SIST EN ISO 19141:2009

SLOVENSKI STANDARD
01-september-2009
*HRJUDIVNHLQIRUPDFLMH6KHPD]DSUHPLNDMRþHHQRWH,62
Geographic information - Schema for moving features (ISO 19141:2008)
Information géographique - Schéma des entités mobiles (ISO 19141:2008)
Ta slovenski standard je istoveten z: EN ISO 19141:2009
ICS:
35.240.70 Uporabniške rešitve IT v IT applications in science
znanosti
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

EUROPEAN STANDARD
EN ISO 19141
NORME EUROPÉENNE
EUROPÄISCHE NORM
August 2009
ICS 35.240.70
English Version
Geographic information - Schema for moving features (ISO
19141:2008)
Information géographique - Schéma des entités mobiles Geoinformation - Schema für sich bewegende Objekte (ISO
(ISO 19141:2008) 19141:2008)
This European Standard was approved by CEN on 30 July 2009.
CEN members are bound to comply with the CEN/CENELEC Internal Regulations which stipulate the conditions for giving this European
Standard the status of a national standard without any alteration. Up-to-date lists and bibliographical references concerning such national
standards may be obtained on application to the CEN Management Centre or to any CEN member.
This European Standard exists in three official versions (English, French, German). A version in any other language made by translation
under the responsibility of a CEN member into its own language and notified to the CEN Management Centre has the same status as the
official versions.
CEN members are the national standards bodies of Austria, Belgium, Bulgaria, Cyprus, Czech Republic, Denmark, Estonia, Finland,
France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal,
Romania, Slovakia, Slovenia, Spain, Sweden, Switzerland and United Kingdom.
EUROPEAN COMMITTEE FOR STANDARDIZATION
COMITÉ EUROPÉEN DE NORMALISATION
EUROPÄISCHES KOMITEE FÜR NORMUNG
Management Centre: Avenue Marnix 17, B-1000 Brussels
© 2009 CEN All rights of exploitation in any form and by any means reserved Ref. No. EN ISO 19141:2009: E
worldwide for CEN national Members.

Contents Page
Foreword .3

Foreword
The text of ISO 19141:2008 has been prepared by Technical Committee ISO/TC 211 “Geographic
information/Geomatics” of the International Organization for Standardization (ISO) and has been taken over
as EN ISO 19141:2009 by Technical Committee CEN/TC 287 “Geographic Information” the secretariat of
which is held by NEN.
This European Standard shall be given the status of a national standard, either by publication of an identical
text or by endorsement, at the latest by February 2010, and conflicting national standards shall be withdrawn
at the latest by February 2010.
Attention is drawn to the possibility that some of the elements of this document may be the subject of patent
rights. CEN [and/or CENELEC] shall not be held responsible for identifying any or all such patent rights.
According to the CEN/CENELEC Internal Regulations, the national standards organizations of the following
countries are bound to implement this European Standard: Austria, Belgium, Bulgaria, Cyprus, Czech
Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia,
Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Romania, Slovakia, Slovenia, Spain,
Sweden, Switzerland and the United Kingdom.
Endorsement notice
The text of ISO 19141:2008 has been approved by CEN as a EN ISO 19141:2009 without any modification.

INTERNATIONAL ISO
STANDARD 19141
First edition
2008-06-01
Geographic information — Schema for
moving features
Information géographique — Schéma des entités mobiles

Reference number
ISO 19141:2008(E)
©
ISO 2008
ISO 19141:2008(E)
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©  ISO 2008
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ii © ISO 2008 – All rights reserved

ISO 19141:2008(E)
Contents Page
Foreword. v
Introduction . vi
1 Scope . 1
2 Conformance. 1
2.1 Conformance classes. 1
2.2 Requirements . 2
3 Normative references . 2
4 Terms, definitions, and abbreviated terms . 3
4.1 Terms and definitions. 3
4.2 Abbreviated terms . 5
5 Package – Moving Features. 6
5.1 Semantics . 6
5.2 Package structure. 7
5.3 Class hierarchy . 7
6 Package – Geometry Types . 9
6.1 Package semantics. 9
6.2 Type – MF_OneParamGeometry . 9
6.3 Type – MF_TemporalGeometry . 11
6.4 Type – MF_Trajectory. 12
6.5 Type – MF_TemporalTrajectory. 14
6.6 Class – MF_PositionExpression . 20
6.7 Type – MF_SecondaryOffset . 20
6.8 Type – MF_MeasureFunction . 21
7 Package – Prism Geometry . 22
7.1 Package structure. 22
7.2 CodeList – MF_GlobalAxisName. 23
7.3 Type – MF_LocalGeometry . 25
7.4 Type – MF_PrismGeometry . 27
7.5 Type – MF_RigidTemporalGeometry . 28
7.6 Type – MF_RotationMatrix . 29
7.7 Type – MF_TemporalOrientation. 30
8 Moving features in application schemas.30
8.1 Introduction . 30
8.2 Representing the spatial characteristics of moving features . 31
8.3 Associations of moving features . 31
8.4 Operations of moving features. 31
Annex A (normative) Abstract test suite. 32
A.1 Application schemas for data transfer . 32
A.2 Application schemas for data with operations. 32
Annex B (informative) UML Notation. 34
B.1 Introduction . 34
B.2 Class. 34
B.3 Stereotype . 34
B.4 Attribute . 35
B.5 Operation . 35
B.6 Constraint . 36
B.7 Note . 36
ISO 19141:2008(E)
B.8 Association. 36
B.9 Role name . 36
B.10 Multiplicity. 37
B.11 Navigability . 37
B.12 Aggregation . 37
B.13 Composition . 38
B.14 Dependency. 38
B.15 Generalization. 38
B.16 Realization . 39
Annex C (informative) Interpolating between orientations . 40
C.1 Introduction . 40
C.2 Euler rotations and gimbal lock. 40
C.3 Interpolating between two orientation matrices . 42
C.4 Interpolating between other orientation representations . 44
C.5 Sample interpolation. 45
Bibliography . 49

iv © ISO 2008 – All rights reserved

ISO 19141:2008(E)
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies
(ISO member bodies). The work of preparing International Standards is normally carried out through ISO
technical committees. Each member body interested in a subject for which a technical committee has been
established has the right to be represented on that committee. International organizations, governmental and
non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely with the
International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of technical committees is to prepare International Standards. Draft International Standards
adopted by the technical committees are circulated to the member bodies for voting. Publication as an
International Standard requires approval by at least 75 % of the member bodies casting a vote.
Attention is drawn to the possibility that some of the elements of this document may be the subject of patent
rights. ISO shall not be held responsible for identifying any or all such patent rights.
ISO 19141 was prepared by Technical Committee ISO/TC 211, Geographic information/Geomatics.

ISO 19141:2008(E)
Introduction
This International Standard specifies a conceptual schema that addresses moving features, i.e., features
whose locations change over time. This schema includes classes, attributes, associations and operations that
provide a common conceptual framework that can be implemented to support various application areas that
deal with moving features, including:
⎯ Location Based Services,
⎯ Intelligent Transportation Systems,
⎯ Tracking and navigation (land-based, marine, or space), and
⎯ Modeling and simulation.
The schema specifies mechanisms to describe motion consisting of translation and/or rotation of the feature,
but not including deformation of the feature. The schema is based on the concept of a one parameter set of
geometries that may be viewed as a set of leaves or a set of trajectories, where a leaf represents the
geometry of the moving feature at a particular value of the parameter (e.g., a point in time) and a trajectory is
a curve that represents the path of a point in the geometry of the moving feature as it moves with respect to
the parameter.
vi © ISO 2008 – All rights reserved

INTERNATIONAL STANDARD ISO 19141:2008(E)

Geographic information — Schema for moving features
1 Scope
This International Standard defines a method to describe the geometry of a feature that moves as a rigid body.
Such movement has the following characteristics.
a) The feature moves within any domain composed of spatial objects as specified in ISO 19107.
b) The feature may move along a planned route, but it may deviate from the planned route.
c) Motion may be influenced by physical forces, such as orbital, gravitational, or inertial forces.
d) Motion of a feature may influence or be influenced by other features, for example:
1) The moving feature might follow a predefined route (e.g. road), perhaps part of a network, and might
change routes at known points (e.g. bus stops, waypoints).
2) Two or more moving features may be “pulled” together or pushed apart (e.g. an airplane will be
refuelled during flight, a predator detects and tracks a prey, refugee groups join forces).
3) Two or more moving features may be constrained to maintain a given spatial relationship for some
period (e.g. tractor and trailer, convoy).
This International Standard does not address other types of change to the feature. Examples of changes that
are not adressed include the following:
⎯ The deformation of features.
⎯ The succession of either features or their associations.
⎯ The change of non-spatial attributes of features.
⎯ The feature’s geometric representation cannot be embedded in a geometric complex that contains the
geometric representations of other features, since this would require the other features’ representations to
be updated as the feature moves.
Because this International Standard is concerned with the geometric description of feature movement, it does
not specify a mechanism for describing feature motion in terms of geographic identifiers. This is done, in part,
in ISO 19133.
2 Conformance
2.1 Conformance classes
2.1.1 Introduction
This International Standard specifies four conformance classes (Table 1). They are differentiated on the basis
of two criteria: purpose and level of complexity.
ISO 19141:2008(E)
2.1.2 Purpose
This International Standard may be used in support of data transfer. Operations defined for objects are
irrelevant to data transfer, which requires only descriptions of the state of the objects at the time of transfer.
Thus, two conformance classes require only the implementation of attributes and associations of the classes
specified in the schema. The other two conformance classes support the object-oriented implementation of
systems or interfaces; they require implementation of operations as well as implementation of attributes and
associations.
2.1.3 Complexity
Many applications do not need a complete description of the geometry of a feature and its orientation at any
point in time. Their requirements are satisfied by describing the movement of a single reference point on the
feature using its trajectory as specified in Clause 6. One pair of conformance classes supports these simple
applications.
Other applications need knowledge of the positions at each time of all points or a significant subset of the
points on a moving feature. They require the full description provided by the prism geometry specified in
Clause 7.
Table 1 — Conformance classes
Purpose
Complexity
Data Transfer Data with operations
Trajectory A.1.1 A.2.1
Prism Geometry A.1.2 A.2.2
2.2 Requirements
To conform to this International Standard, an application schema shall satisfy the requirements of the Abstract
Test Suite in Annex A.
3 Normative references
The following referenced documents are indispensable for the application of this document. For dated
references, only the edition cited applies. For undated references, the latest edition of the referenced
document (including any amendments) applies.
ISO/TS 19103, Geographic information — Conceptual schema language
ISO 19107, Geographic information — Spatial schema
ISO 19108, Geographic information — Temporal schema
ISO 19109, Geographic information — Rules for application schema
ISO 19133, Geographic information — Location-based services — Tracking and navigation
2 © ISO 2008 – All rights reserved

ISO 19141:2008(E)
4 Terms, definitions, and abbreviated terms
4.1 Terms and definitions
For the purposes of this document, the following terms and definitions apply.
4.1.1
base representation
〈moving features〉 representation, using a local origin and local ordinate vectors, of a geometric object at a
given reference time
NOTE 1 A rigid geometric object may undergo translation or rotation, but remains congruent with its base
representation.
NOTE 2 The local origin and ordinate vectors establish an engineering coordinate reference system (ISO 19111), also
called a local frame or a local Euclidean coordinate system.
4.1.2
curve
1-dimensional geometric primitive, representing the continuous image of a line
[ISO 19107:2003, definition 4.23]
NOTE The boundary of a curve is the set of points at either end of the curve. If the curve is a cycle, the two ends are
identical, and the curve (if topologically closed) is considered to not have a boundary. The first point is called the start
point, and the last is the end point. Connectivity of the curve is guaranteed by the "continuous image of a line" clause. A
topological theorem states that a continuous image of a connected set is connected.
4.1.3
design coordinate reference system
engineering coordinate reference system in which the base representation of a moving object is specified
4.1.4
feature
abstraction of real world phenomena
[ISO 19101:2002, definition 4.11]
NOTE A feature may occur as a type or an instance. Feature type or feature instance shall be used when only one is
meant.
4.1.5
feature association
relationship that links instances of one feature type with instances of the same or a different feature type
[ISO 19110:2004, definition 4.2]
NOTE Feature associations include aggregation of features.
4.1.6
feature attribute
characteristic of a feature
[ISO 19101:2002, definition 4.12]
4.1.7
feature operation
operation that every instance of a feature type may perform
[ISO 19110:2004, definition 4.5]
ISO 19141:2008(E)
4.1.8
foliation
one parameter set of geometries such that each point in the prism of the set is in one and only one
trajectory and in one and only one leaf
4.1.9
geometric object
spatial object representing a geometric set
[ISO 19107:2003, definition 4.47]
4.1.10
geometric primitive
geometric object representing a single, connected, homogeneous element of space
[ISO 19107:2003, definition 4.48]
NOTE Geometric primitives are non-decomposed objects that present information about geometric configuration.
They include points, curves, surfaces, and solids.
4.1.11
instant
0-dimensional geometric primitive representing position in time
[ISO 19108:2002, definition 4.1.17]
4.1.12
leaf
〈one parameter set of geometries〉 geometry at a particular value of the parameter
4.1.13
location-based service
LBS
service whose return or other property is dependent on the location of the client requesting the service or of
some other thing, object or person
[ISO 19133:2005, definition 4.11]
4.1.14
network
abstract structure consisting of a set of 0-dimensional objects called junctions, and a set of 1-dimensional
objects called links that connect the junctions, each link being associated with a start (origin, source) junction
and end (destination, sink) junction
[ISO 19133:2005, definition 4.17]
NOTE The network is essentially the universe of discourse for the navigation problem. Networks are a variety of 1-
dimensional topological complex. In this light, junction and topological node are synonyms, as are link and directed edge.
4.1.15
one parameter set of geometries
function f from an interval t ∈ [a, b] such that f(t) is a geometry and for each point P ∈ f(a) there is a one
parameter set of points (called the trajectory of P) P(t) : [a, b] →P(t) such that P(t) ∈ f(t)
EXAMPLE A curve C with constructive parameter t is a one parameter set of points c(t).
4.1.16
period
one-dimensional geometric primitive representing extent in time
[ISO 19108:2002, definition 4.1.27]
NOTE A period is bounded by two different temporal positions.
4 © ISO 2008 – All rights reserved

ISO 19141:2008(E)
4.1.17
point
0-dimensional geometric primitive, representing a position
[ISO 19107:2003, definition 4.61]
NOTE The boundary of a point is the empty set.
4.1.18
prism
〈one parameter set of geometries〉 set of points in the union of the geometries (or the union of the
trajectories) of a one parameter set of geometries
NOTE This is a generalization of the concept of a geometric prism that is the convex hull of two congruent polygons
in 3D-space. Such polyhedrons can be viewed as a foliation of congruent polygons.
4.1.19
temporal coordinate system
temporal reference system based on an interval scale on which distance is measured as a multiple of a
single unit of time
[ISO 19108:2002, definition 4.1.31]
4.1.20
temporal position
location relative to a temporal reference system
[ISO 19108:2002, definition 4.1.34]
4.1.21
temporal reference system
reference system against which time is measured
[ISO 19108:2002, definition 4.1.35]
4.1.22
trajectory
path of a moving point described by a one parameter set of points
4.1.23
vector
quantity having direction as well as magnitude
[ISO 19123:2005, definition 4.1.43]
4.2 Abbreviated terms
CRS Coordinate Reference System (ISO 19111)
SLERP Spherical Linear Interpolation
LRS Linear Referencing System (ISO 19133)
OCL Object Constraint Language (ISO/IEC 19501)
UML Unified Modelling Language (ISO/IEC 19501)
ISO 19141:2008(E)
5 Package – Moving Features
5.1 Semantics
A moving feature can be modelled as a combination of movements. The overall motion can be expressed as
the temporal path or trajectory of some reference point on the object (the “origin”), such as its center of gravity.
Once the origin’s trajectory has been established, the position along the trajectory can be described using a
linear reference system (as defined in ISO 19133). The “parameterization by length” for curves (as defined in
ISO 19107) can be used as a simple linear reference if no other is available. The relationship between time (t)
and measure value (m) can be represented as the graph of the t→ m function in a plane with coordinates
(t, m). This separation of the geometry of the path and the actual “time to position” function allows the moving
feature to be tracked along existing geometry.
Figure 1 illustrates how the concepts of foliation, prism, trajectory, and leaf relate to one another. In this
illustration, a 2D rectangle moves and rotates. Each representation of the rectangle at a given time is a leaf.
The path traced by each corner point of the rectangle (and by each of its other points) is a trajectory. The set
of points contained in all of the leaves, and in all of the trajectories, forms a prism. The set of leaves also
forms a foliation.
Figure 1 — Feature movement as foliation
These two object representations, of the path and the position along that path, give the general position of the
moving feature. The other variable in describing the position of the feature is the rotation about the chosen
reference point. To describe this, a local engineering coordinate system is established using the object
reference point as its origin. The geometry of the feature is described in the engineering coordinate system
and the real-world orientation of the feature is given by mapping of the local coordinate axes to the global
coordinate system (the CRS of the trajectory of the reference point). This can be given as a matrix that maps
the unit vectors of the local coordinate system to vectors in the global CRS.
If the global CRS and local CRS have the same dimension, then each point within the local CRS can be
traced in time through the global CRS by combinations of these various mappings. The map would trace from
time (t) to the measure (m) to a position on the reference point's path using the LRS. Then using the rotation
matrix, the calculated offset from this point gives a direct position in the global CRS.
This means that the ‘prism’ of the moving feature (defined as all the points which part of the feature passes
through) can be viewed (and calculated to whatever degree of accuracy needed) as a bundle of trajectories of
6 © ISO 2008 – All rights reserved

ISO 19141:2008(E)
points on the local engineering representation of the feature's geometry. If viewed in a 4 dimensional spatio-
temporal coordinate system, the points on the feature at different times are different points. Then the pre-
image of the prism (points on the trajectories augmented by a time coordinate) is a foliation, meaning that
there is a complete and separate representation of the geometry of the feature for each specific time (called a
“leaf”). These names come from a 3D metaphor of a book, where each page or leaf is a slice of time in the
“folio”.
This might form the basis for an extension of this standard to non-rigid, mutable objects. Each leaf in the 4D
foliation is a separate representation of the object, and by creating methods to describe the change through
time of the shape and form of the feature, the existing machinery in this International Standard can be used to
place those representations in positions with respect to the global coordinate system.
5.2 Package structure
This clause presents a conceptual schema for describing moving features that is specified using the Unified
Modelling Language (UML) [ISO/IEC 19501], following the guidance of ISO/TS 19103. Annex B describes
UML notation as used in this International Standard.
The schema is contained in the UML package Moving Features. Names of classes included in this package
carry the prefix "MF_". The package is subdivided into two leaf packages (Figure 2), Geometry Types and
Prism Geometry. The classes in these two packages are derived from classes included in the Geometry
Package specified in ISO 19107. Classes from the packages Basic Types [ISO/TS 19103], Geometry
[ISO 19107], Temporal Objects, and Temporal Reference System [ISO 19108] are used as data types in the
schema.
5.3 Class hierarchy
The classes of the moving features schema form an inheritance hierarchy that has its source in the classes
GM_Object and GM_Curve specified in ISO 19107 (Figure 3). This allows the subclasses specific to this
schema to be used as feature attributes in compliance with the General Feature Model specified in ISO 19109.
The second level of the hierarchy consists of a set of classes that describe a one-parameter geometry. These
might be used to describe the movement of a feature with respect to any single variable such as pressure,
temperature, or time. The third level specializes these classes to describe motion in time. The classes are
specified fully in Clauses 6 and 7.
ISO 19141:2008(E)
<>
Geometry
(from ISO 19107 Spatial Schema)
ISO 19141 Moving Features
(from Logical View)
<>
Geometry Types
Coordinate geometry
(from Geometry)
+ MF_MeasureFunction
+ MF_OneParamGeometry
+ MF_PositionExpression
+ MF_SecondaryOffset
+ MF_TemporalGeometry
<>
+ MF_TemporalTrajectory
Geometry root
+ MF_Trajectory
(from Geometry)
<>
Geometric primitive
(from Geometry)
Prism Geometry
+ MF_GlobalAxisName
+ MF_LocalGeometry
+ MF_PrismGeometry
+ MF_RigidTemporalGeometry
<>
+ MF_RotationMatrix
Temporal Reference System
+ MF_TemporalOrientation
(from ISO 19108 Temporal)
Figure 2 — Moving Feature Package
<> <>
GM_Object GM_Curve
(from Geometry root) (from Geometric primitive)
<> <>
MF_OneParamGeometry MF_Trajectory
<> <>
MF_TemporalGeometry MF_TemporalTrajectory

Figure 3 — Components of the Geometry Types Package
8 © ISO 2008 – All rights reserved

ISO 19141:2008(E)
6 Package – Geometry Types
6.1 Package semantics
The Geometry Types package contains seven types. Two classes – MF_OneParamGeometry and
MF_Trajectory – specify one-parameter geometry types based on the geometric objects specified in
ISO 19107 (see Figure 3). Two other classes – MF_TemporalGeometry and MF_TemporalTrajectory –
specialize the first classes in order to specify a one-parameter set of geometries in which the parameter is
time. The other three classes – MF_MeasureFunction, MF_SecondaryOffset and MF_PositionExpression
(Figure 4) – are used to extend the concepts of linear reference systems as defined in ISO 19133. Description
of movement in terms of geographic identifiers is out of scope, and is partly covered in ISO 19133.
<>
LR_LinearReferenceMethod
(from Linear Reference Systems)
+LRM
<>
LR_OffsetReference
(from Linear Reference Systems)
+ centerline
+ edgeOfTravel
+ rightOfWay
+referenceElement 1.*
+ curb
+ edgeOfBerm
+ sidewalkInside
<>
+referenceDomain
+ sidewalkOutside
LR_Element
(from Linear Reference Systems)
{ordered}
+datumMarkers
0.* 0.* +marker
<> <>
0.1
LR_PositionExpression LR_ReferenceMarker
(from Linear Reference Systems) (from Linear Reference Systems)
+referent
<>
MF_PositionExpression 0.*
MF_SecondaryOffset
+secondaryOffset + offsetVector : Vector
<>
+offset
<>
LR_OffsetExpression
MF_MeasureFunction
0.1 (from Linear Reference Systems)
+ graphOfMeasure : GM_Curve[1.*]
+ graphOfOffset[0.1] : GM_Curve[1.*]
+ graphOfSecondaryOffsets [0.1] : GM_Curve[1.*]
+ geometry : LR_Element
+ range : LR_LinearReferenceMethod

Figure 4 — Use of Linear Reference System by Moving Features
6.2 Type – MF_OneParamGeometry
6.2.1 Class semantics
A one parameter set of geometries is a function f from an interval t ∈ [a, b] such that f(t) is a geometry and for
each point P ∈ f(a) there is a one parameter set of points (called the trajectory of P) P(t) : [a, b] → P(t) such
that P(t) ∈ f(t). A leaf of a one parameter set of geometries is the geometry f(t) at a particular value of the
ISO 19141:2008(E)
parameter. The set of geometries forms a prism that is the set of points in the union of the geometries (or the
union of the trajectories).
EXAMPLE A curve C with constructive parameter t is a one parameter set of points c(t).
6.2.2 Inheritance from GM_Object
The type "MF_OneParamGeometry" (Figure 5) inherits from the type "GM_Object." As such it shall implement
all attributes, operations and associations inherited from that type as specified in ISO 19107, as well as those
specified in this subclause.
<> <>
GM_Object Number
(from Geometry root) (from Numerics)
<>
MF_OneParamGeometry
+ beginDomain : Number
+ endDomain : Number
+ leafGeometry(p : Number) : GM_Object
+ trajectory(point : DirectPosition, p : Number) : MF_Trajectory
+ prism() : GM_Object
<> <>
MF_Trajectory MF_TemporalGeometry

Figure 5 — Context Diagram: MF_OneParamGeometry
6.2.3 Attribute – beginDomain
The attribute "beginDomain" shall contain the value of the parameter at the start of the domain of the one-
parameter geometry. The data type Number is specified in ISO/TS 19103.
MF_OneParamGeometry::beginDomain: Number
6.2.4 Attribute – endDomain
The attribute "endDomain" shall contain the value of the parameter at the end of the domain of the one-
parameter geometry.
MF_OneParamGeometry::endDomain: Number
6.2.5 Operation – leafGeometry
The operation leafGeometry shall accept a value of the parameter as input and return the leaf associated with
that value as an instance of GM_Object.
MF_OneParamGeometry::leafGeometry( p: Number ): GM_Object
10 © ISO 2008 – All rights reserved

ISO 19141:2008(E)
6.2.6 Operation – trajectory
The operation trajectory shall accept the position of a point on a leaf (identified by a value of the parameter p)
of the one parameter set of geometries and return the trajectory of that point.
MF_OneParamGeometry::trajectory( point: DirectPosition, p: Number ):
MF_Trajectory
6.2.7 Operation – prism
The operation prism shall return an instance of GM_Object that is the prism formed by the union of all the
leaves of this instance of MF_OneParamGeometry.
MF_OneParamGeometry::prism( ): GM_Object
6.3 Type – MF_TemporalGeometry
6.3.1 Class semantics
MF_TemporalGeometry (Figure 6) is a specialization of MF_OneParamGeometry in which the parameter is
time as expressed by TM_Coordinate. TM_Coordinate is specified in ISO 19108; it expresses time as a
multiple of a single unit of measure such as year, day, or second.
<>
MF_OneParamGeometry
<>
MF_TemporalGeometry
+ leafGeometry(m : TM_Coordinate) : GM_Object
+ trajectory(point : DirectPosition, p : TM_Coordinate) : MF_TemporalTrajectory
+ startTime() : TM_Coordinate
+ endTime() : TM_Coordinate
+ nearestApproach(object : GM_Object, timeInterval : TM_Period) : Distance, TM_GeometricPrimitive[1.*]
+ intersection(object : GM_Object, timeInterval : TM_Period) : TM_TemporalGeometry
<>
<>
MF_PrismGeometry
MF_TemoporalTrajectory
(from Prism Geometry)
Figure 6 — Context Diagram: MF_TemporalGeometry
6.3.2 Inheritance from MF_OneParamGeometry
The type "MF_TemporalGeometry" inherits from the type "MF_OneParamGeometry". As such it shall
implement all inherited attributes, operations and associations.
ISO 19141:2008(E)
6.3.3 Operation – leafGeometry
The operation leafGeometry shall accept a time as input and return the instance of GM_Object that describes
the leaf of the temporal geometry at that time.
MF_TemporalGeometry::leafGeometry( m: TM_Coordinate ): GM_Object
6.3.4 Operation – trajectory
The operation trajectory shall accept the position of a point on a leaf of the MF_TemporalGeometry at a
specified time and return the temporal trajectory of that point.
MF_TemporalGeometry::trajectory( point: DirectPosition, p: TM_Coordinate ):
MF_TemporalTrajectory
6.3.5 Operation – startTime
The operation startTime shall return the time at which the temporal geometry begins. This shall correspond to
the value of "beginDomain".
MF_TemporalGeometry::startTime( ): TM_Coordinate
6.3.6 Operation – endTime
The operation endTime shall return the time at which the temporal geometry ends. This shall correspond to
the value of "endDomain".
MF_TemporalGeometry::endTime( ): TM_Coordinate
6.3.7 Operation – nearestApproach
The operation "nearestApproach" shall return the distance and time of the nearest approach of the temporal
geometry to any other geometric object. If the other geometric object is also a temporal geometry, then this
operation is symmetric. The parameter "timeInterval" shall restrict the search to a particular period of time.
MF_TemporalGeometry::nearestApproach( object: GM_Object, timeInterval:
TM_Period ): Distance, TM_GeometricPrimitive[1.*]
6.3.8 Operation – intersection
The operation "intersection" shall return the temporal geometry of the intersection of the temporal geometry to
any other geometric object. If the other geometric object is also a temporal geometry, then this operation is
symmetric. The parameter "timeInterval" shall restrict the search to a particular period of time.
MF_TemporalGeometry::intersection( object: GM_Object, timeInterval:
TM_Period ): MF_TemporalGeometry
6.4 Type – MF_Trajectory
6.4.1 Class semantics
MF_Trajectory (Figure 7) describes a one-parameter geometry whose cross section is a point. The class is
subject to the constraint that the position of the GM_Point returned by the leafGeometry operation equals the
position returned by the leaf operation for the same value of the parameter m. This is expressed by the OCL:
{leafGeometry(m).position = leaf(m)}
The attributes of the class are derived using inherited operations as well as those specified for the class.
12 © ISO 2008 – All rights reserved

ISO 19141:2008(E)
<>
<>
GM_Curve
MF_OneParamGeometry
(from Geometric primitive)
<>
MF_Trajectory
/+ pathGeometry : GM_Curve
/+ graphParameterToPoint : GM_Curve[1.*]
/+ graphParameterToMeasure : Set
+ leaf(p : Number) : DirectPosition
+ leafGeometry(p : Number) : GM_Point
+ prism() : GM_Curve
+ parameterToMeasure() : Set
+ positionAtParameter(p: Number) : MF_PositionExpression
<> <>
MF_TemporalTrajectory MF_MeasureFunction

Figure 7 — Context Diagram: MF_Trajectory
6.4.2 Inheritance from MF_OneParamGeometry
The type "MF_Trajectory" inherits from the type "MF_OneParamGeometry". As such it shall implement all
inherited attributes, operations and associations.
6.4.3 Inheritance from GM_Curve
The type "MF_Trajectory" inherits from the type "GM_Curve". As such it shall implement all inherited attributes,
operations and associations. GM_Curve is described using both spatial and temporal coordinates.
6.4.4 Attr
...