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The FAQ tells us there is no the Pythagorean theorem in Three Dimensions movement calculating, so just add the vertical distance to horizontal distance to ascertain the distance between 2 targets.
When goes to Threatened Squares calculating, a creature adjacent to another in horizontal plane but 5ft.-high is still counted as adjacent(5ft. in Three Dimensions) or 10ft.? They can attack each other with melee（not reach）weapon or not？
That's not what that FAQ says.
When a character or creature is flying, and that creature decides to ascend at a 45 degree angle, the rules states that it moves at half speed? Do the rules for diagonal square counting still apply when moving up diagonally in this way?
No. Since the game is generally assumed to be played in two dimensions, even when representing three dimensional combat, the rules for ascending are handled by the speed reduction instead of asking players and GMs to ascertain the diagonal vertical movement.
It's saying that when you halve a character's speed for ascending while flying, you don't also reduce the speed by counting the vertical movement as squares - no double jeopardy.
There is no reason not to treat a target adjacent horizontally and 5' higher as out of reach IMO.
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There is no reason not to treat a target adjacent horizontally and 5' higher as out of reach IMO.
It depends on what we intend with 5' higher.
A medium creature in Pathfinder is a cube with a 5' side. If it is adjacent to a character and 5' higher, its foot, base, ventral part, or whatever is still adjacent to the character higher part. It will have a height advantage, but it will be adjacent for AoO, attacks, and other things.
If the base of the creature is 5' higher than the top of the character it isn't adjacent.
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It is in the flying rules. You move normally, but pay double for each square moved when ascending, while you gain movement when descending.
It is very approximative.
Note that you aren't necessarily ascending all the time. As we work in 5' cubes, you ascend only when you move in a cube that is higher than the previous one.
So, if move up to 45° degree upward, you pay the linear movement and double the cost of the squares where you move upward.
As an example, you want to move 30' forward and rise by 10':
you pay 5' of movement, then 5', then 10' (and rise by 5'), then 5', 5', and 10' (and rise by another 5').
If the angle is 45° or higher you are ascending in every square, possibly more than once for square. At that point, you pay the ascending movement and add the cost of the lateral movement when you do it.
On the map, you aren't moving in a linear way but end moving along a staircase. It is an approximation.
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As you said, the three dimensional distance is horizontal + vertical. So 5ft higher and 5ft in horizon is 10ft away, (if calculating the range for range attack) but they can attack each other with melee weapon. Am I get you right?
And what about 10ft-reach weapon? Can I attack a creature which is 10ft away in horizon and 5ft higher (or 5ft in horizon and 10ft higher)?
If the answer is Yes, is that means I should always minus 5ft. when calculating the melee attack Threatened Squares?
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If you look at your character from the side, the flying creature is in the corner of your square. If it was on the terrain map you would be able to attack it. Why changing it to a height map would change that?
To me, it seems that you are confusing the movement cost to reach a position with the distance to the target.
Plain map, without flying. You are in the lowest square on a stair, your opponent is in the adjacent, higher square. Do you think that it will be outside your threatened area?
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Let's just go with RAW and logic.
CRB 180：Threatened Squares: You threaten all squares into which
you can make a melee attack, even when it is not your turn. Generally, that means everything in all squares adjacent to your space (including diagonally).
Now, the question goes to the bold "diagonally" apply to three dimension or not. I believe most people will say Yes, while I still wanna make some digging within the area of RAW.
XXO
XOX
XXX
Why is that any different from
XXX
XOO
XXX
You have 5ft reach. Up, down, left, right, up and to the left a little, up and to the right a little, it doesn't make an appreciable difference.
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I got another rule.
Aquatic Adventures 44(Aquatic Rules, Thinking in Three Dimensions):
Flanking: All the usual rules for flanking still apply in three dimensions, but now there are far more ways for a target to potentially be flanked. It also means a larger number of weaker creatures can gang up on a stronger creature, since there are 26 valid spaces adjacent to a given square, rather than 8.
"26 valid spaces" under Flanking implies all the 26 squares can threaten the creature. Let's assume the designer means 5ft- melee attack, So It is proven that the diagonally threatened squares rules can be apply in three dimensions.