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Text S1
Model (1b): Cost Minimization
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familiar route at the start of search. This is in contrast to the earlier analysis which weighted
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the cost uniformly over the linear spatial extent of the familiar route.
In this section, the cost of searching is weighted uniformly over the angular extent of the
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For convenience, the distance r between the start of search,   X , Y  , and the position of a
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familiar location  x, 0  is defined  : arctan  Y , xL   X     arctan  Y ,  X  as
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r  ;  X , Y  
 Y
sin 
(S1)
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where arctan(y,x) is the 4-quadrant arctangent function. Note that these conventions are
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suitable for Y  0 (Figure S2). It is clear from the construction of Fig S2 that the angular
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extent of the familiar route from   X , Y  is maximum when  X  xL 2 , because
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2Y
(S2)
xL
This property itself may be a potential advantage for beginning a search at B as compared to
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B ' since the familiar route occupies a wider azimuthal space at B . However, without
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specifying a search strategy a priori, expected cost minimization is not guaranteed by
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maximizing total angular extent.
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OB ' A '  OBA '  tan 1
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The expected cost of search with respect to angular extent of the familiar route is found as
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follows. Since OBA  O ' B ' A ' (congruent triangles), OBA  O ' B ' A ' , whose angular
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magnitude is denoted 0 in Fig S2. Since the distribution of distances between OA and B is
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exactly the same as between O ' A ' and B ' , the expected cost of search over these two
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angular extents are the same, both denoted as g 0 .
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As before, the cost of search for any point along the familiar route is assumed to be a
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monotonically increasing function of its distance from the start of search. Now, all points
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along OO ' are further from B ' than any point along AA ' to B . Therefore, the expected cost
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of searching for any set of points along OO ' from B ' , denoted g OO ' , must be greater than the
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expected cost of searching for any set of points along AA ' from B , denoted g AA ' , i.e.,
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gOO '  g AA ' .
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In order to compare the expected cost of search with respect to angular space, it is necessary
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to restrict the angular extent at B which is used for the comparison, since    ' . In other
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words, because OBA '  OB ' A ' (or equivalently    ' ), a raw comparison of the average
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cost of search is unfair since the target (familiar route) subtends a smaller angle at the start of
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search, if searching begins at B ' rather than at B . To account for this difference, we consider
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only part of OBA ' whose angular extent is equivalent to OB ' A ' . The corresponding
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expected cost of search for the fraction of points along AA ' which subtends angle  ' at B is
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denoted g AA' ' . It is then straightforward to show that g AA' '  g AA '  gOO ' . With angular
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adjustment, the expected cost of search from B is thus
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g
B

0 g0   ' g AA' '
0   '
(S3)

 0 g 0   ' gOO '
0   '
(S4)
and
g
B'
Thus with adjustment for angular extent at the start of search,
g
B'
 g
B
(S5)
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Hence the expected cost of search with respect to the angular extent of the familiar route is
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minimized when  X  xL 2 (or equivalently when x  0 ). This result is consistent with the
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analysis of search cost with respect to the linear spatial extent of the familiar route.
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