Rain in the real world and GURPS MAGIC

[whswhs]

Originally Posted by maximara

Bill Stoddard seems to be saying the circle is inside the hexagon ie A= 2 * √(3) * R^2

I don't see how you reach that conclusion. A formula for the area of any circle must include π. The formula I gave does not include π. It implicitly includes the square root of 3, which would not be in a formula for circles, but which emerges naturally from the analysis of a regular hexagon as partitioned into twelve 30° right triangles.

The "radius" of a hexagon in this analysis is one-half the distance from the hexagon's central point to the central point of an adjacent hexagon. That seems the most natural extension of the concept.

[whswhs]

Originally Posted by StevenH

So, spells use circles as radius of effect, but they are represented on the hex-based battle mat as a hex surrounded by other hexes, as appropriate. In the case of area effect spells, Pi*R^2 (times the appropriate thickness, which is usually 2 yards, but that makes no sense in a weather context). Makes the math easier.

I don't think that π is any easier to use than the square root of three.

[maximara]

Originally Posted by whswhs

I don't see how you reach that conclusion. A formula for the area of any circle must include π. The formula I gave does not include π. It implicitly includes the square root of 3, which would not be in a formula for circles, but which emerges naturally from the analysis of a regular hexagon as partitioned into twelve 30° right triangles.

The "radius" of a hexagon in this analysis is one-half the distance from the hexagon's central point to the central point of an adjacent hexagon. That seems the most natural extension of the concept.

I don't see that being any different from the A= 2 * √(3) * R^2 I gave as there is no π in that equation either and it uses √(3) as well.

It all boils down to how spell area within a hex is handled. If it exactly covers a hex ("because its magic") then you have these things that look like somebody used hexagons instead of squares in Minecraft.

But if the area actually is a circle then it either doesn't cover the entire area or there is spill over.

The easiest way to see this is to use nothing more than a hex grid, a straight edge, and a compass and do the following:

1. Take the straight edge and find the center - either with the compass only method compass only method or by breaking it down to its 6 triangles.
2. Take the compass and using the distance from the center to a side and draw a circle.
3. Take the compass and using the distance from the center to a vertex draw a circle.

The distance between the two circles is the area in question.