Integrated generic resource: Geometric and topological representation ISO 10303-42:2021(E)
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Cover page
Table of contents
Copyright
Foreword
Introduction
1 Scope
2 Normative references
3 Terms, definitions and abbreviated terms
    3.1 Terms and definitions
    3.2 Abbreviated terms

4 Geometry
   4.1 General
   4.2 Fundamental concepts and assumptions
   4.3 Geometry constant definition
   4.4 Geometry type definitions
   4.5 Geometry entity definitions
   4.6 Geometry function definitions
   4.7 Geometry rule definitions
5 Topology
   5.1 General
   5.2 Fundamental concepts and assumptions
   5.3 Topology constant definition
   5.4 Topology type definitions
   5.5 Topology entity definitions
   5.6 Topology function definitions
6 Geometric model
   6.1 General
   6.2 Fundamental concepts and assumptions
   6.3 Geometric model type definitions
   6.4 Geometric model entity definitions
   6.5 Geometric model function definitions
7 Scan data 3d shape model
   7.1 General
   7.2 Fundamental concepts and assumptions
   7.3 Scan data 3d shape model type definition
   7.4 Scan data 3d shape model entity definitions
   7.5 Scan data 3d shape model function definitions

A Short names of entities
B Information object registration
C Computer interpretable listings
D EXPRESS-G diagrams
E Change history
Bibliography
Index

3 Terms, definitions and abbreviated terms

3.1 Terms and definitions

3.1.1 Terms defined in ISO 10303-1

For the purposes of this document, the following terms defined in ISO 10303-1 apply:

3.1.2 Other terms and definitions

For the purposes of this document, the following terms and definitions apply:

3.1.2.1
d -manifold with boundary

domain which is the union of its d -dimensional interior and its boundary

3.1.2.2
arcwise connected

such that any pair of distinct points in the relevant domain may be connected by a continuous arc entirely contained within that domain

3.1.2.3
axi-symmetric

invariant under all rotations about a central axis

3.1.2.4
boundary

subset of the points x in a domain X having the property that any open ball U centred on x satisfies U ⋂ X ≠ U

NOTE    Any open ball centred on the boundary of the domain will contain both points inside the domain and points outside the domain.

3.1.2.5
boundary representation solid model

B-rep

type of geometric model in which the size and shape of a solid is defined in terms of the faces, edges and vertices which make up its boundary

3.1.2.6
bounds

limits of a topological entity

NOTE    Bounds are the topological entities of lower dimensionality which mark the limits of a topological entity. The bounds of a face are loops, and the bounds of an edge are vertices.

3.1.2.7
closed curve

curve such that both end points are the same

3.1.2.8
closed surface

connected 2-manifold that divides space into exactly two connected components, one of which is finite

3.1.2.9
completion of a topological entity

set consisting of the entity in question together with all the faces, edges and vertices referenced, directly or indirectly, in the definition of the bounds of that entity

3.1.2.10
connected

synonym for arcwise connected

3.1.2.11
connected component

maximal connected subset of a domain

3.1.2.12
constructive solid geometry

CSG

type of geometric modelling in which a solid is defined as the result of a sequence of regularised Boolean operations operating on solid models

3.1.2.13
coordinate space

reference system that associates a unique set of n parameters with each point in an n-dimensional space

3.1.2.14
curve

set of mathematical points which is the image, in two- or three-dimensional space, of a continuous function defined over a connected subset of the real line R 1 , and which is not a single point

3.1.2.15
cycle

chain of alternating vertices and edges in a graph such that the first and last vertices are the same

3.1.2.16
dimensionality

number of independent coordinates in the parameter space of a geometric entity

NOTE    A curve has dimensionality 1, a surface has dimensionality 2. The dimensionality of topological entities which need not have domains is specified in the entity definitions. The dimensionality of a list or set is the maximum of the dimensionalities of the elements of that list or set.

3.1.2.17
domain

mathematical point set in model space corresponding to an entity

3.1.2.18
euler equations

equations used to verify the topological consistency of objects

NOTE    Various equalities relating topological properties of entities are derived from the invariance of a number known as the Euler characteristic. Typically, these are used as quick checks on the integrity of the topological structure. A violation of an Euler condition signals an "impossible" object. Two special cases are important in this document. The Euler equation for graphs is discussed in 5.2.3. Euler conditions for surfaces are discussed in 5.4.25 and 5.2,27.

3.1.2.19
extent

measure of the domain of a geometric entity in units appropriate to the dimensionality of the entity

NOTE    Length, area and volume are used for dimensionalities 1, 2, and 3, respectively. Where necessary, the symbol Ξ will be used to denote extent.

3.1.2.20
facet

a planar triangle

3.1.2.21
finite

capable of being completely counted or measured

NOTE    An entity is finite (or alternatively bounded) if there is a finite upper bound on the distance between any two points in its domain.

3.1.2.22
genus of a graph

integer-valued invariant defined algorithmically by the graph traversal algorithm

NOTE    The graph traversal algorithm is described in the note in 5.2.3.

3.1.2.23
genus of a surface

number of handles that are added to a sphere to produce a surface homeomorphic to the surface in question

NOTE    Handle is defined below.

3.1.2.24
geometric coordinate system

underlying global rectangular Cartesian coordinate system to which all geometry refers

3.1.2.25
geometrically founded

having an associated coordinate space

NOTE    Geometric founding is a property of geometric_representation_item s (see 4.4.2) asserting their relationship to a coordinate space in which the coordinate values of points and directions on which they depend for position and orientation are measured.

3.1.2.26
geometrically related

related by being in the same geometric context

NOTE    If two geometric_representation_item s (see 4.4.2) are geometricaslly related then the concepts of distance and direction between them are defined.

3.1.2.27
graph

set of vertices and edges

NOTE    The graphs discussed in this document are generally called pseudographs in the technical literature because they allow self-loops and also multiple edges connecting the same two vertices.

3.1.2.28
handle

structure distinguishing a torus from a sphere, which can be viewed as a tubular surface connecting two holes in a surface

3.1.2.29
homeomorphic

in one to one correspondence

NOTE    Domains X and Y are homeomorphic if there is a continuous function f from X to Y which is a one-to-one correspondence, so that the inverse function f  -1 exists, and f  -1 is also continuous.

3.1.2.30
inside

completely included within

NOTE    A domain X is inside domain Y if both domains are contained in the same Euclidean space, R m , and Y separates R m into exactly two connected components, one of which is finite, and X is contained in the finite component.

3.1.2.31
interior

point set resulting from exclusion of all boundary points from a bounded point set

NOTE    The d-dimensional interior of a d-dimensional domain X contained in R m is the set of mathematical points x in X for which there is an open ball U in R m containing x such that the intersection U ⋂ X is homeomorphic to an open ball in R d .

3.1.2.32
list

ordered homogeneous collection with possibly duplicate members

3.1.2.33
model space

space with dimensionality 2 or 3 in which the geometry of a representation of a physical object, or any of its elements, is defined

3.1.2.34
open curve

curve which has two distinct end points

3.1.2.35
open surface

surface which is a manifold with boundary, but is not closed

NOTE    Either it is not finite, or it does not divide space into exactly two connected components.

3.1.2.36
orientable

capable of being oriented in space

NOTE    A surface is orientable if a consistent, continuously varying choice can be made of the sense of the normal vectors to the surface. This does not require a continuously varying choice of the values of the normal vectors; the surface may have tangent plane discontinuities.

3.1.2.37
overlap

area or range shared in common by two or more geometric elements

NOTE    Two topological entities overlap when they have shells, faces, edges, or vertices in common.

3.1.2.38
parameter range

range of valid parameter values for a curve, surface, or volume

3.1.2.39
parameter space

one-dimensional space associated with a curve via its uniquely defined parameterisation, or the two-dimensional space associated with a surface

3.1.2.40
parametric volume

bounded region of three dimensional model space with an associated parametric coordinate system such that every interior point is associated with a list (u,v,w) of parameter values

3.1.2.41
placement coordinate system

rectangular Cartesian coordinate system associated with the placement of a geometric entity in space

NOTE    This is used to describe the interpretation of the attributes and to associate a unique parameterisation with curve and surface entities

3.1.2.42
self-intersect

intersection of a geometric element with itself

NOTE    A curve or surface self-intersects if there is a mathematical point in its domain which is the image of at least two points in the object's parameter range, and one of those two points lies in the interior of the parameter range. A vertex, edge or face self-intersects if its domain does.

NOTE    A curve or surface is not considered to be self-intersecting just because it is closed.

3.1.2.43
self-loop

edge that has the same vertex at both ends

3.1.2.44
set

unordered collection in which no two members are equal

3.1.2.45
space dimensionality

number of parameters required to define the location of a point in the coordinate space

3.1.2.46
surface

set of mathematical points which is the image of a continuous function defined over a connected subset of the plane R 2

3.1.2.47
tessellated geometry

geometry composed of a large number of planar tiles, usually of triangular shape

NOTE    Tessellated geometry is frequently used as an approximation to the exact shape of an object.

3.1.2.48
topological sense

sense of a topological entity as derived from the order of its attributes

EXAMPLE 1   The topological sense of an edge is from the edge start vertex to the edge end vertex.

EXAMPLE 2   The topological sense of a path follows the edges in their listed order.

3.2 Abbreviated terms

For the purposes of this document, the following abbreviated terms apply:

URL uniform resource locator
B-rep boundary representation solid model
CSG constructive solid geometry


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