132 4 Geometry standards
sons the defining points of a parametric curve surface often lie in a square pattern
and thus form a quadrilateral grid. The resulting surface is called a gridded surface
(class GM_GriddedSurface). The ISO 19107 standardises six types of gridded surface: cone, cylinder, sphere, bilinear grid, bicubic grid, and B-spline surface (classes
GM_Cone, GM_Cylinder, GM_Sphere, GM_BilinearGrid, GM_BicubicGrid, and
GM_BSplineSurface respectively). Abilinear grid uses line strings as horizontal and
vertical curves while a bicubic grid uses cubic polynomial splines as horizontal and
vertical curves. The names bilinear and bicubic grid will not be confused with the respective terms used for the interpolation methods.
A surface with only polygon surface patches is called a polyhedral surface (class
GM _PolyhedraISurface). If the polygons are triangles it is called a triangulated surface (GM_TriangulatedSurface) with no restriction on how the triangulation is derived.
A TIN ( class GM_Tin) is a triangulated surface that uses the Delaunay algorithm
or a similar algorithm complemented with consideration for breaklines, stoplines,
and maximum length of triangle sides. These networks satisfy the Delaunay criterion
away from the modifications. For each triangle in the network, the circle passing
through its vertexes does not contain the vertex of any other triangle in its interior.
The TIN is also covered by the ISO 19123 (Schema for coverage geometry and
functions). The ISO 19107 standardises the description ofan existing TIN. The ISO
19123 addresses the computation of a TIN and the interpolation of elevations.
4.3.3 Detailed description of the geometry classes of ISO 19107
The following ehapter gives an exhaustive list of the geometrie elements of ISO
19107.
4.3.3.1 Dimensions and map projection
The ISO 19107 standardises geometries in the 3-dimensional space as a base for
the all ISO 19100 standards. The theory of ISO 19107 also allows its use in an ndimensional space. However, the application of ISO 19107 leads to some inconsistancies.
Dimensions
The base geometries are points, eurves, surfaces, and solids. Not all geometries
are weIl defined in a 3-dimensional space.
Arcs and circles have their shape on one plane only. Any projection to a coordinate plane changes them to an elliptic are or an ellipse other than the exceptional
ease that the plane ofthe geometry is parallel to a eoordinate plane.
Cones keep their properties as being cones but change their shape.
The mathematies of splines and clothoids are defined in two dimensions only. In
the ease where they are used in three dimensions, the formulas are usually expressed
in the parameterized form. This simply means that an additional dimension "t " is in-
sons the defining points of a parametric curve surface often lie in a square pattern
and thus form a quadrilateral grid. The resulting surface is called a gridded surface
(class GM_GriddedSurface). The ISO 19107 standardises six types of gridded surface: cone, cylinder, sphere, bilinear grid, bicubic grid, and B-spline surface (classes
GM_Cone, GM_Cylinder, GM_Sphere, GM_BilinearGrid, GM_BicubicGrid, and
GM_BSplineSurface respectively). Abilinear grid uses line strings as horizontal and
vertical curves while a bicubic grid uses cubic polynomial splines as horizontal and
vertical curves. The names bilinear and bicubic grid will not be confused with the respective terms used for the interpolation methods.
A surface with only polygon surface patches is called a polyhedral surface (class
GM _PolyhedraISurface). If the polygons are triangles it is called a triangulated surface (GM_TriangulatedSurface) with no restriction on how the triangulation is derived.
A TIN ( class GM_Tin) is a triangulated surface that uses the Delaunay algorithm
or a similar algorithm complemented with consideration for breaklines, stoplines,
and maximum length of triangle sides. These networks satisfy the Delaunay criterion
away from the modifications. For each triangle in the network, the circle passing
through its vertexes does not contain the vertex of any other triangle in its interior.
The TIN is also covered by the ISO 19123 (Schema for coverage geometry and
functions). The ISO 19107 standardises the description ofan existing TIN. The ISO
19123 addresses the computation of a TIN and the interpolation of elevations.
4.3.3 Detailed description of the geometry classes of ISO 19107
The following ehapter gives an exhaustive list of the geometrie elements of ISO
19107.
4.3.3.1 Dimensions and map projection
The ISO 19107 standardises geometries in the 3-dimensional space as a base for
the all ISO 19100 standards. The theory of ISO 19107 also allows its use in an ndimensional space. However, the application of ISO 19107 leads to some inconsistancies.
Dimensions
The base geometries are points, eurves, surfaces, and solids. Not all geometries
are weIl defined in a 3-dimensional space.
Arcs and circles have their shape on one plane only. Any projection to a coordinate plane changes them to an elliptic are or an ellipse other than the exceptional
ease that the plane ofthe geometry is parallel to a eoordinate plane.
Cones keep their properties as being cones but change their shape.
The mathematies of splines and clothoids are defined in two dimensions only. In
the ease where they are used in three dimensions, the formulas are usually expressed
in the parameterized form. This simply means that an additional dimension "t " is in-