611:) or regularized set operations. Terminal nodes are primitive leaves that represent closed regular sets. The semantics of CSG representations is clear. Each subtree represents a set resulting from applying the indicated transformations/regularized set operations on the set represented by the primitive leaves of the subtree. CSG representations are particularly useful for capturing design intent in the form of features corresponding to material addition or removal (bosses, holes, pockets etc.). The attractive properties of CSG include conciseness, guaranteed validity of solids, computationally convenient Boolean algebraic properties, and natural control of a solid's shape in terms of high level parameters defining the solid's primitives and their positions and orientations. The relatively simple data structure and elegant
262:
implicitly recognizes the possibility of several computer representations of the same physical object as long as any two such representations are consistent. It is impossible to computationally verify informational completeness of a representation unless the notion of a physical object is defined in terms of computable mathematical properties and independent of any particular representation. Such reasoning led to the development of the modeling paradigm that has shaped the field of solid modeling as we know it today.
628:
relative motion, motion planning, and even in computer graphics applications such as tracing the motions of a brush moved on a canvas. Most commercial CAD systems provide (limited) functionality for constructing swept solids mostly in the form of a two dimensional cross section moving on a space trajectory transversal to the section. However, current research has shown several approximations of three dimensional shapes moving across one parameter, and even multi-parameter motions.
1345:
1187:
1054:
868:
934:
297:
508:. For example, a family of bolts is a generic primitive, and a single bolt specified by a particular set of parameters is a primitive instance. The distinguishing characteristic of pure parameterized instancing schemes is the lack of means for combining instances to create new structures which represent new and more complex objects. The other main drawback of this scheme is the difficulty of writing
91:
36:
1156:. Constraints are relationships between entities that make up a particular shape. For a window, the sides might be defined as being parallel, and of the same length. Parametric modeling is obvious and intuitive. But for the first three decades of CAD this was not the case. Modification meant re-draw, or add a new cut or protrusion on top of old ones. Dimensions on engineering drawings were
1130:). Here, surfaces are defined, trimmed and merged, and filled to make solid. The surfaces are usually defined with datum curves in space and a variety of complex commands. Surfacing is more difficult, but better applicable to some manufacturing techniques, like injection molding. Solid models for injection molded parts usually have both surfacing and sketcher based features.
249:, and so on. A central problem in all these applications is the ability to effectively represent and manipulate three-dimensional geometry in a fashion that is consistent with the physical behavior of real artifacts. Solid modeling research and development has effectively addressed many of these issues, and continues to be a central focus of
534:. Spatial arrays are unambiguous and unique solid representations but are too verbose for use as 'master' or definitional representations. They can, however, represent coarse approximations of parts and can be used to improve the performance of geometric algorithms, especially when used in conjunction with other representations such as
850:, and references to other features. Features also provide access to related production processes and resource models. Thus, features have a semantically higher level than primitive closed regular sets. Features are generally expected to form a basis for linking CAD with downstream manufacturing applications, and also for organizing
480:. All representation schemes are organized in terms of a finite number of operations on a set of primitives. Therefore, the modeling space of any particular representation is finite, and any single representation scheme may not completely suffice to represent all types of solids. For example, solids defined via
1118:
A solid part model generally consists of a group of features, added one at a time, until the model is complete. Engineering solid models are built mostly with sketcher-based features; 2-D sketches that are swept along a path to become 3-D. These may be cuts, or extrusions for example. Design work on
546:
This scheme follows from the combinatoric (algebraic topological) descriptions of solids detailed above. A solid can be represented by its decomposition into several cells. Spatial occupancy enumeration schemes are a particular case of cell decompositions where all the cells are cubical and lie in a
1148:
Example: A shaft is created by extruding a circle 100 mm. A hub is assembled to the end of the shaft. Later, the shaft is modified to be 200 mm long (click on the shaft, select the length dimension, modify to 200). When the model is updated the shaft will be 200 mm long, the hub will
1144:
Parametric modeling uses parameters to define a model (dimensions, for example). Examples of parameters are: dimensions used to create model features, material density, formulas to describe swept features, imported data (that describe a reference surface, for example). The parameter may be modified
627:
out volume (a solid) that may be represented by the moving set and its trajectory. Such a representation is important in the context of applications such as detecting the material removed from a cutter as it moves along a specified trajectory, computing dynamic interference of two solids undergoing
594:
Constructive solid geometry (CSG) is a family of schemes for representing rigid solids as
Boolean constructions or combinations of primitives via the regularized set operations discussed above. CSG and boundary representations are currently the most important representation schemes for solids. CSG
1029:
as a standalone product – the only commercial 3D modeling kernel from Russia. Other contributions came from Mäntylä, with his GWB and from the GPM project which contributed, among other things, hybrid modeling techniques at the beginning of the 1980s. This is also when the
Programming Language of
386:
sets). In addition, solids are required to be closed under the
Boolean operations of set union, intersection, and difference (to guarantee solidity after material addition and removal). Applying the standard Boolean operations to closed regular sets may not produce a closed regular set, but this
475:
Based on assumed mathematical properties, any scheme of representing solids is a method for capturing information about the class of semi-analytic subsets of
Euclidean space. This means all representations are different ways of organizing the same geometric and topological data in the form of a
261:
The notion of solid modeling as practised today relies on the specific need for informational completeness in mechanical geometric modeling systems, in the sense that any computer model should support all geometric queries that may be asked of its corresponding physical object. The requirement
1422:
Because CAD programs running on computers "understand" the true geometry comprising complex shapes, many attributes of/for a 3‑D solid, such as its center of gravity, volume, and mass, can be quickly calculated. For instance, the cube with rounded edges shown at the top of this article
529:
are cubes of a fixed size and are arranged in a fixed spatial grid (other polyhedral arrangements are also possible but cubes are the simplest). Each cell may be represented by the coordinates of a single point, such as the cell's centroid. Usually a specific scanning order is imposed and the
1145:
later, and the model will update to reflect the modification. Typically, there is a relationship between parts, assemblies, and drawings. A part consists of multiple features, and an assembly consists of multiple parts. Drawings can be made from either parts or assemblies.
658:
to consist of all the points that satisfy the condition specified by the predicate. The simplest form of a predicate is the condition on the sign of a real valued function resulting in the familiar representation of sets by equalities and inequalities. For example, if
1110:. Solid modelers have become commonplace in engineering departments in the last ten years due to faster computers and competitive software pricing. Solid modeling software creates a virtual 3D representation of components for machine design and analysis. A typical
1412:
488:
of a primitive moving according to a space trajectory, except in very simple cases. This forces modern geometric modeling systems to maintain several representation schemes of solids and also facilitate efficient conversion between representation schemes.
575:
Similar to boundary representation, the surface of the object is represented. However, rather than complex data structures and NURBS, a simple surface mesh of vertices and edges is used. Surface meshes can be structured (as in triangular meshes in
512:
for computing properties of represented solids. A considerable amount of family-specific information must be built into the algorithms and therefore each generic primitive must be treated as a special case, allowing no uniform overall treatment.
1168:
part may have a thousand features, and modifying an early feature may cause later features to fail. Skillfully created parametric models are easier to maintain and modify. Parametric modeling also lends itself to data re-use. A whole family of
193:
The use of solid modeling techniques allows for the automation process of several difficult engineering calculations that are carried out as a part of the design process. Simulation, planning, and verification of processes such as
825:
allow conversions of such representations into a single function inequality for any closed semi analytic set. Such a representation can be converted to a boundary representation using polygonization algorithms, for example, the
1114:
includes programmable macros, keyboard shortcuts and dynamic model manipulation. The ability to dynamically re-orient the model, in real-time shaded 3-D, is emphasized and helps the designer maintain a mental 3-D image.
378:⊂ ℝ can be turned into a closed regular set or "regularized" by taking the closure of its interior, and thus the modeling space of solids is mathematically defined to be the space of closed regular subsets of ℝ (by the
429:
elements is an example of a stratification that is commonly used. The combinatorial model of solidity is then summarized by saying that in addition to being semi-analytic bounded subsets, solids are three-dimensional
147:
438:
of the combinatorial boundary of the polyhedron is 2. The combinatorial manifold model of solidity also guarantees the boundary of a solid separates space into exactly two components as a consequence of the
1423:
measures 8.4 mm from flat to flat. Despite its many radii and the shallow pyramid on each of its six faces, its properties are readily calculated for the designer, as shown in the screenshot at right.
446:
The point-set and combinatorial models of solids are entirely consistent with each other, can be used interchangeably, relying on continuum or combinatorial properties as needed, and can be extended to
854:
for design data reuse. Parametric feature based modeling is frequently combined with constructive binary solid geometry (CSG) to fully describe systems of complex objects in engineering.
364:
must be consistently three dimensional; points with lower-dimensional neighborhoods indicate a lack of solidity. Dimensional homogeneity of neighborhoods is guaranteed for the class of
1018:(CIS), ASCON began internal development of its own solid modeler in the 1990s. In November 2012, the mathematical division of ASCON became a separate company, and was named
202:
were one of the main catalysts for the development of solid modeling. More recently, the range of supported manufacturing applications has been greatly expanded to include
710:
815:
780:
745:
944:
409:
subsets of
Euclidean space is closed under Boolean operations (standard and regularized) and exhibits the additional property that every semi-analytic set can be
500:
This scheme is based on notion of families of object, each member of a family distinguishable from the other by a few parameters. Each object family is called a
492:
Below is a list of techniques used to create or represent solid models. Modern modeling software may use a combination of these schemes to represent a solid.
580:
or quad meshes with horizontal and vertical rings of quadrilaterals), or unstructured meshes with randomly grouped triangles and higher level polygons.
451:
dimensions. The key property that facilitates this consistency is that the class of closed regular subsets of ℝ coincides precisely with homogeneously
387:
problem can be solved by regularizing the result of applying the standard
Boolean operations. The regularized set operations are denoted ∪, ∩, and −.
1149:
relocate to the end of the shaft to which it was assembled, and the engineering drawings and mass properties will reflect all changes automatically.
1747:
54:
185:, and in general, support the creation, exchange, visualization, animation, interrogation, and annotation of digital models of physical objects.
1762:
959:
1498:
821:. More complex functional primitives may be defined by Boolean combinations of simpler predicates. Furthermore, the theory of
1855:
1830:
459:-dimensional solid may be unambiguously represented by its boundary and the boundary has the combinatorial structure of an
181:, by its emphasis on physical fidelity. Together, the principles of geometric and solid modeling form the foundation of 3D-
1805:
1153:
1015:
1904:
1392:
1234:
1093:
981:
915:
131:
72:
1374:
1216:
897:
563:
for the numerical solution of partial differential equations. Other cell decompositions such as a
Whitney regular
402:
so that the cells provide finite spatial addresses for points in an otherwise innumerable continuum. The class of
1366:
1208:
1071:
963:
889:
17:
1899:
1370:
1212:
1075:
893:
647:
564:
1164:. Parametric modeling is very powerful, but requires more skill in model creation. A complicated model for an
1894:
999:
998:, the key milestones being the development of the research system BUILD followed by its commercial spin-off
150:
The geometry in solid modeling is fully described in 3‑D space; objects can be viewed from any angle.
410:
1107:
443:
theorem, thus eliminating sets with non-manifold neighborhoods that are deemed impossible to manufacture.
273:
material that could be added or removed. These postulated properties can be translated into properties of
589:
535:
481:
1647:
1123:
methods. An assembly model incorporates references to individual part models that comprise the product.
559:(number of holes). Cell decompositions in the form of triangulations are the representations used in 3D
1406:
1252:
250:
161:) is a consistent set of principles for mathematical and computer modeling of three-dimensional shapes
221:
routing, etc. Beyond traditional manufacturing, solid modeling techniques serve as the foundation for
1522:
Requicha, A.A.G & Voelcker, H. (1983). "Solid
Modeling: Current Status and Research Directions".
1248:
1127:
1026:
604:
101:
434:, specifically three-dimensional orientable manifolds with boundary. In particular this implies the
1355:
1197:
1011:
878:
414:
1331:
will be necessary to generate an accurate and realistic geometrical description of the scan data.
662:
1782:
1458:
1359:
1201:
1064:
882:
637:
623:
The basic notion embodied in sweeping schemes is simple. A set moving through space may trace or
600:
379:
310:
246:
1442:
1437:
785:
750:
548:
1557:
Tilove, R.B.; Requicha, A.A.G. (1980), "Closure of
Boolean operations on geometric entities",
842:
such as intrinsic geometric parameters (length, width, depth etc.), position and orientation,
1741:
1463:
1307:
1043:
995:
818:
431:
182:
146:
715:
229:
by reconstructing solids from sampled points on physical objects, mechanical analysis using
1586:
Requicha, A.A.G. (1980). "Representations for Rigid Solids: Theory, Methods, and
Systems".
1416:
440:
435:
8:
1452:
1328:
1133:
847:
843:
556:
332:
326:
320:
266:
226:
163:
951:
269:, so initially the focus was on mathematically modeling rigid parts made of homogeneous
113:
50:
1729:
1688:
1603:
1539:
1432:
1324:
1289:
1120:
370:
349:
293:
respectively. Both models specify how solids can be built from simple pieces or cells.
289:
283:
242:
168:
1800:. CreateSpace Independent Publishing Platform (24 December 2014). p. Back cover.
1622:
1851:
1826:
1801:
1774:
1607:
1570:
1474:
1447:
1300:
1267:
can be used to create point clouds or polygon mesh models of external body features.
1165:
577:
222:
210:
172:
1692:
1543:
994:
The historical development of solid modelers has to be seen in context of the whole
1733:
1719:
1680:
1595:
1566:
1531:
1260:
822:
337:
1277:
Visualization of specific body tissues (just blood vessels and tumor, for example)
1313:
Computational analysis of complex biological processes, e.g. air flow, blood flow
560:
552:
422:
278:
234:
230:
547:
regular grid. Cell decompositions provide convenient ways for computing certain
955:
827:
567:
or Morse decompositions may be used for applications in robot motion planning.
477:
304:
According to the continuum point-set model of solidity, all the points of any
1888:
1778:
1255:
scanners can be used to create solid models of internal body features called
403:
383:
206:
199:
1535:
1684:
1411:
1296:
418:
1724:
1707:
1599:
1281:
1264:
1119:
components is usually done within the context of the whole product using
596:
485:
426:
406:
203:
177:
1078: in this section. Unsourced material may be challenged and removed.
399:
238:
1323:
If the use goes beyond visualization of the scan data, processes like
167:. Solid modeling is distinguished within the broader related areas of
1285:
1003:
612:
509:
270:
195:
1344:
1186:
1053:
867:
851:
608:
1705:
1136:
can be created semi-automatically and reference the solid models.
1288:, and other medical and dental devices (this is sometimes called
214:
107:
1303:(to aid surgeons preparing for difficult surgeries, for example)
300:
Regularization of a 2D set by taking the closure of its interior
218:
615:
algorithms have further contributed to the popularity of CSG.
296:
265:
All manufactured components have finite size and well behaved
1468:
1316:
Computational simulation of new medical devices and implants
1310:
solid modeling (design of hip replacement parts, for example)
1256:
1170:
1031:
838:
Features are defined to be parametric shapes associated with
650:
that can be evaluated at any point in space. In other words,
526:
413:
into a collection of disjoint cells of dimensions 0,1,2,3. A
1879:
1671:
Ziegler, M. (2004). "Computable Operators on Regular Sets".
1106:
The modeling of solids is only the minimum requirement of a
1874:
1521:
1007:
1111:
1023:
1019:
281:. The two common approaches to define "solidity" rely on
833:
1839:
1550:
817:
represent, respectively, a plane and two open linear
788:
753:
718:
665:
530:
corresponding ordered set of coordinates is called a
455:-dimensional topological polyhedra. Therefore, every
1708:"Challenges in feature based manufacturing research"
962:, and by adding encyclopedic content written from a
504:, and individual objects within a family are called
417:
of a semi-analytic set into a collection of points,
1152:Related to parameters, but slightly different, are
1126:
Another type of modeling technique is 'surfacing' (
45:
may be too technical for most readers to understand
1639:
809:
774:
739:
704:
642:A very general method of defining a set of points
356:to be considered solid, every neighborhood of any
1814:
1581:
1579:
1886:
1585:
1492:
1490:
516:
1515:
470:
336:points. Assuming ℝ is endowed with the typical
1645:
1576:
1556:
1496:
1002:which went on to influence the development of
583:
525:occupied by the solid. The cells, also called
482:combinations of regularized Boolean operations
1798:Geometric Modeling: The mathematics of shapes
1487:
1022:. It was assigned the task of developing the
521:This scheme is essentially a list of spatial
463:-dimensional polyhedron having homogeneously
27:Set of principles for modeling solid geometry
1746:: CS1 maint: multiple names: authors list (
1652:. MIT press, ACM doctoral dissertation award
1173:can be contained in one model, for example.
857:
390:The combinatorial characterization of a set
1706:Mantyla, M., Nau, D., and Shah, J. (1996).
1670:
1620:
1373:. Unsourced material may be challenged and
1259:-based models, with images generated using
1215:. Unsourced material may be challenged and
896:. Unsourced material may be challenged and
256:
1664:
631:
1795:
1723:
1471:– Programming Language of Solid Modeling.
1393:Learn how and when to remove this message
1235:Learn how and when to remove this message
1176:
1094:Learn how and when to remove this message
1034:was conceived at the University of Rome.
1014:. One of the first CAD developers in the
982:Learn how and when to remove this message
916:Learn how and when to remove this message
595:representations take the form of ordered
484:cannot necessarily be represented as the
308:⊂ ℝ can be classified according to their
132:Learn how and when to remove this message
73:Learn how and when to remove this message
57:, without removing the technical details.
1845:
1820:
1410:
1037:
603:represent either rigid transformations (
570:
295:
145:
1699:
1649:The Complexity of Robot Motion Planning
1524:IEEE Computer Graphics and Applications
495:
14:
1887:
1614:
1139:
943:contains content that is written like
1760:
1415:Mass properties window of a model in
834:Parametric and feature-based modeling
541:
394:⊂ ℝ as a solid involves representing
55:make it understandable to non-experts
1530:(7). IEEE Computer Graphics: 25–37.
1371:adding citations to reliable sources
1338:
1213:adding citations to reliable sources
1180:
1076:adding citations to reliable sources
1047:
927:
894:adding citations to reliable sources
861:
106:In particular, it has problems with
84:
29:
1306:Combining polygon mesh models with
24:
1016:Commonwealth of Independent States
382:it is implied that all solids are
25:
1916:
1868:
1343:
1270:Uses of medical solid modeling;
1185:
1052:
932:
866:
89:
34:
1846:LaCourse, Donald (1995). "11".
1789:
1063:needs additional citations for
368:, defined as sets equal to the
277:, subsets of three-dimensional
1880:The Solid Modeling Association
1850:. McGraw Hill. p. 111.2.
1821:LaCourse, Donald (1995). "2".
1754:
1334:
798:
792:
763:
757:
728:
722:
98:This article needs editing to
13:
1:
1480:
517:Spatial occupancy enumeration
1825:. McGraw Hill. p. 2.5.
1673:Mathematical Logic Quarterly
1627:. Cambridge University Press
1571:10.1016/0010-4485(80)90025-1
705:{\displaystyle f=ax+by+cz+d}
471:Solid representation schemes
467:-dimensional neighborhoods.
340:, a neighborhood of a point
225:, digital data archival and
7:
1796:Golovanov, Nikolay (2014).
1426:
618:
590:Constructive Solid Geometry
584:Constructive solid geometry
536:constructive solid geometry
188:
10:
1921:
1848:Handbook of Solid Modeling
1823:Handbook of Solid Modeling
1761:Yares, Evan (April 2013).
1407:Computer-aided engineering
1404:
1253:magnetic resonance imaging
1041:
635:
587:
251:computer-aided engineering
237:and NC path verification,
1712:Communications of the ACM
1249:computed axial tomography
1128:Freeform surface modeling
1108:CAD system's capabilities
1027:geometric modeling kernel
858:History of solid modelers
810:{\displaystyle f(p)<0}
775:{\displaystyle f(p)>0}
1905:Euclidean solid geometry
1112:graphical user interface
1012:Solid Modeling Solutions
257:Mathematical foundations
100:comply with Knowledge's
1646:Canny, John F. (1987).
1536:10.1109/MCG.1983.263271
1497:Shapiro, Vadim (2001).
1459:boundary representation
638:Function representation
632:Implicit representation
555:(number of pieces) and
374:of their interior. Any
1773:(4). WTWH Media, LLC.
1685:10.1002/malq.200310107
1679:(45). Wiley: 392–404.
1443:Computational geometry
1438:Free-surface modelling
1419:
1177:Medical solid modeling
811:
776:
741:
740:{\displaystyle f(p)=0}
706:
551:of solids such as its
549:topological properties
301:
151:
1900:Computer-aided design
1875:sgCore C++/C# library
1725:10.1145/230798.230808
1600:10.1145/356827.356833
1588:ACM Computing Surveys
1559:Computer-Aided Design
1464:List of CAx companies
1414:
1044:Computer-aided design
1038:Computer-aided design
964:neutral point of view
812:
777:
742:
707:
571:Surface mesh modeling
432:topological polyhedra
348:takes the form of an
299:
183:computer-aided design
149:
1895:3D computer graphics
1621:Hatcher, A. (2002).
1433:Wire frame modelling
1367:improve this section
1209:improve this section
1134:Engineering drawings
1072:improve this article
890:improve this section
844:geometric tolerances
786:
751:
716:
663:
496:Primitive instancing
436:Euler characteristic
1785:on 30 January 2015.
1453:Engineering drawing
1329:image-based meshing
1140:Parametric modeling
956:promotional content
848:material properties
599:where non-terminal
506:primitive instances
380:Heine-Borel theorem
366:closed regular sets
227:reverse engineering
114:improve the content
1624:Algebraic Topology
1420:
1325:image segmentation
1290:mass customization
958:and inappropriate
807:
772:
737:
702:
542:Cell decomposition
302:
290:algebraic topology
284:point-set topology
169:geometric modeling
152:
1857:978-0-07-035788-4
1832:978-0-07-035788-4
1475:Technical drawing
1448:Computer graphics
1403:
1402:
1395:
1301:rapid prototyping
1245:
1244:
1237:
1121:assembly modeling
1104:
1103:
1096:
992:
991:
984:
926:
925:
918:
502:generic primitive
398:as an orientable
223:rapid prototyping
211:injection molding
173:computer graphics
142:
141:
134:
83:
82:
75:
16:(Redirected from
1912:
1862:
1861:
1843:
1837:
1836:
1818:
1812:
1811:
1793:
1787:
1786:
1781:. Archived from
1758:
1752:
1751:
1745:
1737:
1727:
1703:
1697:
1696:
1668:
1662:
1661:
1659:
1657:
1643:
1637:
1636:
1634:
1632:
1618:
1612:
1611:
1583:
1574:
1573:
1554:
1548:
1547:
1519:
1513:
1512:
1510:
1508:
1494:
1398:
1391:
1387:
1384:
1378:
1347:
1339:
1261:volume rendering
1240:
1233:
1229:
1226:
1220:
1189:
1181:
1166:injection molded
1099:
1092:
1088:
1085:
1079:
1056:
1048:
987:
980:
976:
973:
967:
945:an advertisement
936:
935:
928:
921:
914:
910:
907:
901:
870:
862:
816:
814:
813:
808:
781:
779:
778:
773:
746:
744:
743:
738:
711:
709:
708:
703:
646:is to specify a
338:Euclidean metric
314:with respect to
243:dynamic analysis
137:
130:
126:
123:
117:
93:
92:
85:
78:
71:
67:
64:
58:
38:
37:
30:
21:
1920:
1919:
1915:
1914:
1913:
1911:
1910:
1909:
1885:
1884:
1871:
1866:
1865:
1858:
1844:
1840:
1833:
1819:
1815:
1808:
1794:
1790:
1759:
1755:
1739:
1738:
1704:
1700:
1669:
1665:
1655:
1653:
1644:
1640:
1630:
1628:
1619:
1615:
1584:
1577:
1555:
1551:
1520:
1516:
1506:
1504:
1495:
1488:
1483:
1429:
1409:
1399:
1388:
1382:
1379:
1364:
1348:
1337:
1241:
1230:
1224:
1221:
1206:
1190:
1179:
1142:
1100:
1089:
1083:
1080:
1069:
1057:
1046:
1040:
1030:Solid Modeling
988:
977:
971:
968:
949:
937:
933:
922:
911:
905:
902:
887:
871:
860:
836:
787:
784:
783:
752:
749:
748:
717:
714:
713:
712:the conditions
664:
661:
660:
640:
634:
621:
592:
586:
573:
561:finite elements
544:
519:
498:
473:
279:Euclidean space
259:
235:motion planning
231:finite elements
191:
159:solid modelling
138:
127:
121:
118:
111:
102:Manual of Style
94:
90:
79:
68:
62:
59:
51:help improve it
48:
39:
35:
28:
23:
22:
18:Solid modelling
15:
12:
11:
5:
1918:
1908:
1907:
1902:
1897:
1883:
1882:
1877:
1870:
1869:External links
1867:
1864:
1863:
1856:
1838:
1831:
1813:
1807:978-1497473195
1806:
1788:
1753:
1698:
1663:
1638:
1613:
1594:(4): 437–464.
1575:
1565:(5): 219–220,
1549:
1514:
1500:Solid Modeling
1485:
1484:
1482:
1479:
1478:
1477:
1472:
1466:
1461:
1455:
1450:
1445:
1440:
1435:
1428:
1425:
1405:Main article:
1401:
1400:
1351:
1349:
1342:
1336:
1333:
1321:
1320:
1314:
1311:
1304:
1293:
1278:
1275:
1243:
1242:
1193:
1191:
1184:
1178:
1175:
1141:
1138:
1102:
1101:
1060:
1058:
1051:
1042:Main article:
1039:
1036:
996:history of CAD
990:
989:
960:external links
940:
938:
931:
924:
923:
874:
872:
865:
859:
856:
835:
832:
828:marching cubes
806:
803:
800:
797:
794:
791:
771:
768:
765:
762:
759:
756:
736:
733:
730:
727:
724:
721:
701:
698:
695:
692:
689:
686:
683:
680:
677:
674:
671:
668:
636:Main article:
633:
630:
620:
617:
588:Main article:
585:
582:
572:
569:
565:stratification
543:
540:
518:
515:
497:
494:
478:data structure
472:
469:
441:Jordan-Brouwer
258:
255:
190:
187:
155:Solid modeling
140:
139:
97:
95:
88:
81:
80:
42:
40:
33:
26:
9:
6:
4:
3:
2:
1917:
1906:
1903:
1901:
1898:
1896:
1893:
1892:
1890:
1881:
1878:
1876:
1873:
1872:
1859:
1853:
1849:
1842:
1834:
1828:
1824:
1817:
1809:
1803:
1799:
1792:
1784:
1780:
1776:
1772:
1768:
1764:
1763:"Russian CAD"
1757:
1749:
1743:
1735:
1731:
1726:
1721:
1717:
1713:
1709:
1702:
1694:
1690:
1686:
1682:
1678:
1674:
1667:
1651:
1650:
1642:
1626:
1625:
1617:
1609:
1605:
1601:
1597:
1593:
1589:
1582:
1580:
1572:
1568:
1564:
1560:
1553:
1545:
1541:
1537:
1533:
1529:
1525:
1518:
1502:
1501:
1493:
1491:
1486:
1476:
1473:
1470:
1467:
1465:
1462:
1460:
1456:
1454:
1451:
1449:
1446:
1444:
1441:
1439:
1436:
1434:
1431:
1430:
1424:
1418:
1413:
1408:
1397:
1394:
1386:
1376:
1372:
1368:
1362:
1361:
1357:
1352:This section
1350:
1346:
1341:
1340:
1332:
1330:
1326:
1319:
1315:
1312:
1309:
1305:
1302:
1298:
1294:
1291:
1287:
1283:
1279:
1276:
1274:Visualization
1273:
1272:
1271:
1268:
1266:
1262:
1258:
1254:
1250:
1239:
1236:
1228:
1218:
1214:
1210:
1204:
1203:
1199:
1194:This section
1192:
1188:
1183:
1182:
1174:
1172:
1167:
1163:
1160:, instead of
1159:
1155:
1150:
1146:
1137:
1135:
1131:
1129:
1124:
1122:
1116:
1113:
1109:
1098:
1095:
1087:
1077:
1073:
1067:
1066:
1061:This section
1059:
1055:
1050:
1049:
1045:
1035:
1033:
1028:
1025:
1021:
1017:
1013:
1009:
1005:
1001:
997:
986:
983:
975:
965:
961:
957:
953:
947:
946:
941:This section
939:
930:
929:
920:
917:
909:
899:
895:
891:
885:
884:
880:
875:This section
873:
869:
864:
863:
855:
853:
849:
845:
841:
831:
829:
824:
820:
804:
801:
795:
789:
769:
766:
760:
754:
734:
731:
725:
719:
699:
696:
693:
690:
687:
684:
681:
678:
675:
672:
669:
666:
657:
653:
649:
645:
639:
629:
626:
616:
614:
610:
606:
602:
598:
591:
581:
579:
568:
566:
562:
558:
554:
553:connectedness
550:
539:
537:
533:
532:spatial array
528:
524:
514:
511:
507:
503:
493:
490:
487:
483:
479:
468:
466:
462:
458:
454:
450:
444:
442:
437:
433:
428:
424:
421:, triangular
420:
419:line segments
416:
415:triangulation
412:
408:
405:
404:semi-analytic
401:
397:
393:
388:
385:
381:
377:
373:
372:
367:
363:
359:
355:
351:
347:
343:
339:
335:
334:
329:
328:
323:
322:
317:
313:
312:
311:neighborhoods
307:
298:
294:
292:
291:
286:
285:
280:
276:
272:
268:
263:
254:
252:
248:
244:
240:
236:
232:
228:
224:
220:
216:
212:
208:
207:manufacturing
205:
201:
197:
186:
184:
180:
179:
174:
170:
166:
165:
160:
156:
148:
144:
136:
133:
125:
115:
109:
105:
103:
96:
87:
86:
77:
74:
66:
56:
52:
46:
43:This article
41:
32:
31:
19:
1847:
1841:
1822:
1816:
1797:
1791:
1783:the original
1770:
1767:Design World
1766:
1756:
1742:cite journal
1718:(2): 77–85.
1715:
1711:
1701:
1676:
1672:
1666:
1654:. Retrieved
1648:
1641:
1629:. Retrieved
1623:
1616:
1591:
1587:
1562:
1558:
1552:
1527:
1523:
1517:
1505:. Retrieved
1499:
1421:
1389:
1383:January 2012
1380:
1365:Please help
1353:
1322:
1317:
1297:polygon mesh
1269:
1246:
1231:
1225:January 2012
1222:
1207:Please help
1195:
1161:
1157:
1151:
1147:
1143:
1132:
1125:
1117:
1105:
1090:
1084:January 2012
1081:
1070:Please help
1065:verification
1062:
993:
978:
969:
954:by removing
950:Please help
942:
912:
906:January 2012
903:
888:Please help
876:
839:
837:
655:
651:
643:
641:
624:
622:
597:binary trees
593:
574:
545:
531:
522:
520:
505:
501:
499:
491:
474:
464:
460:
456:
452:
448:
445:
400:cell complex
395:
391:
389:
375:
369:
365:
361:
357:
353:
345:
341:
331:
325:
319:
315:
309:
305:
303:
288:
282:
274:
264:
260:
192:
176:
162:
158:
154:
153:
143:
128:
119:
112:Please help
99:
69:
63:January 2021
60:
44:
1335:Engineering
1299:models for
1282:prosthetics
1265:3D scanners
1154:constraints
830:algorithm.
823:R-functions
654:is defined
607:preserving
605:orientation
427:tetrahedral
204:sheet metal
178:3D modeling
1889:Categories
1503:. Elsevier
1481:References
1280:Designing
1263:. Optical
952:improve it
840:attributes
819:halfspaces
656:implicitly
609:isometries
510:algorithms
411:stratified
267:boundaries
247:mechanisms
175:, such as
1779:1941-7217
1608:207568300
1354:does not
1295:Creating
1286:orthotics
1196:does not
1171:capscrews
1004:Parasolid
972:June 2015
877:does not
852:databases
648:predicate
613:recursive
578:STL files
350:open ball
271:isotropic
239:kinematic
196:machining
1693:17579181
1656:20 April
1631:20 April
1544:14462567
1507:20 April
1427:See also
1020:C3D Labs
619:Sweeping
333:boundary
327:exterior
321:interior
200:assembly
189:Overview
164:(solids)
122:May 2024
1734:3340804
1375:removed
1360:sources
1318:in vivo
1247:Modern
1217:removed
1202:sources
1158:created
1000:Romulus
898:removed
883:sources
407:bounded
384:compact
371:closure
275:regions
215:welding
108:MOS:BBB
49:Please
1854:
1829:
1804:
1777:
1732:
1691:
1606:
1542:
1457:Euler
1417:Cobalt
782:, and
527:voxels
425:, and
352:. For
1730:S2CID
1689:S2CID
1604:S2CID
1540:S2CID
1469:PLaSM
1257:voxel
1162:shown
1032:PLaSM
625:sweep
601:nodes
557:genus
523:cells
486:sweep
423:faces
330:, or
1852:ISBN
1827:ISBN
1802:ISBN
1775:ISSN
1748:link
1658:2010
1633:2010
1509:2010
1358:any
1356:cite
1327:and
1251:and
1200:any
1198:cite
1010:and
1008:ACIS
881:any
879:cite
802:<
767:>
287:and
241:and
219:pipe
198:and
171:and
157:(or
1720:doi
1681:doi
1596:doi
1567:doi
1532:doi
1369:by
1308:CAD
1211:by
1074:by
1024:C3D
892:by
465:n−1
461:n−1
318:as
245:of
53:to
1891::
1769:.
1765:.
1744:}}
1740:{{
1728:.
1716:39
1714:.
1710:.
1687:.
1677:50
1675:.
1602:.
1592:12
1590:.
1578:^
1563:12
1561:,
1538:.
1526:.
1489:^
1284:,
1006:,
846:,
747:,
538:.
324:,
253:.
233:,
217:,
213:,
209:,
1860:.
1835:.
1810:.
1771:8
1750:)
1736:.
1722::
1695:.
1683::
1660:.
1635:.
1610:.
1598::
1569::
1546:.
1534::
1528:3
1511:.
1396:)
1390:(
1385:)
1381:(
1377:.
1363:.
1292:)
1238:)
1232:(
1227:)
1223:(
1219:.
1205:.
1097:)
1091:(
1086:)
1082:(
1068:.
985:)
979:(
974:)
970:(
966:.
948:.
919:)
913:(
908:)
904:(
900:.
886:.
805:0
799:)
796:p
793:(
790:f
770:0
764:)
761:p
758:(
755:f
735:0
732:=
729:)
726:p
723:(
720:f
700:d
697:+
694:z
691:c
688:+
685:y
682:b
679:+
676:x
673:a
670:=
667:f
652:X
644:X
457:n
453:n
449:n
396:X
392:X
376:X
362:X
360:∈
358:p
354:X
346:X
344:∈
342:p
316:X
306:X
135:)
129:(
124:)
120:(
116:.
110:.
104:.
76:)
70:(
65:)
61:(
47:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
