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the M-line, the arm endpoint can continue through points b9 , b10 , b11 , meeting the M-line again at point b12 as under option 2. In fact, when starting at point b2 under any of the two options, if one continues rotating the arm around obstacle A while keeping in contact with it, the arm endpoint will make a complete closed curve, passing through the points b2 , b3 , b4 . . . , b8 , b9 . . . , b13 , b14 and eventually arriving at the same point b2 . This indicates that the paths produced under both options are complementary to each other: When added together, they form a closed curve. Regarding this curve, consider the area whose curvilinear boundary passes through points b , b2 , b3 , b4 , then the segment b4 , b10 of the workspace boundary, then points b10 , b9 , b of our curve, and nally the smaller part of the obstacle A boundary between points b and b . This area is called the shadow of obstacle A: Though this is a part of free space, no point (such as P1 ) inside this area can be reached by the arm endpoint. This suggests that an obstacle shadow will be perceived by the arm as an obstacle, as real as an actual physical obstacle. The arm cannot penetrate either

1

always @(posedge Clk) for(i=0;i<32;i=i+1) Out[i] = Y[i*2] ^ X[i];

Figure 5.4 An example of interaction between obstacles. The shadow (shaded area) behind obstacle A is the result of interaction between obstacles A and B. If the arm moves through the positions (a1 , b1 ), (a2 , b2 ), . . . , (a17 , b17 ), at any moment it is in contact with either obstacle A or B. This means that the arm will perceive these two obstacles as one obstacle. Because of obstacle C, link l1 cannot realize any angle values 1 in the range 1 < 1 < 1 .

Because they had tied up their capital in the losing position, they couldn t take any more trades So they were locked into a bad spot What did they do They started to hedge the position What does that mean Using their relationships with banks, they actually traded the other side of the position They sold Pounds Right About one hundred points lower than they had bought it, they started to sell it Meaning, with separate liquidity providers, they opened the opposite position This means that they were now locked into a onehundred-point loss What good would that do First, it prevented them from having to be wrong Closing the trade completely would mean they had to admit they were wrong And they didn t want, above all else, to close the position and then see the trade go in their favor.

of them. The shape of a shadow depends on the shape, size, and position in W -space of the corresponding actual obstacle that creates the shadow, as well as on the arm links shapes and dimensions. An obstacle can form disconnected shadows, as in the case of obstacle B (Figure 5.3). Or, obstacles can interact in forming shadows; this happens, for example, when two or more points of the arm body touch two or more actual obstacles simultaneously, as at position (a8 , b8 ) in Figure 5.4. De nition 5.2.1. A virtual obstacle X is an area (or areas) in W -space, no points of which can be reached by the arm endpoint because of the arm s possible interference with the actual obstacle X. Thus a virtual obstacle consists of the corresponding actual obstacles and their shadows. In W -space a virtual obstacle forms one or more compact areas (see Figure 5.4). Whereas topologically this combination presents little of interest in W -space, we will see below that it possesses interesting properties in the arm s C-space.

As can be seen from the above example, there are no feedback mechanisms in the loop. Rather, the for-loop is used to condense similar operations.

De nition 522 Passing around an obstacle presents a continuous motion of the arm, during which the arm is constantly in contact with the corresponding physical obstacle(s) It is clear from Figure 54 that two or more actual obstacles may be interpreted by the arm as a single virtual obstacle In Figure 54, at any position from the set (a1 , b1 ), (a2 , b2 ), , (a17 , b17 ) the arm is in contact with at least one of the actual obstacles A and B Hence the two obstacles will be interpreted as one De nition 523 A virtual line is a curve in W -space that the arm endpoint follows when passing around an obstacle The virtual line forms the boundary of a virtual obstacle in W -space A virtual line is not necessarily a smooth curve.

Think about it! They were afraid of losing money that they hadn t earned! They were afraid of missing out on a move that hadn t happened and had become less and less likely By the time they hedged the losing position with another set of trades, they had way overextended themselves on their capital, they had removed the chance of opening new trades, and they were locked into a sure loss This sounds so unlike George Unlike the George you met today But totally like the George you would have met back then I scratched my head When did you come in Not yet By this time, I knew about George and his positions I was helping the trader who took the other side of his hedged positions.