### Table 1: Correspondence between singularities of tangents of the manifold, the 2-parameter family of height functions, and the pedal surface. There are two singularities of co-dimension one: curves of cusps and curves of self-intersections (xings). There are six singu- larities of co-dimension two.

"... In PAGE 4: ... In each case, the singularity is defined by two pairs of critical points and we get two types each because these pairs may be disjoint or share one of the points. See Table1 for the features on C5 that correspond to the six types of co-dimension two singularities. We can now be more precise about what we mean by a generic 2-manifold.... In PAGE 4: ... GENERICITY ASSUMPTION A. The 2-parameter family of height functions on C5 has no violations of Conditions I and II for Morse functions other than the ones mentioned above (and enumerated in Table1 below). Some of these violations will be discussed in more detail later as they can be locations of maximum elevation.... In PAGE 5: ... The only remaining possibility for discontinuous elevation is thus at interchanges, which happen when two points share the same tangent plane. As mentioned in Table1 , this corresponds to a point at which the pedal surface intersects itself. Figure 6 shows that discontinuities in the elevation can indeed arise at co-tangent points.... In PAGE 6: ... We know that DC is not a flat point of C5 , else its elevation would be zero. This simple observation eliminates five of the eight singularities in Table1 . Furthermore, the assumption of a generic 2-manifold C5 implies that a multiplicity three point can only be paired with a multiplicity one point.... ..."

### Table 1: Correspondence between singularities of tangents of the manifold, the 2-parameter family of height func- tions, and the pedal surface. There are two singularities of co-dimension one: curves of cusps and curves of self- intersections (xings). There are six singularities of co-

"... In PAGE 5: ... In each case, the singular- ity is defined by two pairs of critical points and we get two types each because these pairs may be disjoint or share one of the points. See Table1 for the features on a0 that corre- spond to the six types of co-dimension two singularities. We can now be more precise about what we mean by a generic 2-manifold.... In PAGE 5: ... GENERICITY ASSUMPTION A. The 2-parameter family of height functions on a0 has no violations of Conditions I and II for Morse functions other than the ones men- tioned above (and enumerated in Table1 below). Some of these violations will be discussed in more detail later as they can be locations of maximum elevation.... In PAGE 6: ... The only re- maining possibility for discontinuous elevation is thus at in- terchanges, which happen when two points share the same tangent plane. As mentioned in Table1 , this corresponds to a point at which the pedal surface intersects itself. Figure 6 shows that discontinuities in the elevation can indeed arise at co-tangent points.... In PAGE 8: ... We know that a7 is not a flat point of a0 , else its elevation would be zero. This simple observation eliminates five of the eight singularities in Table1 . Furthermore, the assumption of a generic 2-manifold a0 implies that a multi- plicity three point can only be paired with a multiplicity one point.... ..."

### Table 1: Parameters of a separable and a non-separable mapping. HDm: Hypercube Dimension, GCM: Separable Mapping, based on binaryreflectedGraycodes(23 23 ! 8 8), FGM:Non-separable, mapping based on two folded grids (26 ! 64).

"... In PAGE 3: ... HDm: Hypercube Dimension, GCM: Separable Mapping, based on binaryreflectedGraycodes(23 23 ! 8 8), FGM:Non-separable, mapping based on two folded grids (26 ! 64). In Table1 GCM, the Gray code mapping, is obtained by Gray coded processor indices for two ring mappings. The hypercube dimensions alternate in both mesh dimensions.... ..."

### Table 2 also shows how many triangles NAR are generated in AR-improving procedure after -triangulation, where the number N of triangles is 222. We have also performed the triangulation in fully rounded interval arithmetic (RIA). Table 3 shows the result. As far as the time cost is concerned, RIA is one order of magnitude more expensive than FPA. However, we need to keep in mind that crucial computations such as intersection test between the edges of an approximately developed surface net, point mapping between the triangulation domain and the parametric space, Delaunay test and self-intersection check of the approximating triangles, may result in system failure if the algorithm is operating purely in FPA. Global behavior of the other realistic examples shown in Figure 9 is consistent with that of our rst example, where we set = 10?3 and AR = 4. Figure 9-(a) shows the result of meshing (Ntotal = 7832) for a surface of revolution, a typical example of rational B-spline surfaces. We have also implemented the method with a composite integral B-spline surface representing a part of a ship hull with a bulbous bow { see Figure 9-(b), where Ntotal = 2580. The next example is another composite integral B-spline surface that represents an airfoil. Figure 9-(c) illustrates the performance (Ntotal = 3310). In these examples, we can verify the well-conditioned meshing as well as local adaptivity near high curvature regions.

"... In PAGE 10: ...1 4.9 Table2 : Trimmed bi-cubic NURBS surface patch; AR-improving triangulation, = 10?2, N = 222 ble and dominant with respect to Cinit. Also in Table 1, CAR, Cfin and Ctotal show the time cost of the AR-improving procedure, intersection test/ nal triangulation, and whole process, respectively.... In PAGE 10: ... It is worthwhile to mention that the satisfactory average aspect ratio ARavg results from the coupled e ects of the features of Delaunay based point insertion algorithm; AR-improving procedure in the planar domain of triangulation; and preservation of triangles apos; shape during the mapping process from the triangulation domain into three-dimensional space. Table2 shows the dependency of AR-improving procedure upon the AR-threshold AR for a xed approx- imation tolerance = 10?2. In this example, 3% of Ntotal have AR gt; AR = 3:5 due to slight deformation occurred in the process of approximate locally isometric mapping.... ..."

### Table 1 compares the computational complexity of all the MAP-based decoding algorithms for a Ti data rate.

"... In PAGE 8: ...Table1 . Complexity comparison between different MAP algorithms.... ..."

### Table 1. Correspondence between STC and mesh entities.

"... In PAGE 6: ... Sheets with more than one loop and with self-intersections are discussed in more detail in Section 4. The one-to-one correspondence between dual (STC) entities and mesh entities is shown in Table1 . A mesh facet of dimension corresponds to a STC facet of dimension .... ..."

### Table 5 Results and comparisons of change maps based on the unsupervised and MOLA classifications summarized in three ways: a) change in the percentage of 5-km pixels in each land-cover type, b) change in percentage of land-cover types over sub-regions, and c) change in land-cover percentages over states

2005

"... In PAGE 11: ...assessment of the change information indicates that the MOLA classification improved on the unsupervised approach in every test ( Table5 ). The RMSEs can be interpreted as the error in the estimated change in percentages within the 5-km pixels.... ..."

### Table 2. Comparison of a MAP based on class-specific contrast classifier and a standard 3-class neural network in classification with unbiased labeled data Class

2003

"... In PAGE 4: ... For this experiment, a total of 1500 labeled examples and 150,000 unlabeled examples were generated with different class proportions. In Table2 we report their accuracies obtained in experiments with 3 different class proportions. The ... ..."

Cited by 6

### Table 1. Summary of a survey on ten multimodal user studies. The ve columns in the upper table deal with map-based domains, whereas the lower ve columns deal with other kinds of applications.

1998

"... In PAGE 2: ... To have a global view of existing user studies for multimodal interfaces, we analyzed ten of them as they were described in the literature. Table1 summarises this survey. 4 Existing multimodal studies and TYCOON We have compiled a list of questions based on our framework.... ..."

Cited by 12

### Table 1. Summary of a survey on ten multimodal user studies. The ve columns in the upper table deal with map-based domains, whereas the lower ve columns deal with other kinds of applications.

1998

"... In PAGE 2: ... To have a global view of existing user studies for multimodal interfaces, we analyzed ten of them as they were described in the literature. Table1 summarises this survey. 4 Existing multimodal studies and TYCOON We have compiled a list of questions based on our framework.... ..."

Cited by 12