ISO 10303-42:2000

INTERNATIONAL ISO
STANDARD 10303-42
Second edition
2000-09-01
Industrial automation systems and
integration — Product data representation
and exchange —
Part 42:
Integrated generic resource: Geometric and
topological representation
Systèmes d'automatisation industrielle et intégration — Représentation et
échange de données de produits —
Partie 42: Ressource générique intégrée: Représentation géométrique et
topologique
Reference number
©
ISO 2000
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ii © ISO 2000 – All rights reserved

Contents Page
1 Scope . . . . . 1
1.1 Geometry . . . . . 1
1.2 Topology . . . . . 2
1.3 GeometricShapeModels . . . . 2
2 Normativereferences . . . . 3
3 Terms,definitions,symbolsandabbreviations . . . 3
3.1 Terms defined in ISO 10303-1 . . . . 3
3.2 Othertermsanddefinitions. . . . 4
3.3 Symbols. . . . . . 9
3.4 Abbreviations. . . . 10
4 Geometry . . . . . 13
4.1 Introduction. . . . 14
4.2 Fundamental concepts and assumptions . . . 14
4.2.1 Space dimensionality . . . . . . 14
4.2.2 Geometricrelationships . . . . 15
4.2.3 Parametrisation of analytic curves and surfaces . . 15
4.2.4 Curves . . . . 15
4.2.5 Surfaces . . . . 15
4.2.6 Preferredform . . . . 16
4.3 Geometryconstantandtypedefinitions . . . 16
4.3.1 dummy_gri . . . . 16
4.3.2 dimension_count . . . . 17
4.3.3 b_spline_curve_form. . . . 17
4.3.4 b_spline_surface_form . . . . . 18
4.3.5 extent_enumeration . . . . 19
4.3.6 knot_type. . . . 19
4.3.7 preferred_surface_curve_representation . . . 20
4.3.8 transition_code . . . . 21
4.3.9 trimming_preference . . . . . . 21
4.3.10 axis2_placement . . . . 22
4.3.11 curve_on_surface. . . . 22
4.3.12 pcurve_or_surface . . . . 23
4.3.13 surface_boundary . . . . 23
4.3.14 trimming_select . . . . 23
4.3.15 vector_or_direction. . . . 24
4.4 Geometry entity definitions . . . . . . 24
4.4.1 geometric_representation_context . . . 24
4.4.2 geometric_representation_item . . . 25
4.4.3 point . . . . 27
4.4.4 cartesian_point . . . . 27
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4.4.5 cylindrical_point . . . . 28
4.4.6 spherical_point . . . . 29
4.4.7 polar_point . . . . 30
4.4.8 point_on_curve. . . . 31
4.4.9 point_on_surface . . . . 32
4.4.10 point_in_volume . . . . 32
4.4.11 point_replica . . . . 33
4.4.12 degenerate_pcurve . . . . 34
4.4.13 evaluated_degenerate_pcurve . . . 35
4.4.14 direction . . . . 35
4.4.15 vector . . . . 36
4.4.16 placement. . . . 37
4.4.17 axis1_placement . . . . 37
4.4.18 axis2_placement_2d . . . . . . 38
4.4.19 axis2_placement_3d . . . . . . 39
4.4.20 cartesian_transformation_operator . . . 41
4.4.21 cartesian_transformation_operator_3d. . . 43
4.4.22 cartesian_transformation_operator_2d. . . 45
4.4.23 curve . . . . 47
4.4.24 line . . . . . 47
4.4.25 conic . . . . 48
4.4.26 circle . . . . 49
4.4.27 ellipse. . . . 50
4.4.28 hyperbola. . . . 52
4.4.29 parabola . . . . 53
4.4.30 clothoid. . . . 55
4.4.31 bounded_curve . . . . 56
4.4.32 polyline. . . . 57
4.4.33 b_spline_curve. . . . 58
4.4.34 b_spline_curve_with_knots . . . 61
4.4.35 uniform_curve . . . . 63
4.4.36 quasi_uniform_curve . . . . . . 63
4.4.37 bezier_curve . . . . 64
4.4.38 rational_b_spline_curve . . . . 65
4.4.39 trimmed_curve . . . . 67
4.4.40 composite_curve . . . . 69
4.4.41 composite_curve_segment . . . 71
4.4.42 reparametrised_composite_curve_segment . . . 72
4.4.43 pcurve . . . . 73
4.4.44 bounded_pcurve . . . . 74
4.4.45 surface_curve. . . . 75
4.4.46 intersection_curve . . . . 77
4.4.47 seam_curve . . . . 77
4.4.48 bounded_surface_curve . . . . 78
4.4.49 composite_curve_on_surface . . . 78
4.4.50 offset_curve_2d . . . . 80
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4.4.51 offset_curve_3d . . . . 81
4.4.52 curve_replica . . . . 82
4.4.53 surface . . . . 83
4.4.54 elementary_surface. . . . 84
4.4.55 plane . . . . 84
4.4.56 cylindrical_surface . . . . 85
4.4.57 conical_surface. . . . 86
4.4.58 spherical_surface. . . . 88
4.4.59 toroidal_surface . . . . 89
4.4.60 degenerate_toroidal_surface . . . 91
4.4.61 dupin_cyclide_surface . . . . . 93
4.4.62 swept_surface . . . . 97
4.4.63 surface_of_linear_extrusion . . . 98
4.4.64 surface_of_revolution . . . . . 98
4.4.65 surface_curve_swept_surface . . . 99
4.4.66 fixed_reference_swept_surface . . . 101
4.4.67 bounded_surface . . . . 103
4.4.68 b_spline_surface . . . . 103
4.4.69 b_spline_surface_with_knots . . . 106
4.4.70 uniform_surface . . . . 108
4.4.71 quasi_uniform_surface . . . . . 109
4.4.72 bezier_surface . . . . 110
4.4.73 rational_b_spline_surface . . . 110
4.4.74 rectangular_trimmed_surface . . . 111
4.4.75 curve_bounded_surface . . . . 113
4.4.76 boundary_curve . . . . 115
4.4.77 outer_boundary_curve . . . . . 115
4.4.78 rectangular_composite_surface . . . 116
4.4.79 surface_patch. . . . 118
4.4.80 offset_surface. . . . 119
4.4.81 oriented_surface . . . . 120
4.4.82 surface_replica . . . . 121
4.4.83 volume . . . . 121
4.4.84 block_volume . . . . 122
4.4.85 wedge_volume . . . . 123
4.4.86 pyramid_volume . . . . 125
4.4.87 tetrahedron_volume . . . . . . 126
4.4.88 hexahedron_volume . . . . . . 127
4.4.89 spherical_volume. . . . 129
4.4.90 cylindrical_volume. . . . 130
4.4.91 eccentric_conical_volume . . . 131
4.4.92 toroidal_volume . . . . 132
4.4.93 ellipsoid_volume . . . . 133
4.4.94 b_spline_volume . . . . 134
4.4.95 b_spline_volume_with_knots . . . 136
4.4.96 bezier_volume . . . . 138
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4.4.97 uniform_volume . . . . 139
4.4.98 quasi_uniform_volume . . . . . 140
4.4.99 rational_b_spline_volume . . . 141
4.5 Geometryschemaruledefinition:compatible_dimension . . 142
4.6 Geometryfunctiondefinitions . . . . 143
4.6.1 dimension_of. . . . 143
4.6.2 acyclic_curve_replica . . . . . 145
4.6.3 acyclic_point_replica . . . . . . 145
4.6.4 acyclic_surface_replica . . . . 146
4.6.5 associated_surface . . . . 147
4.6.6 base_axis . . . . 147
4.6.7 build_2axes. . . . 149
4.6.8 build_axes . . . . 150
4.6.9 orthogonal_complement . . . . 150
4.6.10 first_proj_axis . . . . 151
4.6.11 second_proj_axis. . . . 152
4.6.12 cross_product. . . . 153
4.6.13 dot_product. . . . 154
4.6.14 normalise . . . . 156
4.6.15 scalar_times_vector . . . . 157
4.6.16 vector_sum . . . . 158
4.6.17 vector_difference. . . . 160
4.6.18 default_b_spline_knot_mult . . . 161
4.6.19 default_b_spline_knots . . . . . 162
4.6.20 default_b_spline_curve_weights. . . 163
4.6.21 default_b_spline_surface_weights . . . 163
4.6.22 constraints_param_b_spline . . . 164
4.6.23 curve_weights_positive . . . . 166
4.6.24 constraints_composite_curve_on_surface . . . 166
4.6.25 get_basis_surface. . . . 167
4.6.26 surface_weights_positive . . . . 168
4.6.27 volume_weights_positive . . . 169
4.6.28 constraints_rectangular_composite_surface . . . . 170
4.6.29 list_to_array . . . . 171
4.6.30 make_array_of_array . . . . . . 172
4.6.31 make_array_of_array_of_array . . . 173
4.6.32 above_plane . . . . 174
4.6.33 same_side . . . . 175
5 Topology . . . . . 177
5.1 Introduction. . . . 177
5.2 Fundamental concepts and assumptions . . . 177
5.2.1 Geometricassociations. . . . 178
5.2.2 Associationswithparameterspacegeometry . . . 179
5.2.3 Graphs,cycles,andtraversals . . . 181
5.3 Topology constant and type definitions. . . 182
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5.3.1 dummy_tri . . . . 182
5.3.2 shell. . . . . 182
5.3.3 reversible_topology_item . . . 183
5.3.4 list_of_reversible_topology_item . . . 183
5.3.5 set_of_reversible_topology_item . . . 184
5.3.6 reversible_topology . . . . . . 184
5.4 Topology entity definitions . . . . . . 184
5.4.1 topological_representation_item . . . 184
5.4.2 vertex . . . . 185
5.4.3 vertex_point . . . . 186
5.4.4 edge. . . . . 186
5.4.5 edge_curve . . . . 188
5.4.6 oriented_edge . . . . 189
5.4.7 seam_edge . . . . 190
5.4.8 subedge. . . . 191
5.4.9 path . . . . . 191
5.4.10 oriented_path. . . . 192
5.4.11 open_path . . . . 193
5.4.12 loop. . . . . 194
5.4.13 vertex_loop . . . . 195
5.4.14 edge_loop . . . . 196
5.4.15 poly_loop. . . . 197
5.4.16 face_bound . . . . 198
5.4.17 face_outer_bound . . . . 198
5.4.18 face . . . . . 199
5.4.19 face_surface . . . . 201
5.4.20 oriented_face . . . . 202
5.4.21 subface . . . . 203
5.4.22 connected_face_set. . . . 204
5.4.23 vertex_shell. . . . 205
5.4.24 wire_shell . . . . 205
5.4.25 open_shell . . . . 207
5.4.26 oriented_open_shell . . . . . . 209
5.4.27 closed_shell . . . . 210
5.4.28 oriented_closed_shell . . . . 212
5.4.29 connected_face_sub_set . . . . 213
5.4.30 connected_edge_set . . . . 214
5.5 Topology function definitions . . . . 214
5.5.1 conditional_reverse. . . . 214
5.5.2 topology_reversed . . . . 215
5.5.3 edge_reversed . . . . 216
5.5.4 path_reversed. . . . 217
5.5.5 face_bound_reversed . . . . . . 217
5.5.6 face_reversed . . . . 218
5.5.7 shell_reversed . . . . 219
5.5.8 closed_shell_reversed . . . . 219
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5.5.9 open_shell_reversed . . . . 220
5.5.10 set_of_topology_reversed . . . 221
5.5.11 list_of_topology_reversed . . . 222
5.5.12 boolean_choose . . . . 222
5.5.13 path_head_to_tail . . . . 223
5.5.14 list_face_loops . . . . 224
5.5.15 list_loop_edges. . . . 224
5.5.16 list_shell_edges. . . . 225
5.5.17 list_shell_faces . . . . 225
5.5.18 list_shell_loops. . . . 226
5.5.19 mixed_loop_type_set . . . . . . 227
5.5.20 list_to_set. . . . 228
5.5.21 edge_curve_pcurves . . . . 228
5.5.22 vertex_point_pcurves . . . . . . 230
6 Geometricmodels . . . . . 231
6.1 Introduction. . . . 231
6.2 Fundamental concepts and assumptions . . . 232
6.3 Geometricmodeltypedefinitions . . . 232
6.3.1 boolean_operand . . . . 232
6.3.2 boolean_operator . . . . 233
6.3.3 csg_primitive . . . . 233
6.3.4 csg_select. . . . 234
6.3.5 geometric_set_select . . . . 234
6.3.6 surface_model . . . . 235
6.3.7 wireframe_model. . . . 235
6.4 Geometric model entity definitions. . . 235
6.4.1 solid_model . . . . 235
6.4.2 manifold_solid_brep . . . . 236
6.4.3 brep_with_voids . . . . 239
6.4.4 faceted_brep . . . . 239
6.4.5 brep_2d. . . . 240
6.4.6 csg_solid . . . . 241
6.4.7 boolean_result . . . . 242
6.4.8 block . . . . 243
6.4.9 right_angular_wedge . . . . . . 244
6.4.10 rectangular_pyramid . . . . . . 245
6.4.11 faceted_primitive. . . . 246
6.4.12 tetrahedron . . . . 247
6.4.13 convex_hexahedron . . . . . . 248
6.4.14 sphere. . . . 249
6.4.15 right_circular_cone. . . . 250
6.4.16 right_circular_cylinder. . . . . 251
6.4.17 eccentric_cone . . . . 252
6.4.18 torus . . . . 253
6.4.19 ellipsoid . . . . 254
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6.4.20 cyclide_segment_solid . . . . 255
6.4.21 half_space_solid . . . . 256
6.4.22 boxed_half_space . . . . 257
6.4.23 box_domain . . . . 258
6.4.24 primitive_2d . . . . 259
6.4.25 circular_area . . . . 259
6.4.26 elliptic_area . . . . 260
6.4.27 rectangular_area . . . . 260
6.4.28 polygonal_area . . . . 261
6.4.29 half_space_2d . . . . 261
6.4.30 rectangled_half_space . . . . . 262
6.4.31 rectangle_domain . . . . 263
6.4.32 swept_face_solid . . . . 263
6.4.33 extruded_face_solid . . . . . . 264
6.4.34 revolved_face_solid . . . . . . 265
6.4.35 surface_curve_swept_face_solid. . . 266
6.4.36 swept_area_solid . . . . 268
6.4.37 extruded_area_solid . . . . 268
6.4.38 revolved_area_solid . . . . 269
6.4.39 surface_curve_swept_area_solid. . . 270
6.4.40 trimmed_volume. . . . 271
6.4.41 solid_replica . . . . 273
6.4.42 shell_based_surface_model . . . 273
6.4.43 face_based_surface_model. . . 274
6.4.44 shell_based_wireframe_model. . . 275
6.4.45 edge_based_wireframe_model. . . 276
6.4.46 geometric_set. . . . 277
6.4.47 geometric_curve_set . . . . 277
6.4.48 sectioned_spine. . . . 278
6.4.49 geometric_set_replica . . . . . 279
6.5 Geometricmodelfunctiondefinitions . . . 280
6.5.1 acyclic_solid_replica. . . . 280
6.5.2 acyclic_set_replica . . . . 281
6.5.3 constraints_geometry_shell_based_surface_model . . 281
6.5.4 constraints_geometry_shell_based_wireframe_model. . 282
6.5.5 build_transformed_set . . . . 283
6.5.6 msb_shells . . . . 284
Annex A (normative) Short names of entities. . . 286
AnnexB(normative) Informationobjectregistration . . . 294
B.1 Documentidentification . . . . 294
B.2 Schemaidentification. . . . 294
AnnexC(informative) Computer-interpretablelistings . . . 295
AnnexD(informative) EXPRESS-Gdiagrams . . . 296
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Bibliography . . . . . 317
Index . . . . . 318
Figures
Figure 1 Spherical_point attributes . . . . . 30
Figure 2 Axis2_placement_3d . . . . 41
Figure 3 (a) Cartesian_transformation_operator_3d . . . 44
Figure 3 (b) Cartesian_transformation_operator_3d . . . 45
Figure 3 (c) Cartesian_transformation_operator_3d . . . 46
Figure4 Circle . . . . . 50
Figure 5 Ellipse . . . . . 51
Figure6 Hyperbola. . . . 53
Figure7 Parabola . . . . 54
Figure8 Clothoidcurve. . . . 57
Figure9 B-splinecurve . . . . 60
Figure10 Composite_curve . . . . 71
Figure11 Conical_surface . . . . 87
Figure12 Crosssectionofdegenerate_toroidal_surface. . . 92
Figure 13 Cross-sections of a Dupin cyclide with C=0. . . 94
Figure14 ADupinringcyclide . . . . 95
Figure15 ADupinhornedcyclide. . . . 95
Figure16 ADupinspindlecyclide. . . . 96
Figure17 Fixed_reference_swept_surface. . . 102
Figure 18 Curve bounded surface . . . . . . 114
Figure19 Wedge_volumeanditsattributes . . . 124
Figure20 Edgecurve . . . . 187
Figure 21 Right angular wedge and its attributes . . . 244
Figure 22 Convex_hexahedron. . . . 250
Figure23 Crosssectionofcyclide_segment_solid . . . 256
Figure24 Revolvedfacesolid . . . . 265
FigureD.1 Geometry_schemaEXPRESS-Gdiagram1of13 . . 297
FigureD.2 Geometry_schemaEXPRESS-Gdiagram2of13 . . 298
FigureD.3 Geometry_schemaEXPRESS-Gdiagram3of13 . . 299
FigureD.4 Geometry_schemaEXPRESS-Gdiagram4of13 . . 300
FigureD.5 Geometry_schemaEXPRESS-Gdiagram5of13 . . 301
FigureD.6 Geometry_schemaEXPRESS-Gdiagram6of13 . . 302
FigureD.7 Geometry_schemaEXPRESS-Gdiagram7of13 . . 303
FigureD.8 Geometry_schemaEXPRESS-Gdiagram8of13 . . 304
FigureD.9 Geometry_schemaEXPRESS-Gdiagram9of13 . . 305
FigureD.10 Geometry_schemaEXPRESS-Gdiagram10of13. . 306
FigureD.11 Geometry_schemaEXPRESS-Gdiagram11of13. . 307
FigureD.12 Geometry_schemaEXPRESS-Gdiagram12of13. . 308
FigureD.13 Geometry_schemaEXPRESS-Gdiagram13of13. . 309
Figure D.14 Topology_schema EXPRESS-G diagram 1 of 3 . . . 310
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Figure D.15 Topology_schema EXPRESS-G diagram 2 of 3 . . . 311
Figure D.16 Topology_schema EXPRESS-G diagram 3 of 3 . . . 312
FigureD.17 Geometric_model_schemaEXPRESS-Gdiagram1of4. . 313
FigureD.18 Geometric_model_schemaEXPRESS-Gdiagram2of4. . 314
FigureD.19 Geometric_model_schemaEXPRESS-Gdiagram3of4. . 315
FigureD.20 Geometric_model_schemaEXPRESS-Gdiagram4of4. . 316
Tables
Table1 Geometrymathematicalsymbology . . . 9
Table 2 Topology symbol definitions . . . 11
Table A.1 Short names of entities . . . . . . 286
c ISO 2000 – All rights reserved xi

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 com-
mittee 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 stan-
dardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 3.
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.
AttentionisdrawntothepossibilitythatsomeoftheelementsofthispartofISO10303maybesubject to
patent rights. ISO shall not be held responsible for any or all such patent rights.
International Standard ISO 10303-42 was prepared by Technical Committee ISO/TC 184, Industrial
automation systems and integration, Subcommittee SC 4, Industrial data.
This second edition constitutes a technical revision of the first edition (ISO 10303-42:1994), which
is provisionally retained in order to support the continued use and maintenance of implementations
based of the first edition and to satisfy the normative references of other parts of ISO 10303.
It incorporates the corrections published in ISO 10303-42:1994/Cor.1:1999, 10303-42:1994/Cor.2:1999
1)
and ISO 10303-42:1994/Cor.3:–
This International Standard is organised as a series of parts, each published separately. The structure of
this International Standard is described in ISO 10303-1. The numbering of the parts of this International
Standard reflects its structure:
— Parts 11 to 14 specify the description methods;
— Parts 21 to 29 specify the implementation methods;
— Parts 31 to 35 specify the conformance testing methodology and framework;
— Parts 41 to 50 specify the integrated generic resources;
— Parts 101 to 107 specify the integrated application resources;
— Parts 201 to 237 specify the application protocols;
1)
To be published
xiic ISO 2000 – All rights reserved

— Parts 301 to 307 specify the abstract test suites;
— parts 501 to 520 specify the application interpreted constructs.
A complete list of parts of ISO 10303 is available from Internet:

Should further parts of ISO 10303 be published, they will follow the same numbering pattern.
This part of ISO 10303 is a member of the integrated resources series. The integrated resources specify
a single conceptual product data model.
Annexes A and B form a normative part of this part of ISO 10303. Annexes C and D are for information
only.
c ISO 2000 – All rights reserved xiii

Introduction
ISO 10303 is an International Standard for the computer-interpretable representation and exchange of
product data. The objective is to provide a neutral mechanism capable of describing product data
throughout the life cycle of a product independent from any particular system. The nature of this de-
scription makes it suitable not only for neutral file exchange, but also as a basis for implementing and
sharing product databases and archiving.
This International Standard is organized as a series of parts, each published separately. The parts of
ISO 10303 fall into one of the following series: description methods, integrated resources, application
interpreted constructs, application protocols, abstract test suites, implementation methods, and confor-
mance testing. The series are described in ISO 10303–1. This part of ISO 10303 is a member of the
integrated generic resource series.
This part of ISO 10303 specifies the integrated resources used for geometric and topological representa-
tion. Their primary application is for explicit representation of the shape or geometric form of a product
model. The shape representation presented here has been designed to facilitate stable and efficient com-
munication when mapped to a physical file.
The geometry in clause 4 is exclusively the geometry of parametric curves and surfaces. It includes
the curve and surface entities and other entities, functions and data types necessary for their definition.
A common scheme has been used for the definition of both two-dimensional and three-dimensional
geometry. All geometry is defined in a coordinate system which is established as part of the context of
the item which it represents. These concepts are fully defined in ISO 10303 Part 43.
The topology in clause 5 is concerned with connectivity relationships between objects rather than with
the precise geometric form of objects. This clause contains the basic topological entities and specialised
subtypes of these. In some cases the subtypes have geometric associations. Also included are functions,
particularly constraint functions, and data types necessary for the definitions of the topological entities.
The geometric models in clause 6 provide basic resources for the communication of data describing
the precise size and shape of three-dimensional solid objects. The geometric shape models provide a
complete representation of the shape which in many cases includes both geometric and topological data.
Included here are the two classical types of solid model, constructive solid geometry (CSG) and boundary
representation (B-rep). Other entities, providing a rather less complete description of the geometry of a
product, and with less consistency constraints, are also included.
This edition incorporates modifications that are upwardly compatible with the previous edition. Modifi-
cations to EXPRESS specifications are upwardly compatible if:
— instances encoded according to ISO 10303-21 and that conform to an ISO 10303 application proto-
col based on the previous edition of this part, also conform to a revision of that application protocol
based on this edition;
xivc ISO 2000 – All rights reserved

— interfaces that conform to ISO 10303-22 and to an ISO 10303 application protocol based on the
previous edition of this part, also conform to a revision of that application protocol based on this
edition;
— the mapping tables of ISO 10303 application protocols based on the previous edition of this part
remain valid in a revision of that application protocol based on this edition.
Technical modifications to ISO 10303-42:1994 are categorised as follows:
— changes to the EXPRESS declarations,
— new EXPRESS declarations.
The following EXPRESS declarations have been modified:
geometry schema:
— axis1_placement;
— base_axis;
— build_axes;
— build_2axes;
— cartesian_transformation_operator_3d;
— cartesian_transformation_operator_2d;
— composite_curve_segment;
— constraints_param_b_spline;
— cross_product;
— curve_bounded_surface;
— default_b_spline_curve_weights;
— default_b_spline_knot_mult;
— default_b_spline_knots;
— default_b_spline_surface_weights;
— geometric_representation_item;
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— get_basis_surface;
— list_to_array;
— make_array_of_array;
— orthogonal_complement;
—point;
— rectangular_composite_surface;
— scalar_times_vector;
— surface_of_revolution;
— surface_patch;
— swept_surface;
— trimmed_curve;
— vector_sum;
— vector_difference;
topology schema:
—edge;
— edge_reversed;
— face_bound_reversed;
— face_reversed;
— face_surface;
— mixed_loop_type_set;
— path_head_to_tail;
— path_reversed;
— shell_reversed;
geometric model schema:
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— boolean_operand;
— build_transformed_set;
— csg_primitive;
— csg_solid;
— revolved_area_solid;
— revolved_face_solid;
— solid_model;
— swept_area_solid;
— swept_face_solid.
The following EXPRESS declarations have been added:
geometry schema:
— above_plane;
— b_spline_volume;
— b_spline_volume_with_knots;
— bezier_volume;
— block_volume;
—clothoid;
— cylindrical_point;
— cylindrical_volume;
— dummy_gri;
— dupin_cyclide_surface;
— eccentric_conical_volume;
— ellipsoid_volume;
— oriented_surface;
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— hexahedron_volume;
— make_array_of_array_of_array;
— point_in_volume;
— polar_point;
— pyramid_volume;
— quasi_uniform_volume;
— rational_b_spline_volume;
—same_side;
— spherical_point;
— spherical_volume;
— surface_boundary;
— surface_curve_swept_surface;
— tetrahedron_volume;
— toroidal_volume;
— volume;
— wedge_volume;
topology schema:
— closed_shell_reversed;
— connected_face_sub_set;
— dummy_tri;
— open_shell_reversed;
— seam_edge;
— subedge;
geometric model schema:
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— brep_2d;
— circular_area;
— convex_hexahedron;
— cyclide_segment_solid;
— eccentric_cone;
— ellipsoid;
— elliptic_area;
— faceted_primitive;
— half_space_2d;
— polygonal_area;
— primitive_2d;
— rectangular_area;
— rectangular_pyramid;
— sectioned_spine;
— surface_curve_swept_area_solid;
— surface_curve_swept_face_solid;
— tetrahedron;
— trimmed_volume.
Several components of this part of ISO 10303 are available in electronic form. This access is provided
through the specification of Universal Resource Locators (URL’s) that identify the location of these files
on the internet. If there is difficulty in accessing these files, contact the ISO Central Secretariat or the
ISO SC4 Secretariat directly at: sc4@cme.nist.gov.
c ISO 2000 – All rights reserved xix

INTERNATIONAL STANDARD
Industrial automation systems and integration —
Product data representation and exchange —
Part 42:
Integrated generic resource:
Geometric and topological representation
1Scope
This part of ISO 10303 specifies the resource constructs for the explicit geometric and topological
representation of the shape of a product. The scope is determined by the requirements for the explicit
representation of an ideal product model; tolerances and implicit forms of representation in terms of
features are out of scope. The geometry in clause 4 and the topology in clause 5 are available for use
independently and are also extensively used by the various forms of geometric shape model in clause 6.
In addition, this part of ISO 10303 specifies specialisations of the concepts of representation where the
elements of representation are geometric.
1.1 Geometry
The following are within the scope of the geometry schema:
— definition of points, vectors, parametric curves and parametric surfaces;
— definition of finite volumes with internal parametrisation;
— definition of transformation operators;
— points defined directly by their coordinate values or in terms of the parameters of an existing curve
or surface;
— definition of conic curves and elementary surfaces;
— definition of curves defined on a parametric surface;
— definition of general parametric spline curves, surfaces and volumes;
— definition of point, curve and surface replicas;
— definition of offset curves and surfaces;
— definition of intersection curves.
The following are outside the scope of this part of ISO 10303:
— all other forms of procedurally defined curves and surfaces;
— curves and surfaces which do not have a parametric form of representation;
— any form of explicit representation of a ruled surface.
NOTE - For a ruled surface the geometry is critically dependent upon the parametrisation of the
boundary curves and the method of associating pairs of points on the two curves. A ruled surface
with B-spline boundary curves can however be exactly represented by the B-spline surface entity.
1.2 Topology
The following are within the scope of the topology schema:
— definition of the fundamental topological entities vertex, edge, and face, each with a specialised
subtype to enable it to be associated with the geometry of a point, curve, or surface, respectively;
— collections of the basic entities to form topological structures of path, loop and shell and constraints
to ensure the integrity of these structures;
— orientation of topological entities.
1.3 Geometric Shape Models
The following are within the scope of the geometric model schema:
— data describing the precise geometric form of three-dimensional solid objects;
— constructive solid geometry (CSG) models;
— CSG models in two-dimensional space;
— definition of CSG primitives and half-spaces;
— creation of solid models by sweeping operations;
— manifold boundary representation (B-rep) models;
— constraints to ensure the integrity of B-rep models;
— surface models;
— wireframe models;
— geometric sets;
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— creation of a replica of a solid model in a new location.
The following are outside the scope of this part of ISO 10303:
— non-manifold boundary representation models;
— spatial occupancy forms of solid models (such as octree models);
— assemblies and mechanisms.
2 Normative references
The following normative documents contain provisions which, through reference in this text, constitute
provisions of this part of ISO 10303. For dated references, subsequent amendments to, or revisions of,
any of these publications do not apply. However, parties to agreements based on this part of ISO 10303
are encouraged to investigate the possibility of applying the most recent editions of the normative docu-
ments indicated below. For undated references, the latest edition of the normative document referred to
applies. Members of ISO and IEC maintain registers of currently valid International Standards.
ISO/IEC 8824-1:1995, Information technology — Abstract Syntax Notation One (ASN.1): Specification
of basic notation.
ISO 10303-1:1994, Industrial automation systems and integration — Product data representation and
exchange — Part 1: Overview and fundamental principles.
ISO 10303-11:1994, Industrial automation systems and integration — Product data representation and
exchange — Part 11: Description methods: The EXPRESS language reference manual.
2)
ISO 10303-41: — , Industrial automation systems and integration- Product data representation and
exchange- Part 41: Integrated generic resource: Fundamentals of product description and support.
ISO 10303-43: 2000, Industrial automation systems and integration - Product data representation and
exchange- Part 43: Integrated generic resource: Representation structures.
3 Terms, definitions, symbols and abbreviations
3.1 Terms defined in ISO 10303-1
For the purposes of this part of ISO 10303 the following terms defined in ISO 10303-1 apply.
— integrated resource.
2)
To be published. (Revision of ISO 10303-41:1994)
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3.2 Other terms and definitions
For the purposes of this part of ISO 10303, the following terms and definitions apply. A number of
informal definitions are also given here which will later be used to describe and constrain the topological
entities. They are not intended to be mathematically rigourous. The definitions are given in alphabetical,
not logical order.
3.2.1
arcwise connected
an entity is arcwise connected if any two arbitrary points in its domain can be connected by a curve that
lies entirely within the domain.
3.2.2
axi-symmetric
an entity is axi-symmetric if it has an axis of symmetry such that the object is invariant under all rotations
about this axis.
3.2.3
bounds
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.
3.2.4
boundary
m
the set of mathematical points x in a domain X contained in R for which there is an open ball U
m
in R containing x such that the intersection U\X is homeomorphic to an open set in the closed d
d
-dimensional half-space R ,for some dm, where the homeomorphism carries x into the origin in+
d
R .
+
d d
NOTE 1 - R is defined to be the set of all mathematical points(x;:::;x in R with x0.1d)1+
NOTE 2 - For this purpose, the word “open” has its usual mathematical meaning. It does not relate to
“open surface” as defined elsewhere in this part of ISO 10303.
3.2.5
boundary representation solid model (B-rep)
a type of geometric model in which the size and shape of the solid is defined in terms of the faces, edges
and vertices which make up its boundary.
3.2.6
closed curve
a curve such that both end points are the same.
3.2.7
closed surface
a connected 2-manifold that divides space into exactly two connected components, one of which is finite.
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3.2.8
completion of a topological entity
a 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.
3.2.9
connected
equivalent to arcwise connected (see 3.2.1).
3.2.10
connected component
a maximal connected subset of a domain.
3.2.11
constructive s
...