Game Development Reference

In-Depth Information

Glossary of Notation

a
,
b
,
c
,...

vectors, points, quaternions

0

zero vector

e
i

i
th vector of the standard basis

v

·

w
,
v

×

w

dot and cross product of vectors
v
and
w

2

v

,

v

length and squared length of vector
v

A
,
B
,
C
,...

matrices over real numers

[
α
ij
]

matrix with element
α
ij
in the
i
th row and
j
th column

[
a
i
]

matrix with vector
a
i
as
i
th column

A
T
,
A
−
1

transpose and inverse of matrix
A

det(
A
)

determinant of matrix
A

E

identity matrix

F
,
τ

force and torque vector

x
,
x

first and second time derivatives of
x
(velocity and acceleration)

ω
,
α

angular velocity and angular acceleration

I

inertia tensor

i
=1
a
i

a
1
+

···

+
a
n

∂f

∂x

partial derivative of multi-variable function
f
with respect to
x

(
∂f

∂x
1
,...,
∂f

∇

f
(
x
1
,...,x
n
)

),gradientof
f

∂x
n

i
=1

∂
2
f

∂x
2

i

∇

2
f
(
x
1
,...,x
n
)

, Laplacian of
f

q
†
,
q
−
1

conjugate and inverse of quaternion
q

A,B,C,...

point sets (lines, planes, volumes), or predicates

{

x
i
}

set of elements
x
i

{

x
:
P
(
x
)

}

set of elements
x
for which predicate
P
(
x
) holds

∅

empty set

A

∪

B, A

∩

B, A

\

B

union, intersection, and set difference of sets
A
and
B

A

⊆

B

A
is a subset of
B
and is possibly equal to
B

A
+
B
,
A

−

B

Minkowski sum and CSO of objects
A
and
B

[
a, b
]

interval of real numbers containing all
x
for which
a

≤

x

≤

b

aff(
A
), conv(
A
)

affine and convex hull of point set
A

∃x
:
P
(
x
)

there exists an
x
such that predicate
P
(
x
) holds

∀

x
:
P
(
x
)

for all
x
predicate
P
(
x
) holds

f
(
n
)=
O
(
g
(
n
))

constants
k
and
c
exist such that
f
(
n
)

≤

cg
(
n
) for all
n

≥

k