### Table 1: Average positional error accumulated over a simple polyhedral path (ten tours of a square of ap- proximately two feet by two feet). In general, the mag- nitude of the errors depends on various parameters of the trajectory but the relative magnitudes as a function of surface type vary consistently.

"... In PAGE 1: ... over which the robot is moving, an estimate of the rate of error accumulation for dead-reckoning allows us to accurately estimate how often localization, including sensor data acquisition, must be performed. For vari- ous oor coverings in our laboratory, for example, the rate of error accumulation varies by a factor of 10 (See Table1 ). The system we propose uses dead-reckoning and knowledge of the material over which it is mov- ing to maintain an estimate of its position and un- certainty in position.... ..."

### Table 8 Computation statistics of the examples

2004

"... In PAGE 20: ...ig.17 gives another example application of CyberTape in the reverse engineering of a mechanical part. For example, after the point cloud of a mechanical part is obtained from CMMs, the related polyhedral surface is constructed by the algorithm in [2]; then, our CyberTape tool can be applied to determine the measurement curves, which helps us give manufacturing parameters before making physical prototypes. The computation statistics of the examples are listed in Table8 . From the statistics, it is not hard to find that the computation of CyberTape can be finished in real time on a dense polyhedral surface using a standard desktop PC.... ..."

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### Table 4. Polyhedral

"... In PAGE 9: ... Table4 . Polyhedral mesh results Figure 10.... ..."

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### Table 3: Failure Points

"... In PAGE 6: ... Hence no path dependence was considered in the analysis. A set of 6 failure points was calculated as shown in Table3 . These points can be used to construct a polyhedral response surface approximation of the actual failure surface as shown in Fig.... ..."

### Table 7 Solid angle approximations on an L-block at selected v

1995

"... In PAGE 30: ...38E;7 The results for a polyhedral surface were much better. Table7 contains results for the L-block(S#3) at the following representative nodes: v 1 =(0;; 0;; 0);; v 7 =(0;; 0;; 1);; v 9 =(0;; 1;; 1) v 17 =(:5;; 0;; 1:5);; v 20 =(:5;; 0;; 1:5);; v 33 =(:5;; 1;; 1) (73) There is no approximation of the surface in this case, and thus all errors are due to the numerical integration being used. The resulting errors are very small.... ..."

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### Table 1. Update of the polyhedral projections a30

"... In PAGE 3: ... Update of the polyhedral projections a30 when a single vertex a59 a65 is moved from positions a59 a65 to a59 a20 a65 . a0 a21 , and then a0 a96 , which is related to the projection update pro- cedure (see Table1 ). When a single vertice (say a59 a65 ) is moved to a new position a59 a20 a65 a16 a59 a65 a98 a34a33 a59 a65 , we simply extract two local polyhe- dra a0 a32a22 a65 and a0 a20 a22 a65 formed by a59a40a65 (respectively a59 a20 a65 ) and its neighbors, and then compute their exact projections.... ..."

### Table 1: IQ results for Trivalent Polyhedral Clusters

"... In PAGE 5: ... The genetic algorithm had to punish the inherent non-planarity of the polyhedron by clustering some vertices together and essentially turning a trivalent polyhedron with 1 octagon, 4 pentagons, 2 quadrilaterals and 2 triangles into a polyhedron with 8 trivalent vertices and two ve-valent vertices having 7 quadrilaterals and two triangles. Table1 . shows what happens to P14;8 for various coordinatization methods.... ..."

### Table 1: Comparison of Logarithmic and Volumetric Barriers Polyhedral Semide nite

2000

"... In PAGE 3: ... The semide nite generalization of the matrix , which in the polyhedral case is the diagonal matrix = Diag( ). Representations of rV (x) and Q(x) in terms of clearly show the relationship with the polyhedral case (see Table1 , at the end of Section 3). Semide nite generalizations of fundamental inequalities between Q(x) and the Hessian of the logarithmic barrier (see Theorems 4.... In PAGE 4: ... We rst obtain Kronecker product representations for the gradient and Hessian of V ( ), which are then used to prove a variety of results generalizing those in Anstreicher (1996, 1997a). Later in the section the matrix is de ned, and alternative repre- sentations of rV (x) and Q(x) in terms of are obtained (see Table1 ). Section 5 considers the proofs of self-concordance for the volumetric and combined barriers.... In PAGE 22: ...Proof: From the representations in Table1 we easily obtain rV (x)T = ? AT vec( + I); Q(x) + H(x) = A[I ( + I)] A: Let = + I. It follows that rV (x)[Q(x) + H(x)]?1rV (x)T = vec( 1=2 )[I 1=2 ] A A[I ] A ?1 AT[I 1=2 ] vec( 1=2 ) vec( 1=2 )T vec( 1=2 ) = tr( ) = n + m: 2 Using the above results we can now prove the second main result of the paper, characterizing the self-concordance of the combined volumetric-logarithmic barrier for S.... ..."

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### Table 1: Comparison of Logarithmic and Volumetric Barriers Polyhedral Semide nite

2000

"... In PAGE 3: ... The semide nite generalization of the matrix , which in the polyhedral case is the diagonal matrix = diag( ). Representations of rV (x) and Q(x) in terms of clearly show the relationship with the polyhedral case (see Table1 , at the end of Section 3). Semide nite generalizations of fundamental inequalities between Q(x) and the Hessian of the logarithmic barrier (see Theorems 3.... In PAGE 4: ... We rst obtain Kronecker product representations for the gradient and Hessian of V ( ), which are then used to prove a variety of results generalizing those in Anstreicher (1996, 1997a). Later in the section the matrix is de ned, and alternative representations of rV (x) and Q(x) in terms of are obtained (see Table1 ). Section 4 considers the self- concordancy of V ( ).... In PAGE 21: ...emma 5.4 Let x have S(x) 0. Then rV (x)[Q(x) + H(x)]?1rV (x)T n + m. Proof: From the representations in Table1 we easily obtain rV (x)T = ? AT vec( + I); Q(x) + H(x) = A[I ( + I)] A: Let = + I. It follows that rV (x)[Q(x) + H(x)]?1rV (x)T = vec( 1=2 )[I 1=2 ] A A[I ] A ?1 AT[I 1=2 ] vec( 1=2 ) vec( 1=2 )T vec( 1=2 ) = tr( ) = n + m: 2... ..."

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### Table 1: Connectivity and geometry of the convolution of two polyhedral tracings.

1996

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