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Next let's consider variation of gradient along trajectory. Thanks for the smoothness ... Derivative DV ... is a function from B to {+1, -1} and its value on t[i] is defined as follows. That is, if the gradient V(t[i]) of the curremt tile t[i] is equal to the gradient V(t[i-1]) of the previous tile t[i-1], then the value of DV on t[i] is equal to the value of DV on t[i-1]. Otherwise, the value is multiplied by -1. In words, ... Shown below is an example. The left figure shows the previous tile t[k-1] (colored blue) and the right figure shows the current tile t[k] (also colored blue). Because of the smoothness condition, the current tile t[k] could assume one of these two blue slant tiles. Suppose that the value of DV on t[k-1] is -1. Then, in the left case, the gradient of the "blue" current slant tile is equal to the gradient of the "white" previous slant tile. And the value of DV on t[k] is -1. On the other hand, in the right case, the gradient of the "blue" slant tile is not equal to the gradient of the "white" slant tile. Thus, the value of DV is changed from -1 to +1.