% !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}