The concept of a threat radius (or threat range) is already well established in 40k. At its simplest, it is merely looking at the range characteristic of a weapon, and estimating how many targets it could choose to fire at that turn. To be more involved, a player might mentally add his unit's movement to their range, and visualise the area in which they could potentially destroy their foe in a turn or two.
Where the concept is less well established is in the evaluation of different weapon choices in list building. Often, a unit will have several different weapon loadouts to pick from. It is relatively easy to quantify the interaction of S, AP, and number of shots. Weapon type is a more qualitative query, but nonetheless often easily answered by army philosophy. Range can be a tricky factor, sometimes falling between the two. It is a number, after all - but how can we quantify something as intangible as a threat radius?
Firstly, it is worth pointing out that a range is a straight line, but that a unit's shooting is conducted in a two dimensional area. Thus, the area that a unit can choose to fire in increases as the square of the range. If the game was played in three dimensions, then it would increase as the cube of the range, but I digress. Anyway, the point is that a longer range increases a unit's options exponentially, not linearly.
So, on a gaming table devoid of scenery and stretching infinitely in every direction, a gun with a 48" range has double the range of a 24" range weapon, but can choose a target from an area four times as large as the shorter-ranged weapon. Its threat radius is four times as large.
Of course, our gaming tables are not devoid of scenery, and are finite. And frankly even if the above table existed my Dark Eldar wouldn't touch it with a barge pole. Scenery is difficult to model for, given the sheer variety of types and placements, but table width is easy - so let's start there.
One point is immediately obvious. Sure, that lascannon may be able to touch every corner of the table if you stick it in the dead-centre, but why would you? Instead, you'd put it firmly inside your deployment zone, where the extra range will limit an opponent's ability to retaliate. As a consequence, the majority of its threat radius will be removed from consideration, due to it being off the table, behind you.
For that matter, how often are your units called upon to shoot behind them? The majority of most units' effective threat radius is contained in a semi-circle to their front, at best. However, if we assume that the angle of that threat arc or semi-circle or whatever is constant, then the proportionality will hold: a threat area will still increase proportional to the square of the range. A lascannon will still have four times the threat area as the bolter will, despite having only double the range.
Perhaps this suggests a convenient model to rate a weapon's threat radius, for comparison with other weapon choices; by normalising it to the humble bolter. In doing so, at common ranges we get:
48"+: 4.0000
42": 3.0625
36": 2.2500
30": 1.5625
24": 1.0000
18": 0.5625
12": 0.2500
6": 0.0625
So these numbers tell us that an 18" range weapon has 56.25% of the threat range of a (stationary) bolter; likewise a 36" range weapon has 225% of the threat range. Note that there are some irregular ranges, such as 30" and 42"; these are included so that movement may be added to threat radius, if desired. Ranges larger than 48" suffer from diminishing returns, given the size of the average tabletop, and so are not included.
Of course these numbers mean nothing in a vacuum. But perhaps they will come in useful when comparing different weapon options, by factoring into the points per kill or hit equation.
For example, were I to forsake the standard blasters on my Trueborn for more poison - perhaps to combine with the Duke for a nifty amount of 3+ to wound shooting - I may be tempted to compare the shardcarbine option to the splinter cannon, or splinter rifle. Splinter rifles are 12 points (the cost of the dude carrying it), carbines are 17 points, and cannons 22 points. Because they all have the same chance to wound and penetrate armour as each other, I am only really interested in the number of hits I can generate per point.
So:
Rifle (<12"): 1.333 hits, 0.111 hits per point.
Rifle (12 - 24"): 0.667 hits, 0.056 hits per point.
Carbine (<18"): 2 hits, 0.118 hits per point.
Cannon (moving, <36"): 2.667 hits, 0.121 hits per point.
Cannon (stationary, <36"): 4 hits, 0.182 hits per point.
Although the Cannon is a clear winner so far, I am still curious about whether the rifle's longer range may help. So I multiply each category by the threat radius relative to 24", and get:
Rifle (<12"): 0.111 x 0.25 = 0.028
Rifle (12 - 24): 0.056 x 1 = 0.056
Carbine (<18"): 0.118 x 0.5625 = 0.066
Cannon (moving, <36"): 0.121 x 2.25 = 0.273
Cannon (stationary, <36"): 0.182 x 2.25 = 0.410
The carbine's ability to score more hits within 18" win out over the splinter rifle's ability to score some hits at 24" range using this metric. Of course, with its long range, the Cannon shows it is the pre-eminent choice of splinter lovers everywhere. ∫But, I still haven't considered the total threat range, including movement and weapon range. If I were to do so, we would see:
Rifle (<18"): 0.111 x 0.5625 = 0.063
Rifle (18 - 24"): 0.056 x 1 = 0.056
Carbine (<24"): 0.118 x 1 = 0.118
Cannon (moving, <42"): 0.121 x 3.0625 = 0.371
Cannon (stationary, <36"): 0.182 x 2.25 = 0.410
Now that we take into account the ability of a moving carbine to make up the same range as a stationary splinter rifle, we can see a clear benefit to taking the carbine at the points cost listed (again, subject to the validity of the metric being used). But, we can add one more echelon, by putting out Trueborn in a Raider and effectively giving them a movement of 12", provided that they are happy to disembark:
Rifle (<24"): 0.111 x 1 = 0.111
Carbine (<30"): 0.118 x 1.5625 = 0.184
Cannon (moving, <48"): 0.121 x 4 = 0.484
Cannon (stationary, <36"): 0.182 x 2.25 = 0.410
Here, we can see that although the carbine has only a moderately superior output of hits per point than the rapid-firing rifle, its ability to disembark and hit a target up to 30" away from the Raider's start point values it much higher than the rifle - but still nowhere near the Cannon. The path is clear: Cannons for those who can, carbines for the rest. Don't forget the Duke!
Definitely, this is still a work in progress. But if we can reach a greater understanding of the role range plays in weapon power, it will benefit everyone.
Stay tuned for the next installment of TL; DR - I mean, Creeping Darkness - where we will look at interaction between threat radii and opponent moves - or try to!
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