How travel routes works?

Recently I was curious about how cells on the world map affect travel speed and more importantly how to figure out optimal route to build roads on the world map. Look at my second hero to see good illustration :)

At this point I made only obvious conclusions - speed bonus of the cells is the most important factor and traders seems to prefer straight routes.

But what is interesting is the questions - does direction of a movement (vertical, horizontal, diagonal) affect travel time?

And does it matter somehow if route is more straight if everything else is the same?

Example for the last question:

Are this two routes exactly the same in terms of travel time?

CAREFUL THERE IS MATH =)

The fastest route is the one with less costly tiles.

Roughtly for land/naval units (x - multiplier):
Multiplier of movement cost:
- Direct(aligned cell) : x1
- Corner(diagonal cell) : x1.4142..  (square root of 2)
Modifiers from tile types:
- basic : x1
- deep water: x0.66 (1.5 times faster 150% of speed)
- hills/forest: x1.33 (1.25 times slower 75% of speed)
- Lake/river: x2 (2 times slower i.e. 50% of speed)
- mountains: x4 (4 times slower i.e. 25% of speed)
Additional terrain zones crossing:
- sea-land border crossing: x3
- river-land border crossing: x1.5
Air units only affected by diagonal/corner multipliers

In yours examples:
1) Town->Direct on hills->Corner land->Direct land..
2) Town->Corner land ->Direct land->Direct land..
Lets assume bace land cost = 100
Then travel costs(with roads):
- Town->Direct on hills = 100 x 1(direct) x 1.33 (on hills) x 0.5 (roads, depends from technology) = ~66
- Direct on hills->Corner land = 100 x 1.4142(corner) x 1 (on land) x 0.5 (roads) = ~70
- Direct land->Direct land = 100 x 1 x 1 x 0.5 = 50
- Town->Corner land = 100 x 1.4142 x 1 x 0.5 = ~70
- Corner land->Direct land = 100 x 1 x 1 x 0.5 = 50
So, rough calculations for or two cases:
- First one (Town->Direct on hills->Corner land->Direct land): 66 + 70 + 50...
- Second one (Town->Corner land ->Direct land->Direct land): 70 + 50 + 50...

UPD:
Above info was about moving on hostile tile (but without extra penalty due to awareness zone). "Tiles modifiers" do not apply on traders if there is road on them. And they move on hills with same speed as on plain - with basic road speed, without extra tile penalty

Wow, this is the best answer possible :)

It's surprising but also confusing that effect from tile types is actually combined with road because in the game it displayed like every cell with simple road have 150% move speed regardless of natural tile type.

I guess game should tell that there is two numbers in the equation.

Recently I saw this: https://i.imgur.com/X2bpMCt.jpg

It really seem like the path could be way shorter. Looking at my screenshots I also saw this alpha 8: https://i.imgur.com/o6EFG0w.jpg

You can see the artiste purposely avoiding the bridge.

We will take a closer look on issue with bridge. But attacking army has normal route, according to mechanics of how slowdown is working in "awereness" tiles. Also attacking kingdom is locked on shore so it have to reenter twice into awareness zone in order to reach its target.

I forgot about awareness indeed, but would the army take the same way to come back after the attack ? If so, it should probably take the shorter way instead as there is no slowdown when returning (or by manually recalling).

Bridge was also in awareness zone, so army was trying to avoid it..
https://ibb.co/isRH5V  There in yellow is shown where awareness zone should be with radius of 7 (after researching Optics)

Thanks, though, on a town with only 1 cell I would have expected awareness to be like this instead https://i.imgur.com/PDuS2dB.jpg

so, does it in fact count in diagonal too as 1,41 cell, rounding up ?

my question would be, why army chooses to go over several cells of deadwood which has travel penalties, whereas the spot at the railroad end could be reached with the same amount of travelled cells, but instead on railroads, built on plains.

https://imgur.com/a/8fnkesB

2nd screenshot is taken immidiately once snails kick in, so im assuming that's when awareness zone is reached, which is the yellow line there (give or take a zigzag somewhere).

Even if the route chosen by army helps to avoid one awareness cell, does it really compensate going over several cells of terrain with penalties, opposed to going over plains with railroad for that part of the trip (while total amount of cells to travel is equal)?

Thanks, though, on a town with only 1 cell I would have expected awareness to be like this instead https://i.imgur.com/PDuS2dB.jpg

so, does it in fact count in diagonal too as 1,41 cell, rounding up ?

Due to armies can move by diagonal - in your case they would be able to reach town in 3 steps with "7 tile" awareness radius. It would definitely be wrong.
Not so purely as diagonal with 1.41 and rounding up, but quite similar.

...

Even if the route chosen by army helps to avoid one awareness cell, does it really compensate going over several cells of terrain with penalties, opposed to going over plains with railroad for that part of the trip (while total amount of cells to travel is equal)?

Diagonal movement has higher costs compared to direct ones. So I'm assuming that summary result would show lower costs by traveling directly using forest tile. Compared to several diagonal movement on plains but with slowdown penalties. Also improvement tiles bonuses don't work inside awareness/enemy zone