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Inverse Square Law Issues

Agnaa

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The way we go about Inverse Square Law feats has a few flaws. While this mathematical principle works well for long distances and tiny surface areas, it starts to show its inaccuracies when those limits are pushed in the other direction.

Flaw 1: Road to Infinity
We essentially go about these feats by figuring out how much energy each square unit of the surface area of the sphere has, and then multiply that by the target's cross-section. But practically that amounts to wrapping a character around the sphere. When this is done for explosions a small distance away, or for particularly large characters, the joule values accelerate exponentially. For human-sized characters, this starts at 0.23262 meters away from the explosion.

Flaw 2: Slightly Off Radii
When doing ISL calcs we tend to actually represent the explosions like this. As you can tell, this underestimates the radius, inflating the values ever so slightly. While this doesn't matter much for a large radius, it matters more and more the smaller the radius is.

Still, it's not a huge deal, but as I'll show later on, it's something that can be accounted for.

Flaw 3: Chord You See the Problem?
Even if we were to use the correct radii, as per this diagram you may notice that something's off. Because the person's standing upright instead of curved in the manner of the circle, they've actually blocked more of the circle than we give them credit for.

If it seems a bit confusing and hard to tell why they've blocked more, I think I've got a slightly more intuitive way to think about it:

A straight line is the shortest distance between two points, i.e. that red line. However, what was really blocked was the red line, and all the black line of the circle behind it. As it's curved (not a straight line), that part must be longer than the straight line. But since our calcs only use the red line, we're underestimating the result.

Solution: Circle Geometry
Now, I'm not a calc group member myself, but I barely pieced together a 2-D analog of how this could be fixed and done better. Someone better at math than me would be able to come along and finish it.

Essentially, we'll treat the surface area of the being tanking the explosion as a chord, and we'll calculate the length of the arc for the amount blocked.

Practically, this means that we'll need to use some more equations to figure out the CA part of E=I*CA, but the I part will remain the same.

I think that if this is transformed to 3-D it will solve all the issues I've outlined.

Meta: Does It Even Matter?
But at the end of the day, flaws two and three barely change the results. And flaw one can easily be accounted for by simply saying "If the yield is larger than the original yield, use the original yield instead". With the forum move coming up it is probably also unwise to implement a much more difficult variation of a calc and go around nitpicking old calcs to fix it up.

But, an inaccuracy is an inaccuracy. I present the information and it's up to calc group to resolve it; however best they see fit.
 
As you said, flaw one can easily be accounted for by simply saying "If the yield is larger than the original yield, use the original yield instead".

For the other flaws, your suggestion is that the surface area would be chord times length of arc?
 
I'm saying that surface area would be length of arc, and the current surface area would be the chord. We would derive the length of the arc from the length of the chord and the radius as shown on the wikipedia page (except in 3 dimensions).
 
Can you demonstrate, with an existing calc, how much of an effect this makes?

If it's only like, a 50000 joule change on a 9-A Explosion, I'd just say "neat" and add it to the pile of things we do not quite right but are good enough
 
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