The Path VectorA path {$P$} traverses the width, length and height a certain number of times. This one, for example,  traverses the {$a$}-direction three times, the {$b$}-direction twice, and the {$c$}-direction once. We capture this information with a "path vector", {$\vec P=(p_a,p_b,p_c)$}, in this case, {$(3,2,1)$}. From this we can compute the length of the path: {$$L=\frac12(p_a\cdot a+p_b\cdot b+p_c\cdot c)=\frac12\vec P\cdot(a,b,c).$$} The `{$\frac12$}' is because every diagonal move is traversing two dimensions at the same time. |