### Table 2. Geometric constraints.

2002

"... In PAGE 2: ... Furthermore, auxiliary objects (planes, cylinders, lines, di- rections, positions, lengths, angles) are used to express cer- tain regularities with simple constraints. The geometric constraints are represented as equations listed in Table2 . We require all direction vectors to be nor- malized.... ..."

Cited by 3

### Table 1: The 3D geometric data types

"... In PAGE 35: ...:100,000 {road, river, buildine, ..., lot} Table1 : Default settings for scale ranges and preferred image sources for geographic... In PAGE 75: ...topological projective exception German English German Enelish German English an at hinter behind zwischen between bei near links left fern far neben besid,e NN xn rechts right nahe close to iiber aboae unter below aor in front of Table1 : German spatial relations and their English counterparts 2. COMPUTING THE ELEMENTARY SPATIAL RELATIONS Following [Landau amp; Jackendoff 93], people do not account for every detail of the ob- jects involved when they apply spatial relations.... In PAGE 75: ...e.g., at and near), the projective relations (e.g., in front of , right, and below), and the relation between, which takes an exceptional position in the group of spatial relations.r Table1 shows the 13 different German spatial relations and their English counterparts considered at the moment. lln the sequel only the English expressions for the German prepositions are used.... In PAGE 101: ...einand, A., Gamma, E., amp; Marty, R. (1989). Design and Implementation of ETt*: A searnless Object-Oriented Application Framework. Structured Programming, 10(2),63-87 apos; APPENDIX A: 3D SPATIAL ADT FUNCTIONS This appendix contains the functions specific for one of the following 3D spatial data types: POINT3, POLYLINE3, POLYGON3, and POLYHEDRON3; see Table1 . In Tables 2, 3, 4, and 5, the in- and output functions are omitted.... In PAGE 109: ... The difference with geographic data is that the candidate cells can be based on either the spatial data, the attribute data or a combination of both. Table1 gives an example of primary candidate partitionings in which the predicate is based on some spatial or nonspatial property of the entity. Table 1: Primary Candidate Partitioning Table Entity Predicate a) mapsheet mapsheet_name : apos;Laverton apos; mapsheet_name : apos;Albany apos; mapsheet_name c apos;Laverton apos;, apos;Al- bany apos; b) mlne status : apos;Developing apos; status : apos;Abandoned apos; c) mlne Y-coordinate lt;: 32 Y-coordinate gt; 32... In PAGE 109: ... Table 1 gives an example of primary candidate partitionings in which the predicate is based on some spatial or nonspatial property of the entity. Table1 : Primary Candidate Partitioning Table Entity Predicate a) mapsheet mapsheet_name : apos;Laverton apos; mapsheet_name : apos;Albany apos; mapsheet_name c apos;Laverton apos;, apos;Al- bany apos; b) mlne status : apos;Developing apos; status : apos;Abandoned apos; c) mlne Y-coordinate lt;: 32 Y-coordinate gt; 32... ..."

### Table 3: Normalisation and shape constraints in 3D.

"... In PAGE 23: ... Constraints in three dimensions are thus again of two types: internal constraints on the parameters of the objects, and external constraints expressing relationships between objects. The internal normalisation and shape conditions are given in Table3 . External geometric constraints are represented as in Table 4.... ..."

### Table 1. The Number of Constraints Used in Various Examples

"... In PAGE 23: ... In addition, even a simple visualization requires a considerable number of con- straints. Table1 shows the size of VSR and the number of constraints used in example visualizations created by the TRIP systems. The upper part of the Table 1 shows examples of 2D visualizations generated by TRIP2a.... In PAGE 23: ... Table 1 shows the size of VSR and the number of constraints used in example visualizations created by the TRIP systems. The upper part of the Table1 shows examples of 2D visualizations generated by TRIP2a. The lower part of Table 1 presents the size of VSR and the number of constraints used in 3D visualization examples generated by the TRIP2a/3D System.... In PAGE 23: ... The upper part of the Table 1 shows examples of 2D visualizations generated by TRIP2a. The lower part of Table1 presents the size of VSR and the number of constraints used in 3D visualization examples generated by the TRIP2a/3D System. In general, three-dimensional visualizations require more constraints than two dimensional.... ..."

### Table 2. Geometric Constraints.

2004

"... In PAGE 5: ... They specify rela- tions between geometric objects of the B-rep model, and the auxiliary objects. Any regularities between geometric objects which can be expressed by con- straint types in Table2 can be handled by our system. Note that we do not allow topological changes of the model in this paper|such changes will be considered in a separate paper.... In PAGE 25: ... It may be possible to modify the solvability test to create a decomposition plan for a symbolic constraint solver [7]. The constraint system contains equations of the types in Table2 . We also add one equation per direction vector to ensure that it is a unit vector.... ..."

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### Table 4: External geometric constraints in 3D. 25

"... In PAGE 23: ... Constraints for tori have been omitted for reasons of space, but can easily be obtained from the preceding discussion. There is a subtle di culty not present in the planar case which arises in the rows of Table4 marked (*). Parallelism of two vectors should be expressed by one equation less than the dimension of the space, which is simple in the planar case but not directly possible in 3D.... ..."

### Table 4. Distribution of Geometric Constraints.

2004

"... In PAGE 16: ... We assume that all intersections are generic, as is usual in degrees of freedom analysis. In Table4 we list the di erent options for distribution of constraints. Note that we do not consider all possible distributions of the dimensions, but only those which have a simple geometric meaning.... In PAGE 16: ...eft can also be distributed to the distance parameter, i.e. the xed positions set the value of the distance. Constraints requiring positions to lie on a surface or curve are omitted from Table4 . Surfaces and curves are usually described by a combination of po-... In PAGE 17: ... Algorithm 2 is the overall constraint distribution algorithm. A constraint can be distributed directly if one of the distribution options in Table4 can be applied using the previous reasoning without doing any redistribution (step I). Otherwise, starting at the constraint which should be distributed, we do a breadth- rst search of the constraint graph until a constraint is found which can be redistributed (steps II and III are initialization; step IV does the search).... ..."

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### Table 1: Geometric information

"... In PAGE 20: ... e = QG gt;(G gt;QG) 1e fforcing the constraint G gt; e = eg x01 = 0; x02 = 0; x03 = e fchoose the initial guessg b1 = g; b2 = d = f B gt; e; b3 = 0 fcompute right hand sideg d = K3x0 b fcompute the defectg r0 = D3 ^ L 1 3 d fapply the transformationg p0 = w0 = D 1 3 r0 fpreconditioningg 0 = (w0; r0) for n = 0 step 1 until n quot; 0 do vn = D3 ^ L 1 3 K3pn fmatrix vector multiplication and transformationg = (vn; pn) = n+1= xn+1 = xn pn fupdate the iterateg rn+1 = rn vn fupdate the defectg wn+1 = D 1 3 rn+1 fpreconditioningg n+1 = (wn+1; rn+1) = n+1= n pn+1 = wn+1 + pn fupdate of the search directiong end for Figure 1: Domain decomposition with 8 subdomains In Table1 , the geometric informations about the domain decomposition and the discretization are listed for the re nement levels L used. Starting from the coarsest grid with 192 triangles for the whole domain , the re ned meshes are recursively constructed by subdividing each triangle into four smaller similar triangles.... In PAGE 21: ...Table 1: Geometric information In all following tables, a uni ed notation is used. L again denotes the re nement level and Table1 gives the corresponding information of the grids. t1 and t2 are the measured times in seconds for setting up the corresponding system of linear equations and for their solution.... ..."

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### Table 1.Geometric objects and their features.

2002

"... In PAGE 2: ... The objects are described by a type and a set of appropriate directional, positional, length and angular fea- tures represented as scalars and 3D vectors. There are basic features required to describe the object and extended fea- tures dependent on other features for additional properties as listed in Table1 . Root points for planes are created by taking the average of polygonal edge loops of planar faces.... ..."

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### Table 1 lists a sampling of researchers who have used each imaging model, and suggests how few have generalized to full-perspective imaging. The complexity of computing 3D object pose under full-perspective partially explains this lack of generalization. However, more basic is the common reliance upon local geometric constraints associated with sets of corresponding features. These sets must be larger for full-perspective than for the simpler imaging models. For example, in geometric hashing [LW88], 2 points are required to establish a basis under weak-perspective while full-perspective requires 5 points.

"... In PAGE 6: ... Table1 : Previous work by imaging model. 3 Local Search Matching Local search refers to combinatorial optimization techniques which iteratively search a locally de ned neighborhood until they arrive at locally optimal solutions [PS82].... ..."