except that the definition of diameter is for a figure or body, not a metric space. So the question you are asking is nonsensical.
That's not actually true. If you want to verify this for yourself, take a look at the wikipedia page on diameter and you'll find it explains there are lots of definitions of limited validity, but there is a more general definition stated as this:
The diameter of a subset of a metric space is the least upper bound of the set of all distances between pairs of points in the subset. So, if A is the subset, the diameter is
sup d( x , y ) | x , y ∈ A ....
We did that, but it does not make any sense if the rings are robust enough to act like invested devices, to which these modifiers do not apply. It makes sense when the spells are temporary and have to be recast regularly or irregularly, which is the case if the ring is fragile.
The questions about geometry and considerations of more than three dimensions or examining different metric spaces may warrant further consideration when considering Regios inside of an AotH. Does the AotH protect that Regio? Never? Sometimes? Does it depend on entrances? Maybe a Regio's extent? Etc.
Actually it makes perfect sense to apply [Warping +1 or Safety -1 per Spell] since they are not a Ritual or Enchanted Device. Both of those require the sacrifice of a permanent resource in the form of Vis.
Additionally this is ignoring the further restriction of non-Ritual spells being unable to improve the lab total. The Magus is already presumed to be casting spells as part of lab work and the effects of this are already included in their Lab Total. Included in the presumed casting is Ring/Circle spells when they are appropriate.
It ends with this "A good guiding principle is that anything involving the use of vis (Ritual spells, magic items), or any inspiration or improvement from an external source or non-magical objects, can grant such Lab Total bonuses. Anything that comes directly from the magus’s own magics in the same season cannot."
[NOTE: Yes there is the possibility on a spell by spell basis that the SG might allow a bonus based on a spell if the spell was good enough. However that combined with the test before and after that section means that it is highly unlikely and completely up to the SG. However most SG/Groups would react harshly to that, seeing it as an abuse (as your group did with the house rule).]
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That is exactly the point. If circle/ring spells are played naively by RAW. they would not come from the magus's own magics in the same season. They would be cast once and last season after season, until the magus decides to break the ring.
You could make a lot of lamps, with an engraved ring in them to hold a CrIg effect of ring duration, and they would last for ever. Effectively magical lighting.
So, yes, I agree with the principle you quote, mainly for the sake of game balance and power moderation, and I try to read the other rules so that it make narrative sense.
First of all, there might be many definitions of "diameter", and I didn't come up with any of them: I was considering the one @loke suggested, and for that one he said "Every set in a metric space has a diameter, no matter the shape or number of dimensions." A set is something broader than a figure or a body, and that's exactly why I tried to come up with an example of a set which is neither a figure nor a body.
And finally, I was just trying to explain what metric space I was talking about. If it feels nonsensical then obviously I failed (because it is not), and for that I am sorry.
Let me try an obnoxiously simple explanation, using wood turning as an example since anyone can easily search Youtube for 100s of videos of it. (Hey, they are relaxing to watch)
The "diameter" is the longest distance of all the possible combinations of two points on an object. If you perfectly centered an object at a 90 degree parallel to that line and spun it (clamp it in a wood turning lathe and turn the lathe on), the point where something slowly moving towards it comes into contact will be at one of those two points every time (the point where the tip of the chisel would make contact while wood turning). If you measure that distance from opposite sides it will always equal the "diameter".
By spinning it at an extremely high speed, when viewed in line with the central point of the spin the object will seem to form a circle with a diameter of that longest possible distance from all the combinations of points possible.
Sorry that I missed that point.
I have rather mixed feelings about vertical circles. On one hand, I do not like to spoil the party by flatly stopping creative use, unless there are serious game balancing reasons to do so. On the other, a vertical ring does not feel in line with legends and stories.
The big question, what is the effect you might want to apply on the encircled doorframe? There are many interpretations,
- A standard circular ward around the door, to block certain entities from passing through the door. In this case, I disagree. The target warded against does can pass through the door without passing through the circle in the sense intended in the spell description. Specifically, passing through the door, you do not move in the plane in which the circle exists, and the canon effect is only described within that plane.
- A ward spell on the door, i.e. the door is the target, not the beings to be kept out. I do not really object to that, but I'd like to see the spell description, It is non-canon, and it is not obvious to me how you would design it.
Game balance wise I think it it fair that the vertical application requires a differently designed spell.
I disagree with your disagreement.
I think that a warded door is perfectly fine by the ward design.
ArM p 114:
They prevent things warded against that are within the circle from leaving, and prevent things warded against that are outside from entering. Warded things cannot act across the circle, no matter which side they are on...
So having a doorway warded against "X" would prevent "X" from coming into the room via the door. Yes warding the entire room avoids the loophole of a magical bear tearing down the wall to get in, but a door sized ward could be covered in one action, whereas the room would (depending on its size) take multiple rounds to cover with the same circle ward.
I think that "within the circle" is really key, attempting to walk through a vertical circle would certainly put one within the circle.
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Here I will have to disagree. For me a wizard going round a house and drawing circles around all the windows is very much in line with how I imagine the process of warding a building against X.
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In all the classic stories I can remember, the wizard draws a symbol on the door, rather than a circle around it.
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Attention: using circle for circular disk or disk (like Merriam-Webster circle c.) is - at least mathematically - deprecated.
So - unless you insist on reading disk for circle - you need to rule strictly, how a circle on the ground blocks access into or out of it from above or below, before making the above claim.
Oh, I was not aware of its ever being common,
To my knowledge it was never in mathematics.
I am not able to decode what you are trying to say here, can I convince you to elaborate?
This would go back to the very first posts in this threat, about how high does the ward extend (and by extension, how deep below). I prefer a spherical-ish shape, with a cylindre as second choice. I find the cone less elegant than the other two.
We need go no further than 1.b "a closed plane". Attempting to cross a warded doorway would put the ward-against-individual in this plane, which is forbidden by the ward.
I doubt that, as circle under 1. b. is defined as
a closed plane (see plane entry 6 sense 2b) curve every point of which is equidistant (see equidistant sense 1) from a fixed point within the curve
Exactly. And there you would rely on your troupe's decision.
I don't see what there is to doubt. The circle is defined as a plane, which is closed. Attempting to cross that warded door would mean entering the warded plane.
In math - and tmk whereever precise terminology is required - a circle is the curve enclosing the disk, not the disk proper.
So a coin is said to have the shape of a disk, not of a circle, while a chalk ring has the shape of a circle.
In common English this may be a matter of style.
In math it is important, whether a circle of radius 1 is all the points with a distance of precisely 1 from a common center point, or all the points with a distance of less or equal 1 from that center. The latter notion is strongly discouraged - technically: deprecated.