Symbols
Notation
r, s, t, . . .
α, β, γ, . . .
a, b, c, . . .
A, B, C, . . .
A, B, C, . . .
A, B, C, . . .
a, b, c, . . .
integers
real numbers
points in a space
sets of points
collections of point sets
families of point set collections
real valued functions on Rk
Real valued functions
nP
1P
vP
rP
uA,B
a+b
a−b
a∧b
a
R ∨b
f (x) dx
Rk
I kp
real valued function associated with pattern P
characteristic function of set P
Voronoi surface of P
reflection visibility surface of P
pointwise difference of visibility stars of A and B
pointwise sum of functions a and b
pointwise difference of functions a and b
pointwise minimum of functions a and b
pointwise maximum of functions a and b
Lebesgue integral of f over Rk
special class of real valued functions
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Operations on subsets
vol (P )
diam (P )
Int (P )
k-dimensional volume of set P
diameter of set P
interior of set P
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Cl (P )
Bd (P )
P ∩Q
P ∪Q
P −Q
P 4Q
closure of set P
boundary of set P
intersection of P and Q
union of P and Q
set difference of P and Q
symmetric set difference of P and Q
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Linear and differential geometry
MT
det (M )
Df (x)
δf (x)
δf
transpose of a matrix M
determinant of square matrix M
total derivative of f in x
absolute value of determinant of Df (x)
value for constant δf (x)
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Geometric constructions
Line(x, y)
Seg(x, y)
Ray(x, y)
Bd (x, )
Nd (P, )
B(x, )
N(P, )
C(P )
R (A, B)
SP (x)
VP (x)
V?P (x)
T?P (x)
R?P (x)
VmpS (x)
TmpS (x)
RmpS (x)
line through x and y
open line segment between x and y
open ray emanating from x passing through y
open ball centred at x with radius neighbourhood of P
Euclidean open ball centred at x with radius Euclidean neighbourhood of P
smallest Euclidean ball containing P
relation in definition Hausdorff metric
generic set for point x relative to P
points visible from x relative to P
visibility star for x relative to P
trans visibility star for x relative to P
reflection visibility star for x relative to P
view map for x in S
trans view map for x in S
reflection view map for x in S
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Transformation groups
Hom(X)
Iso(X)
Clos(X, P)
homeomorphisms from X onto itself
isometries in general metric space X
closure group
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Dif k
CDif k
UDif k
Af k
UAf k
Stretk
Thetk
Simk
Isok
Latk
Rotk
Idk
diffeomorphisms from Rk onto itself
ratio of volume preserving diffeomorphisms
volume preserving diffeomorphisms
affine transformations
volume preserving affine transformations
stretch transformations
homotheties
similarity transformations
Euclidean isometries
translations
rotations around the origin
identity group, trivial group
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Congruence, symmetry and normalisation
GA:B
GP
h·i
congruences of A and B in G
symmetries of P in G
normalisation function
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Collections of subsets
℘(X)
F(X)
C(X)
C 0 (X)
K(X)
K0 (X)
K+ (Rk )
KP (X)
KP (X)
KVC1 ,...,Vn (X)
S(X)
S 0 (X)
M(Rk )
T (Rk )
all subsets of X
finite subsets of X
closed and bounded subsets of X
nonempty elements of C(X)
compact subsets of X
nonempty elements of K0 (X)
nonzero volume elements of K(Rk )
elements of K(X) intersecting P
elements of K(X) disjoint with P
equals KC (X) ∩ KV1 (X) ∩ · · · ∩ KVn (X)
solid subsets of X
nonempty elements of S(X)
compact sets with simplex union boundary
elements of M(Rk ) with empty interior
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abstract similarity measure, or (pseudo)metric
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Distances
d
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d.
ρ
k·kp
k·k
lp
k·kp
lp
l∗p
z
c
bρ
fρ
hρ
h.ρ
h∅
ρ
s
s∗
a
a.
s∗.
r
r∗
dS
dG
d|S×S
do
dc
directed similarity measure
base metric
norm for Rk
Euclidean norm for Rk
metric for Rk based on k·kp
pseudonorm for I kp
pseudometric for I kp
normalised pseudometric for I kp
indiscrete pseudometric
discrete metric
Bottleneck distance based on ρ
Fréchet distance based on ρ
Hausdorff metric based on ρ
directed Hausdorff distance based on ρ
Hausdorff metric extended for ∅
volume of symmetric difference
normalised volume of symmetric difference
absolute difference
directed absolute difference
directed version of s∗
reflection visibility distance
normalised reflection visibility distance
quotient metric for d and partition S
orbit pseudometric for d under G
restriction of d to S × S
extension of d with element o
complementation of d
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Geometric branch-and-bound
C
G(C)
l
u
cover
lGlobal
uGlobal
select
refine
coverr
refiner
class of cells
subset of G represented by cell C
lower cell bound
upper cell bound
the cover operation
lower global bound
upper global bound
selection operation for cell collection
refinement operation for cell collection
r times iterated cover operation
r times iterated refine operation
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u
→
τ (C, A)
←
τ (C, B)
σC
fC
fC
increasing binary operator
forward trace of A under C
backward trace of B under C
transformation represented by C
pointwise lower bound for f |G(C)
pointwise upper bound for f |G(C)
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