Figure 1: A sample isotropic cost grid (left) and the corresponding ACS (right) for movement from cell A(2) to cell E(2), based on Queen's moves. The cells forming the LCP from A(2) to E(2) are marked by a grey background. Examples for cost calculations: The costs of the horizontal movement A(2) to B(2) are (1+5)/2; the costs of the diagonal movement from A(2) to B(1) amount to √2*((1+2)/2). The costs of movement from A(2) via B(1) to C(1) is the sum √2*((1+2)/2) + (2+2)/2. Image credit: I. Herzog.
Figure 2: A 7 x 7 neighbourhood of a cell illustrates the raster to graph conversion. The arrows mark the links between the cell in the centre and its neighbours. The red arrows indicate links via Queen's moves, the pale orange arrows represent Knight's moves, and for the blue arrows the terms A- and B-moves are used throughout this article. Red dots indicate cells that can be reached by straight-line connections with Queen's moves, pale orange dots are targets of Knight's moves, and the blue dots mark cells that are targets of A- or B-moves. Image credit: I. Herzog.
Figure 3: A 13 x 13 neighbourhood of a cell with neighbours marked as in Figure 2. If only Queen's moves are supported, some elongation error is introduced when the true optimal path includes cells that are not marked by red dots. Tiny grey dots indicate cells that cannot be reached by a straight-line connection even if Queen's, Knight's, as well as A- and B-moves are implemented. Image credit: I. Herzog.
Figure 4: To avoid decreasing the grid resolution by long moves, the long moves should be subdivided into several submoves: the Knight's move (pale orange) consists of two steps, in an isotropic grid the cost of the intermediate stop is the average of the costs assigned to the two neighbours connected by the black line. A- and B-moves are subdivided into three steps and the cost of an intermediate stop is a weighted average of the costs of the relevant two neighbours, with the weights depending on the distance between the neighbour and the intermediate point. In an elevation grid, intermediate altitudes are derived from the values in the neighbouring cells. Image credit: I. Herzog.
Figure 5: Roof-shaped test landscapes and LCPs calculated on the basis of the straightforward effective slope method with the A- and B-move option. LCPs connecting points 1 and 2, 2 and 3 as well as 1 and 3 are shown. Point 2 is located on the top of the roof, 1 and 3 are on the gutters. The LCPs resulting from the cost function |s|+1 with s percent slope are identical on both a gradient of 15% and 45% (a and b). The quadratic cost function with a critical slope of 12% results in fairly straight-line connections on a slope of 10% (c), whereas with a slope of 15% (d) the LCP includes bends. The example is constructed in such a way that even with A- and B- moves, the LCPs on gradients below the critical slope do not form perfectly straight lines. Image credit: I. Herzog.
Figure 6: LCP results for Tobler's hiking function. (a) and (c) are LCPs generated on roof-shaped test landscapes with a slope of 25%, (b) and (d) are LCPs on a slightly steeper test landscape (30%). Image credit: I. Herzog.
Figure 7: LCP results for Langmuir's cost function. (a) and (c) display LCPs generated on roof-shaped test landscapes with a slope of 20%, (b) and (d) show LCPs on a slightly steeper test landscape (25%). Image credit: I. Herzog.