# B A short dictionary of dimension-based GIS terms

**ambient space** the space in which objects are embedded.

**area** 1. measure of the 2D extent of an object; 2. 2D combinatorial element; 3. 2D geometric object.

**ball** 1. (\(\rightarrow\) same as 3-*ball*) topological definition of the space within a 2-sphere (i.e. 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 \(n\)-*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.

**body** \(n\)-dimensional combinatorial element in an \(n\)D context.

**box** 1. cuboid; 2. orthotope of any dimension.

**cavity** 3D hole.

**cell** 1. (\(\rightarrow\) sometimes \(n\)-*cell*) \(n\)D combinatorial element; 2. 3D combinatorial element.

**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.

**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.

**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.

**congruent** having the same shape and size.

**cube** 1. (\(\rightarrow\) same as \(3\)-*cube*) 3D polyhedron with 6 square facets; 2. (\(\rightarrow\) usually \(n\)-*cube*) \(n\)-orthotope with identical \((n-1)\)-cube facets, akin to a square in 2D and a cube in 3D.

**cuboid** 1. box-shaped 3D polyhedron with 6 rectangular facets, its opposite facets are congruent and parallel; 2. parallelotope.

**curve** 1. 1D geometric object, often with a non-linear geometry; 2. 1D combinatorial element; 3. 1-manifold.

**cut-line** degenerate part of a 2D shape forming a curve and protruding inwards from its boundary.

**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).

**edge** 1D combinatorial element.

**face** 1. 2D combinatorial element; 2. (\(\rightarrow\) sometimes *\(n\)-face of \(X\)*) \(n\)D combinatorial element on the boundary of \(X\); 3. (\(\rightarrow\) usually *face of \(X\)*) \((n-1)\)D combinatorial element on the boundary of an \(n\)D combinatorial element \(X\).

**facet** 1. \((n-1)\)D combinatorial element in an \(n\)D context; 2. (\(\rightarrow\) often *facet of \(X\)*) \((n-1)\)D combinatorial element on the boundary of an \(n\)D combinatorial element \(X\); 3. 2D combinatorial element.

**flat** unbounded geometric object with linear geometry of any dimension, such as a point, line or plane.

**hole** 1. 2D void region; 2. \(n\)D void region.

**hyperball** higher-dimensional ball.

**hypercell** 1. higher-dimensional combinatorial element; 2. 4D combinatorial element.

**hypercube** 1. higher-dimensional cube; 2. 4-cube.

**hyperplane** 1. \((n-1)\)D linear subspace in an \(n\)D context; 2. plane in a higher-dimensional context.

**hyperrectangle** higher-dimensional orthotope.

**hypersphere** higher-dimensional sphere.

**hypersurface** \((n-1)\)D subspace in an \(n\)D context, often curved;

**interval** topological definition of the space between two points in \(\mathbb{R}\) (the endpoints).

**Lebesgue measure** measure of the \(n\)D extent of an object, akin to length in 1D, area in 2D and volume in 3D.

**length** measure of the 1D extent of an object.

**line segment** 1D geometric object.

**manifold** (\(\rightarrow\) sometimes \(n\)-*manifold*) topological space that resembles \(\mathbb{R}^n\) at every point.

**manifold with boundary** (\(\rightarrow\) sometimes \(n\)-*manifold with boundary*) topological space that resembles \(\mathbb{R}^n\) at every point in its interior.

**node** 0D combinatorial element.

**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.

**orthotope** (\(\rightarrow\) sometimes \(n\)-*orthotope*) \(n\)-polytope with congruent parallel facets forming right angles to each other, akin to a rectangle in 2D and a cuboid in 3D.

**parallelepiped** 3D polyhedron bounded by 6 parallelogram-shaped 2D faces.

**parallelogram** polygon bounded by 4 parallel and have the same shape.

**parallelotope** a polytope whose opposite facets are parallel and have the same shape, akin to a parallelogram in 2D and a parallelepiped in 3D.

**peak** 1. \((n-3)\)D combinatorial element in an \(n\)D context; 2. (\(\rightarrow\) *peak of \(X\)*) \((n-3)\)D combinatorial element on the boundary of an \(n\)D combinatorial element \(X\);

**plane** 2D unbounded linear subspace.

**point** 1. 0D geometric element; 2. topological definition of the space at one location.

**polychoron** 4D geometric object with linear geometry.

**polygon** 2D geometric object with linear geometry, possibly with holes.

**polygonal curve** curve formed by a sequence of line segments joined at their endpoints.

**polyhedron** 1. 3D geometric object with linear geometry, possibly with holes; 2. an \(n\)D geometric object with linear geometry.

**polyline** curve formed by a sequence of line segments joined at their endpoints.

**polyteron** 5D geometric object with linear geometry.

**polytope** \(n\)D geometric object with linear geometry.

**puncture** 0D (point) hole, often formed from a degenerate polyline or polygon.

**ridge** 1. \((n-2)\)D combinatorial element in an \(n\)D context; 2. (\(\rightarrow\) *ridge of \(X\)*) \((n-2)\)D combinatorial element on the boundary of an \(n\)D combinatorial element \(X\).

**line string** curve formed by joining a sequence of vertices by straight line segments, represented by these vertices.

**linear ring** 2D geometric object with linear geometry represented as a sequence of vertices.

**ring** 2D combinatorial element represented by its 1D boundary.

**subfacet** 1. \((n-2)\)D combinatorial element in an \(n\)D context; 2. (\(\rightarrow\) *subfacet of \(X\)*) \((n-2)\)D combinatorial element on the boundary of an \(n\)D combinatorial element \(X\).

**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.

**simplex** 1. (\(\rightarrow\) sometimes *\(n\)-simplex*) combinatorial element with \(n+1\) vertices and \(n+1\) facets; 2. (\(\rightarrow\) sometimes *\(n\)-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.

**simplicial complex** 1. (\(\rightarrow\) sometimes *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 *geometric simplicial complex*) abstract simplicial complex where simplices are embedded into Euclidean space, their interiors should not intersect geometrically.

**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 *hollow*).

**spike** degenerate part of a 2D shape forming a curve, often protruding outwards from its boundary.

**sphere** 1. (\(\rightarrow\) same as \(2\)-*sphere*) topological definition of the space at a certain distance (the radius) from a point in \(\mathbb{R}^3\) (the centre); 2. (\(\rightarrow\) often \(n\)-*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.

**surface** 1. 2-manifold; 2. 2D combinatorial element.

**tesseract** 4-cube: polychoron with 8 cubical facets.

**vertex** 0D combinatorial element.

**volume** 1. measure of the 3D extent of an object; 2. 3D combinatorial element.

**wire** 2D combinatorial element represented by its 1D boundary.