Friction and least-cost paths
Euclidean distance in a GIS is the measure of a straight-line distance from pt A to B and is pretty simple.
But a grid-path distance is a different matter altogether. Movement occurs from cell to cell. Following a raindrop down a slope and into a river, you’re not calculating distance as the crow flies, but in increments of one grid cell at a time, either up or down or diagonally.
Here we can see that the use of the topography as an indication of flow direction allows us to not only follow the path down slope, but we can calculate the grid-path distance as well (see the “trace downstream” tool on the Analysis Toolbar).
Grid-path distance on a non-topographic surface
Note; starting with 2.8, the “distance” tools used below are legacy tools. In the current version, Distance Accumulation and Distance Allocation are used to find ‘least cost paths’ and noting that one is not limited a “grid path” distance, but ‘as a polyline” is also allowed. We will explore those after we look at grid paths first. Curious? Have a look at this blog. which still uses the old names for these tools “Euclidean this and that.” This help page for ArcGIS Pro contains some great information to help you understand cost surfaces.
Suppose that movement can occur across a surface, but it occurs in response to other elements besides distance alone. Movement will be greatest where resistance or “friction” is least. This is a good description for movement of groundwater, but might also be used for disease, where resistance is inversely related to health or population density, or for animal migration where friction is a combination of factors including food, predators, climate, and topography. Movement is in inverse proportion to the “cost” of moving in any direction, and we can think of the most likely pattern of movement as being the “least-cost path.” Resistance is used to define a “cost surface” where movement is perpendicular to contours of cost, like water moving downhill perpendicular to contours.
The “friction” ArcGIS Project illustrates this process of creating least cost paths.
Open the. . . \demo\friction\friction project
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- Define the cost of moving across the surface on a per cell basis. This is known as a “friction” layer or theme.
Here the “barrier” is 10 times more costly to movement (10 times the friction) than the background. - Establish the cost surface by calculating the accumulated cost of moving from the midpoint of one cell to the midpoint of the adjacent or diagonal cell
and in that way produce a “cost surface” from any ending point (or multiple ones). In ArcGIS use the Spatial Analyst/Distance/Cost Distance tool 
- Associated with this Cost Surface is a Direction Grid (also known as a “backlink” grid) showing the direction movement occurs on the cost surface.
- Now a least cost path can be made from any point (or multiple points) on the surface to the drain point on the cost surface.use the “shortest Path” from Spatial Analyst\distance menus
- Define the cost of moving across the surface on a per cell basis. This is known as a “friction” layer or theme.
Where would you go on a hike in our topography?
If you used slope as “friction” (or squared, or cubed it), where would your “least-cost path” be from one point on a map to the next through this topography? What if you are finding a path for an animal that avoided south-facing slopes?
from the friction project, topo friction map.
Lastly, how can we use this “friction” idea in a different way. (Open the demo/least_cost_buffer project.)
Go back to the discussion of buffers. Can we make a rubber ruler in a different way?
Imagine: How far could an exotic species migrate from roads (originally carried under bumpers)?. Suppose it is a function of landuse. It migrates five times as far or fast in the woods as in a pasture, and twice as far in a pasture as in developed land. We can reclass the land use grid to get this friction map for migration
20=urban & other
5=ag
1=woods
Next, executing the tool to generate the cost surface to the roads (or “from” them) using the friction above gives us
Notice how the increasing cost (which is a “distance” away) from the road now varies depending on whether the friction is high, or low moving out from the road.
This is another example of a dynamically-scaled buffer (aka, a “rubber ruler”).