ISO/TC 211 Multi-Lingual Glossary of Terms (MLGT)

Note to entry: The free variables mentioned above can usually be thought of as a local coordinate system. In a 3D coordinate space, a plane can be written as ${\displaystyle P(u,v)=A+uX+vY}$ , where ${\displaystyle u}$ and ${\displaystyle v}$ are real numbers and ${\displaystyle A}$ is any point on the plane, and ${\displaystyle X}$ and ${\displaystyle Y}$ are two vectors tangent to the plane. Since the locations on the plane can be distinguished by ${\displaystyle u}$ and ${\displaystyle v}$ (here universally), the plane is 2D and ${\displaystyle (u,v)}$ is a coordinate system for the points on the plane. On generic surfaces, this cannot, in general, be done universally. If we take a plane tangent to the surface, and project points on the surface onto this plane, we will normally get a local isomorphism for small neighbourhoods of the point of tangency. This "local coordinate" system for the underlying surface is sufficient to establish the surface as a 2D topological object. Since this International Standard deals only with spatial coordinates, any 3D object can rely on coordinates to establish its topological dimension. In a 4D model (spatio-temporal), tangent spaces also play an important role in establishing topological dimension for objects up to 3D.