-
Notifications
You must be signed in to change notification settings - Fork 15
/
Copy pathdictionary.tex
278 lines (205 loc) · 9.87 KB
/
dictionary.tex
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
% !TEX root = thesis.tex
\chapter{A short dictionary of dimension-based GIS terms}
\label{ch:dictionary}
\begin{multicols}{2}
\begin{description}
\footnotesize
\item[ambient space]
the space in which objects are embedded.
\item[area]
1.\ measure of the 2D extent of an object;
2.\ 2D combinatorial element;
{\color{gray} 3.\ 2D geometric object}.
\item[ball]
1.\ ($\rightarrow$\ same as 3-\emph{ball}) topological definition of the space within a 2-sphere (\ie\ a sphere), often defined as the space within a certain distance (the radius) from a point in $\mathbb{R}^3$ (the centre);
2.\ ($\rightarrow$\ often $n$-\emph{ball}) topological definition of the space within an $(n-1)$-sphere, often defined as the space within a certain distance (the radius) from a point in $\mathbb{R}^n$ (the centre);
3.\ geometric definition of a perfectly round filled 3D object.
\item[body]
$n$-dimensional combinatorial element in an $n$D context.
\item[box]
1.\ cuboid;
2.\ orthotope of any dimension.
\item[cavity]
3D hole.
\item[cell]
1.\ ($\rightarrow$\ sometimes $n$-\emph{cell}) $n$D combinatorial element;
{\color{gray} 2.\ 3D combinatorial element}.
\item[cell complex]
topological space formed by a set of cells glued along common faces, all faces of a cell in the complex should be in the complex.
\item[circle]
1.\ topological definition of the space at a certain distance (the radius) from a point in $\mathbb{R}^2$ (the centre);
2.\ geometric definition of a perfectly round hollow 2D object.
\item[closed]
1.\ topological definition of an object that includes its boundary;
2.\ geometric definition of an object in $n$D, usually defined using boundary representation, that encloses an $n$D subspace.
\item[congruent]
having the same shape and size.
\item[cube]
1.\ ($\rightarrow$\ same as $3$-\emph{cube}) 3D polyhedron with 6 square facets;
2.\ ($\rightarrow$\ usually $n$-\emph{cube}) $n$-orthotope with identical $(n-1)$-cube facets, akin to a square in 2D and a cube in 3D.
\item[cuboid]
1.\ box-shaped 3D polyhedron with 6 rectangular facets, its opposite facets are congruent and parallel;
{\color{gray} 2.\ parallelotope}.
\item[curve]
1.\ 1D geometric object, often with a non-linear geometry;
2.\ 1D combinatorial element;
3.\ 1-manifold.
\item[cut-line]
degenerate part of a 2D shape forming a curve and protruding inwards from its boundary.
\item[disk]
topological definition of the space within a circle, often defined as the space within a certain distance (the radius) from a point in $\mathbb{R}^2$ (the centre).
\item[edge]
1D combinatorial element.
\item[face]
1.\ 2D combinatorial element;
2.\ ($\rightarrow$\ sometimes \emph{$n$-face of $X$}) $n$D combinatorial element on the boundary of $X$;
{\color{gray} 3.\ ($\rightarrow$\ usually \emph{face of $X$}) $(n-1)$D combinatorial element on the boundary of an $n$D combinatorial element $X$}.
\item[facet]
1.\ $(n-1)$D combinatorial element in an $n$D context;
2.\ ($\rightarrow$\ often \emph{facet of $X$}) $(n-1)$D combinatorial element on the boundary of an $n$D combinatorial element $X$;
{\color{gray} 3.\ 2D combinatorial element}.
\item[flat]
unbounded geometric object with linear geometry of any dimension, such as a point, line or plane.
\item[hole]
1.\ 2D void region;
2.\ $n$D void region.
\item[hyperball]
higher-dimensional ball.
\item[hypercell]
1.\ higher-dimensional combinatorial element;
2.\ 4D combinatorial element.
\item[hypercube]
1.\ higher-dimensional cube;
2.\ 4-cube.
\item[hyperplane]
1.\ $(n-1)$D linear subspace in an $n$D context;
{\color{gray} 2.\ plane in a higher-dimensional context.}
\item[hyperrectangle]
higher-dimensional orthotope.
\item[hypersphere]
higher-dimensional sphere.
\item[hypersurface]
$(n-1)$D subspace in an $n$D context, often curved;
\item[interval]
topological definition of the space between two points in $\mathbb{R}$ (the endpoints).
\item[Lebesgue measure]
measure of the $n$D extent of an object, akin to length in 1D, area in 2D and volume in 3D.
\item[length]
measure of the 1D extent of an object.
\item[line segment]
1D geometric object.
\item[manifold]
($\rightarrow$\ sometimes $n$-\emph{manifold}) topological space that resembles $\mathbb{R}^n$ at every point.
\item[manifold with boundary]
($\rightarrow$\ sometimes $n$-\emph{manifold with boundary}) topological space that resembles $\mathbb{R}^n$ at every point in its interior.
\item[node]
0D combinatorial element.
\item[open]
1.\ topological definition of an object that does not include its boundary;
2.\ geometric definition of an object in $n$D, usually defined using boundary representation, that does not enclose any $n$D subspace.
\item[orthotope]
($\rightarrow$\ sometimes $n$-\emph{orthotope}) $n$-polytope with congruent parallel facets forming right angles to each other, akin to a rectangle in 2D and a cuboid in 3D.
\item[parallelepiped]
3D polyhedron bounded by 6 parallelogram-shaped 2D faces.
\item[parallelogram]
polygon bounded by 4 parallel and have the same shape.
\item[parallelotope]
a polytope whose opposite facets are parallel and have the same shape, akin to a parallelogram in 2D and a parallelepiped in 3D.
\item[peak]
1.\ $(n-3)$D combinatorial element in an $n$D context;
2.\ ($\rightarrow$\ \emph{peak of $X$}) $(n-3)$D combinatorial element on the boundary of an $n$D combinatorial element $X$;
\item[plane]
2D unbounded linear subspace.
\item[point]
1.\ 0D geometric element;
2.\ topological definition of the space at one location.
\item[polychoron]
4D geometric object with linear geometry.
\item[polygon]
2D geometric object with linear geometry, possibly with holes.
\item[polygonal curve]
curve formed by a sequence of line segments joined at their endpoints.
\item[polyhedron]
1.\ 3D geometric object with linear geometry, possibly with holes;
{\color{gray} 2.\ an $n$D geometric object with linear geometry}.
\item[polyline]
curve formed by a sequence of line segments joined at their endpoints.
\item[polyteron]
5D geometric object with linear geometry.
\item[polytope]
$n$D geometric object with linear geometry.
\item[puncture]
0D (point) hole, often formed from a degenerate polyline or polygon.
\item[ridge]
1.\ $(n-2)$D combinatorial element in an $n$D context;
2.\ ($\rightarrow$\ \emph{ridge of $X$}) $(n-2)$D combinatorial element on the boundary of an $n$D combinatorial element $X$.
\item[line string]
curve formed by joining a sequence of vertices by straight line segments, represented by these vertices.
\item[linear ring]
2D geometric object with linear geometry represented as a sequence of vertices.
\item[ring]
2D combinatorial element represented by its 1D boundary.
\item[subfacet]
1.\ $(n-2)$D combinatorial element in an $n$D context;
2.\ ($\rightarrow$\ \emph{subfacet of $X$}) $(n-2)$D combinatorial element on the boundary of an $n$D combinatorial element $X$.
\item[shell]
3D geometric object with linear geometry without 3D holes, usually defined as the volume enclosed by its 2D boundary, often represented as a set of 2D faces.
\item[simplex]
1.\ ($\rightarrow$\ sometimes \emph{$n$-simplex}) combinatorial element with $n+1$ vertices and $n+1$ facets;
2.\ ($\rightarrow$\ sometimes \emph{$n$-simplex}) geometric object based on $n+1$ affinely independent vertices, akin to a point in 0D, a line segment in 1D, a triangle in 2D, or a tetrahedron in 3D.
\item[simplicial complex]
1.\ ($\rightarrow$\ sometimes \emph{abstract simplicial complex}) topological space formed by a set of simplices glued along common faces, all faces of a simplex in the complex should be in the complex;
2.\ ($\rightarrow$\ sometimes \emph{geometric simplicial complex}) abstract simplicial complex where simplices are embedded into Euclidean space, their interiors should not intersect geometrically.
\item[solid]
1.\ 3D geometric object with linear geometry, possibly with 3D holes, often represented as a set of shells;
2.\ object defined based on solid modelling, often as opposed one based on boundary representation;
3.\ an object containing its interior (as opposed to \emph{hollow}).
\item[spike]
degenerate part of a 2D shape forming a curve, often protruding outwards from its boundary.
\item[sphere]
1.\ ($\rightarrow$\ same as $2$-\emph{sphere}) topological definition of the space at a certain distance (the radius) from a point in $\mathbb{R}^3$ (the centre);
2.\ ($\rightarrow$\ often $n$-\emph{sphere}) topological definition of the space at a certain distance (the radius) from a point in $\mathbb{R}^{n+1}$ (the centre);
3.\ geometric definition of a perfectly round hollow 3D object.
\item[surface]
1.\ 2-manifold;
2.\ 2D combinatorial element.
\item[tesseract]
4-cube: polychoron with 8 cubical facets.
\item[vertex]
0D combinatorial element.
\item[volume]
1.\ measure of the 3D extent of an object;
2.\ 3D combinatorial element.
\item[wire]
2D combinatorial element represented by its 1D boundary.
\end{description}
\end{multicols}
% \begin{table}
% \scriptsize
% \begin{tabular}{ccccccc}
% \toprule
% \multicolumn{7}{c}{dimension} \\
% $n$ & 0 & 1 & 2 & 3 & 4 \\
% \midrule
% \multicolumn{7}{c}{topological} \\
% $n$-simplex & vertex & edge & face & volume \\
% $n$-cell & vertex & edge & face & volume \\
% $n$-sphere & pair of points & circle & sphere \\
% $n$-ball & & interval & disk & ball \\
% open $n$-ball & & open interval & open disk & open ball \\
% $n$-manifold & discrete space & curve & surface \\
% \midrule
% \multicolumn{7}{c}{geometric} \\
% $n$-simplex & point & line segment & triangle & tetrahedron & pentachoron \\
% $n$-cube & point & square & cube & tesseract \\
% $n$-orthotope & point & line segment & rectangle & cuboid/box \\
% $n$-parallelotope & point & line segment & parallelogram & parallelepiped \\
% prismatic $n$-polytope & & line segment & rectangle & prism & prismatic polychoron \\
% $n$-polytope & point & line segment & polygon & polyhedron & polychoron \\
% \midrule
% \multicolumn{7}{c}{other} \\
% $n$-th Lebesgue measure & length & area & volume \\
% & & collinear & coplanar \\
% \bottomrule
% \end{tabular}
% \end{table}