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To form a local cycle on this path segment, MA has to return to point H j again Since a point can become a hit point only once (see the proof for Lemma 334), the next time MA returns to point H j it must approach it from the right (see Figure 39), along the obstacle boundary Therefore, after having de ned H j , in order to reach it again, this time from the right, MA must somehow cross the M-line and enter its right semiplane This can take place in one of only two ways: outside or inside the interval (S, T ) Consider both cases 1 The crossing occurs outside the interval (S, T ) This case can correspond only to an in-position con guration (see De nition 332) Theorem 334, therefore, does not apply 2.

The crossing occurs inside the interval (S, T ) We want to prove now that such a crossing of the path with the interval (S, T ) cannot produce local cycles Notice that the crossing cannot occur anywhere within the interval (S, H j ) because otherwise at least a part of the straight-line segment (Lj 1 , H j ) would be included inside the obstacle This is impossible.

The division algorithm discussed in this section is an example of an algorithm that falls into a class of digit recurrence methods. Iterative methods typically refer to successive approximation methods (discussed in the next section), but for consistency with other chapters, we will refer to this as an iterative method as it maps to similar iterative methods used for compact implementations of other functions. Iterative division works much like long division with decimal numbers did in elementary school. One of the advantages in using binary numbers, however, comes from the fact that certain optimizations can be made in the division process. Take the following long division example with the binary representations of 2 divided by 3 as shown in Figure 8.3. The divider for a xed-point number can be architected as shown in Figure 8.4 with a comparator and a subtraction unit. With this architecture, the dividend is normalized to a xed-point value that is necessarily smaller than twice the divisor. By doing this, every subsequent shift operation will produce a new partial quotient that must be less than twice the divisor. This means that the divisor will go into the partial quotient 1 or 0 times. If the divisor is less than or equal to the partial dividend for the current iteration, a logic-1 is shifted into the quotient register, otherwise, a logic-0 is shifted in and the partial dividend is shifted left by 1. After the necessary number of iterations to achieve the desired precision, the output is postnormalized to shift the data to the proper xed-point location (undo the prenormalization).

closed at his profit target of 18600 But I needed to see a chart closer up that would show what had happened today So, against the wishes of old man Wilson, I switched over to a 15-minute chart From my book on technical analysis, I learned that each of the candles I looked at represented the movement of price in a designated period of time On a 15-minute chart, each candle showed how far the price moved in, well, 15 minutes Dark candles meant that price had gone down Light candles meant that price had gone up Easy enough (see Figure 92) Looking at the chart, I realized that the GBP/USD had moved its entire distance for the day 200 pips in the first 15 minutes after the report was released Then it rose back up 100 pips, and then it fell back down 100 pips.

because MA is known to have walked along the whole segment (Lj 1 , H j ). If the crossing occurs within the interval (H j , T ), then at the crossing point MA would de ne the corresponding leave point, Lj , and start moving along the line (S, T ) toward the target T until it de ned the next hit point, H j +1 , or reached the target. Therefore, between points H j and Lj , MA could not have reached into the right semiplane of the M-line (see Figure 3.9). Since the above argument holds for any Q and the corresponding H j , we conclude that in an out-position case MA will never cross the interval (Start, Target) into the right semiplane, which prevents it from producing local cycles. Q.E.D. So far, no constraints on the shape of the obstacles have been imposed. In a special case when all the obstacles in the scene are convex, no in-position con gurations can appear, and the upper bound on the length of paths generated by Bug2 can be improved: Corollary 3.3.4. If all obstacles in the scene are convex, then in the worst case the length of the path produced by algorithm Bug2 is P =D+

(3.10)