# 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.