I recently read an article that gave a table converting the number of contours per cm on the 1:25000 maps with the actual angle of the slope. As I think the steepest slopes I climb as a walker are 45 degrees, I was interested. The first issue is how the slope is being measured. I offer two possible ways below. When the steepness of a road could be given as 1 in 4, I understood that it meant 1 up for every 4 along the slope of the road. That would be a).
But what about maps. I think maps have to be b) because the distance along the slope is further than that along the adjacent on the triangle. It would mean that when the landscape was projected onto a flat map, points in hilly areas would end up further apart. Here's a flat map representation below.
If I try adding relief to make it like it really is, the sides of the map end up 10cm closer.
So to preserve true horizontal displacements when projecting a landscape onto a flat plane, it must be method b). Note that for small angles, either calculation will give the same answer to 2sf.
