### Table 2: Time Complexity for Circular Blending Quaternion Curve.

1995

"... In PAGE 19: ... Table 1 shows a measure for the computation time of the implemented algorithm; the time complexity is given in terms of the number of arithmetic operations and mathematical function calls required for the generation of a single circular quaternion curve approximated with m unit quaternions. Table2 shows a similar measure for the generation of a circular blending quaternion curve which blends two circular curves (with a linear blending function); each circular curve is approximated with m unit quaternions. We have implemented four di erent methods which can interpolate a given sequence of keyframe orientations of a 3D solid: the B ezier method of Shoemake [19], the squad method of Shoemake [20], the circular blending method of this paper, and the cardinal spline method of Pletinckx [17].... ..."

Cited by 18

### Table 2: Time Complexity for Circular Blending Quaternion Curve.

1995

"... In PAGE 19: ... Table 1 shows a measure for the computation time of the implemented algorithm; the time complexity is given in terms of the number of arithmetic operations and mathematical function calls required for the generation of a single circular quaternion curve approximated with m unit quaternions. Table2 shows a similar measure for the generation of a circular blending quaternion curve which blends two circular curves (with a linear blending function); each circular curve is approximated with m unit quaternions. We have implemented four di erent methods which can interpolate a given sequence of keyframe orientations of a 3D solid: the B ezier method of Shoemake [19], the squad method of Shoemake [20], the circular blending method of this paper, and the cardinal spline method of Pletinckx [17].... ..."

Cited by 18

### Table 7.1: The generalization and size of constructed high order perceptrons.

### Table 1: Manually selected curves with high performance in descending order.

2007

"... In PAGE 10: ...n the range a=[0,359]. The selected indices are shown in Table 1. In this experiment we compare the retrieval performance using an increasing number of manually selected profiles and contours. At first we selected the optimal C-contour and determined its retrieval performance, then we added the samples from the second best C-contour from Table1 and tested their effectiveness, and so on until all six contours were used. This experiment was repeated for the XY-contours and Z-contours and for the profiles as well, obtaining the results in Figure 10.... ..."

### Table 3: Compactness 3-D digital curve representations. A: Number of elements per unit length. B: Number of bits per unit length.

1997

"... In PAGE 17: ... The four 3-D digital curve representation schemes were compared according to the four distance measures, and the results are summarized in Table 2. (h) In order to compare the compactness of the four 3-D digital curve representations, the average number of chain elements per unit length of random straight lines is presented in the upper row of Table3 . The values for CQ and TCQ were taken from [27].... In PAGE 17: ... The values for CQ and TCQ were taken from [27]. The average number of bits per unit length is shown in the bottom row of Table3 . It has been obtained by multiplying the average number of chain elements per unit length by log2 6 for CQ and by log2 26 for the other three schemes.... ..."

Cited by 6

### Table 2: Generalized CNF formulas for simple gates

1998

"... In PAGE 8: ... Consequently, the CNF formulas for simple gates given in [22] can be generalized by following the same approach used for deriving (7). These generalized CNF formulas are given in Table2 . As a result, and as was done in Section 2.... ..."

Cited by 8

### Table 2: Generalized CNF formulas for simple gates

1998

"... In PAGE 10: ... Consequently, the CNF formulas for simple gates given in [22] can be generalized by following the same approach used for deriving (7). These generalized CNF formulas are given in Table2 . As a result, and as was done in Section 2.... ..."

### Table 2: Codewords in a rate-9 4 space-time code for 8PSK. The 9 input bits choose one of 32 unit quaternions q and one of 16 unit quaternions p

in Dedication

### Table 1. Ambiguity in reconstruction of symmetric curves: A single view is not enough for re- construction of general symmetric curves, except for case (a) of Figure 2, but sufficient for planar symmetric curves. Two or more views are needed for reconstruction of generally shaped sym- metric curves.

2004

"... In PAGE 8: ... For curves with general shapes, the solution is always unique. To conclude Section 3, we summarize all cases of symmetric curves studied so far in Table1 , in terms of ambiguities in reconstruction from one or two views. 8 We have also tried other distances such as L1-distance and C1-distance.... ..."

Cited by 1

### Table 3: Quaternion Dictionary

"... In PAGE 13: ... gt;From Eqn. (18) and the characterization of diagonal matrices given in Table3 , we see that this would be achieved if the pure quaternion q were rotated into a multiple of i, and the... In PAGE 15: ... (20) and (21), consider the e ect of a similarity by the symplectic orthogonal matrix b R = (x y) on H: b RH b Rt = (x y)( 1u1 v1 + 2u2 v2)(x y) = 1(xu1x yv1y) + 2(xu2x yv2y) = 1(i i) 2( b u2 k) where xu2x = b u2 2 spanfj; kg. Checking Table3 , one quickly observes that b RH b Rt is already 2 2 block-diagonal. Furthermore, since the singular vectors u1; v1 associated with the largest singular value are sent to i, we can use [40, Prop.... In PAGE 16: ... 5.3 Skew-symmetric Hamiltonian If the Hamiltonian matrix H 2 IR4 4 is skew-symmetric, then we can write H = b (1 j) + p 1, where b 2 IR and p 2 IP (see Table3 ). Since (1 j) = J4, it is immediately obvious that similarity by a symplectic orthogonal has no e ect on the rst term of H.... In PAGE 36: ... Unfortunately, there are di culties with this plan; we brie y discuss them here. If W is a 4 4 skew-Hamiltonian matrix, then we can write (see Table3 in x4.5) W = b (1 1) + p j + 1 q ; where b 2 IR; p 2 IP and q = c i+dk 2 IP.... ..."