£ 0}
= the lower halfplane of
R
n
I
n
= {
p
= (p
1
,p
2
,...,p
n
) | p
i
Œ
I
}
= the unit “cube” in
R
n
d
ij
= Kronecker delta (1, if i = j, and 0, otherwise)
e
1
,
e
2
,...,
e
n
= standard (orthonormal) basis of
R
n
, that is,
e
i
= (d
i1
,d
i2
,...,d
in
)
|
v
|
= length of vector
v
= the segment from point
p
to point
q
in
R
n
,
unless p
and
q
are
quaternions in which case this denotes their product
pq
||
pq
||
= signed distance from
p
to
q
-(
u
,
v
)
= angle between vectors
u
and
v
-
s
(
u
,
v
)
= signed angle between vectors
u
and
v
