### Table 1. Simple geometric entities generated from up to three twists

### Table 1: list of the conformal geometric entities

"... In PAGE 2: ...3. The outer product and the basic geometric entities in Conformal Geometric Algebra Table1 lists the two representations of the geometric entities in Conformal Geometric Algebra. Please find details in [8] and [11].... ..."

### Table 1: The proxl functions for some common geometrical entities

1988

"... In PAGE 3: ... It is useful to introduce a prox, function, which is the same function as prox except that a point P and the unit-set {P} are identified. Table1 indicates values of prox, for some common geometric entities. We will also need a direction invariant.... ..."

Cited by 2

### Table 1 Basic topological entities in the CGM geometric model.

2004

"... In PAGE 6: ...epresentations is discussed in section 2.5. 2.1 Topology The basic topological entities used in CGM, along with their dimensions, are shown in Table1 . Note that in the case of Edges and Faces, the most common nomenclature can be confusing in a finite element context, since there are also edge and face element types.... In PAGE 14: ... GeometryModifyTool functions supporting geometry decomposition are summarized in Table 11. Table1 1: Functions supporting geometry decomposition. Function Description GeometryModifyTool::webcut_with_xxx (xxx = plane, sheet, vertices, cylinder, surface, extended_surface, body) Decompose one or more bodies with a given geometry entity or implicitly-defined geometric entity.... In PAGE 15: ... Functions which control merging are summarized in Table 12. Table1 2: MergeTool functions which control merging of manifold into non-manifold geometry. Function Description MergeTool::merge_all_bodies MergeTool::merge_all_refxxx (xxx = vertices, edges, faces) Performs merge check on all entities of the specified type.... In PAGE 15: ... Table 13 summarizes the functions for creating, removing, and performing other interactions with virtual geometry. Table1 3: VirtualGeometryEngine functions for creating, removing and otherwise interacting with virtual geometry. Function Description PartitionTool::partition Partition the given entity using explicit or implicit geometry (implicit geometry is geometry whose geometric data are ... In PAGE 17: ...Table1 4: Topology traversal functions implemented in TopologyEntity. Function Description TopologyEntity::bodies TopologyEntity::ref_xxx (xxx=volumes, faces, edges, vertices) TopologyEntity::shells TopologyEntity::loops TopologyEntity::co_faces TopologyEntity::co_edges Return list of bodies related to the entity from which the function was called.... In PAGE 17: ... Like the traversal functions in TopologyEntity, these functions are called as member functions from a RefEntity object. Table1 5: Parent and child topology traversal functions defined in RefEntity. Function Description RefEntity::get_child_ref_entities Return a list of all immediate children of the RefEntity.... In PAGE 18: ... The purpose of the code in each subdirectory is also described in Table 16. Table1 6: CGM directory structure and purpose of code in each subdirectory. Directory Purpose Depends on: $(CUBIT_BASE_DIR)/util Utility functions for other code in CGM (none) $(CUBIT_BASE_DIR)/util/OtherFiles Configuration files for compiling CGM on various platforms (none) $(CUBIT_BASE_DIR)/geom Core CGM datastructure and tool classes.... In PAGE 19: ... This section gives some additional notes on compiling applications based on CGM. As noted in Table1 6, some of the code in CGM depends on the ACIS solid modeling engine; however, it is not necessary to link CGM applications with ACIS if ACIS-based geometry will not be used. 3.... In PAGE 20: ... In this application, since it exits immediately after these geometry calculations, there is no need to explicitly shut down CGM. Table1 7: Pseudo code for C++ driver code, forward declarations and main. // include tool and datastructure declarations #include GeometryModifyTool.... In PAGE 21: ...8.2 Reading geometry files In the read_geometry() function, shown in Table1 8, each file is opened, then the file pointer is passed to GeometryQueryTool::import_solid_model function. This function reads all the geometry entities defined in that file and stores them in the CGM database.... In PAGE 21: ... The return value from import_solid_model() should be checked for indications of problems reading the geometry file. Table1 8: Pseudo code for C++ driver; read_geometry() function. CubitStatus read_geometry(int num_files, char **argv) { GeometryQueryTool *gqt = GeometryQueryTool::instance(); // For each file, open and read the geometry for (i = 1; i lt; num_files; i++) { file_ptr = fopen(argv[i], r ); if (file_ptr == NULL) PRINT_ERROR( Could not open file %s\n , argv[i]); else { status = gqt- gt;import_solid_model(file_ptr, argv[i], ACIS_SAT ); } } return CUBIT_SUCCESS; } 3.... In PAGE 22: ... GeometryQueryTool::delete_Body() is called at the end of evaluate_overlaps() to delete any bodies produced during the intersection operation. Table1 9: Pseudo code for C++ driver; evaluate_overlaps() function. CubitStatus evaluate_overlaps() { // evaluate overlaps by intersecting bodies pairwise GeometryModifyTool *gmt = GeometryModifyTool::instance(); // make a copy of the body list for use in this function DLIList lt;Body* gt; all_bodies, all_new_bodies; GeometryQueryTool::instance()- gt;bodies(all_bodies); // step forward on this list, extracting the first body and using it // as a tool for remaining bodies for (i = all_bodies.... In PAGE 24: ...Merging and printing results In the main function, after the call to imprint_bodies() (see Table1 7), all bodies in the model are checked for potential merges by calling MergeTool::merge_all_bodies(). This function uses a pre-defined tolerance while checking for coincident geometry; this tolerance can be changed using other functions in MergeTool.... ..."

### Table 2 summarizes the expressions for constructing new geometric entities. entities construction expression eq.

2000

"... In PAGE 4: ... Table2 . Construction of new geometric entities.... ..."

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### TABLE 1.1. list of the conformal geometric entities entity representation 1 representation 2

in geometric

### Table 1 Representational Entities

2003

"... In PAGE 6: ... 4; where M1, M2, and M3 are three MESHSURFACEs, E1, E2, and E3 are their related MESHEDGEs which contain the same number of TRGLEDGEs, and EO is the MESHJOINT containing the adjacent information). The detail description of each entity is shown in Table1 (Pseudo code is shown in Appendix I). Using the data structure, it is easy to carry out any topological and geometrical manipulation on the manifold or non-manifold triangular mesh models.... ..."

Cited by 2

### 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: Statistics for a simple branching structure

"... In PAGE 2: ... For geometry in the 2D case, we characterize divergence from one internode to the next, branching and leaf angles into those turning left, right, and continuing straight, using a mean angle for each angle type. Clustered statistics are subdivided into these geometrical categories (see Table1 ). Spatial positions of leaves and branches are characterized by their depth in the tree structure, calculated by counting the number of internodes between the component and the tree root.... In PAGE 3: ... Figures 1 and 2 show two target plants and their L-system models derived with this evolution method. For the first one, because of its structural simplicity, the whole structure was measured and statistics calculated to determine the target structure ( Table1 ). In the table, entity directions are denoted as left, straight and right and four clusters are involved, G0, G1, G2 and G3.... ..."