imwaesel the strawman arguments does not validate your complete broken system, in opposite it lets you look like a fool, before an audience with knowledge.
You have no knowledge at all about algorithmic and geometry. You can repeat yourself as often as you want, but nonsense stays always nonsense, you cannot calculate the necessary APs to reach a field with a circle. A wall that cuts through the entire circle will invalidates it, but the circle will say that you can reach it. An obstacle like a wall going through the half of the circle will invalidate your calculations, because the reachable fields will not be a circle anymore. Nearly any obstacle within the circle will invalidates you. In your first example you have added 8 additional fields for the following cost: 1 + square root of 2. And that is not 2 but rounded down a 2.41. Therefore it was false, in opposition to toro's example where the cost always were 2.
You are thinking that you are smarter the people (like Dijkstra) with 2x(Your IQ), that have studied for years that kind of problems and have developed algorithms to solve them.
You are making a clown out of yourself.
The accessibility should be determined by a greedy algorithm with no directional preferences.
Also to add is that i have even mentioned that there are reasons for using the square grid, some features can be only applied to it.
HiddenX
Did you knew A* before i mentioned it? If not, then i am glad to have at least not wasted my time. To the example, they are not using the diagonal movement, the resulting diagonal line is the a middle value of the generated possibilities.
Between 0:46 and 0:47 you can see this, if they would use the diagonal movement, then the line should go through the edge connections of the wall. But it displays good the overall working of a A*.
By the way a mad hatter or the hatter or a simple victorian figure?
Or
http://www.youtube.com/watch?v=5JoxISdBzmY :D
tuluse and HiddenX
For the part of the free movement (which is what i would really prefer) there are some solutions but the most common is a gradient solution or better said a potential field.
The problem of a potential field are only the local minima (This looks funny in games, because you have many enemies in specific field that is blocked by an obstacle and they are circling around in that field.).
If you have interest you should look at some videos on youtube.
But there is also the way to make the map into a grid, in which for a accessibility, the fields with obstacles would be marked as not accessible, and the possible movement will be determaind by a greedy algorithm.
FeelTheRads
He does not feel the impacts, lacks the necessary cognitive capability.