128
4 Geometry standards
A knot is a special type of point that is only used for the description of some types
of splines. For example a Bezier spline is defined by four points where the curve
passes through the first and last point and is only controlled in shape by the second
and third point. The four points are the knots ofthe Bezier spline.
Each dimension requires its own sequence of knots. Thus a spline curve has one
sequence and a spline surface has two sequences.
4.3 Spatial schema (ISO 19107)
The ISO 19107 covers a fairly complete world ofvector data and has the following characteristics:
• 3-dimensional space, in theory n-dimensional
• primitives and complexes
• topological relations
• based on Set Theory
Set Theory is the mathematical science of the infinite. It is the study of the properties of sets, abstract objects that pervade the whole of modem mathematics. The language of Set Theory, in its simplicity, is sufficiently universal to formalize all
mathematical concepts and thus Set Theory, along with a few other theories, constitutes the true Foundations ofMathematics (Jech 2002).
The ISO 19107 applies the axioms of Set Theory throughout the standard. An example is a geometrie object that is defined as a set of geometrie points.
The ISO 19107 has two important design criteria: the boundary-criterion and the
complexes. The boundary-criterion means that high-level elements are composed of
a collection of low-Ievel elements. For example, a surface consists of its boundary
curves, and the curves are bounded by their start and end points. The complexes,
both geometrie al and topologieal, consist of geometries, which do not overlap.
4.3.1 Overview over ISO 19107
The following diagram shows the logical tree of the important geometry classes of
ISO 19107.
4 Geometry standards
A knot is a special type of point that is only used for the description of some types
of splines. For example a Bezier spline is defined by four points where the curve
passes through the first and last point and is only controlled in shape by the second
and third point. The four points are the knots ofthe Bezier spline.
Each dimension requires its own sequence of knots. Thus a spline curve has one
sequence and a spline surface has two sequences.
4.3 Spatial schema (ISO 19107)
The ISO 19107 covers a fairly complete world ofvector data and has the following characteristics:
• 3-dimensional space, in theory n-dimensional
• primitives and complexes
• topological relations
• based on Set Theory
Set Theory is the mathematical science of the infinite. It is the study of the properties of sets, abstract objects that pervade the whole of modem mathematics. The language of Set Theory, in its simplicity, is sufficiently universal to formalize all
mathematical concepts and thus Set Theory, along with a few other theories, constitutes the true Foundations ofMathematics (Jech 2002).
The ISO 19107 applies the axioms of Set Theory throughout the standard. An example is a geometrie object that is defined as a set of geometrie points.
The ISO 19107 has two important design criteria: the boundary-criterion and the
complexes. The boundary-criterion means that high-level elements are composed of
a collection of low-Ievel elements. For example, a surface consists of its boundary
curves, and the curves are bounded by their start and end points. The complexes,
both geometrie al and topologieal, consist of geometries, which do not overlap.
4.3.1 Overview over ISO 19107
The following diagram shows the logical tree of the important geometry classes of
ISO 19107.