On Bomb Geometry

The Stealth Bomber is a very polarising and contentious hull in EVE Online. For those who partake in bombing runs, the rush of adrenaline seeing a successful run resulting in a mass of explosions is the highest euphoric state achievable in a video game. For those subjected to it, not so much. Bombing bombers render whole swathes of potential doctrines obsolete because their jobs are too easy, with insufficient counterplay. At the recent EVE Vegas, CCP Larrikin announced a new destroyer module which will specifically target and destroy incoming bombs in an attempt to provide a defense for doctrines susceptible to bomb damage, most notably shield battleships such as the old Rokh fleet.

Though I’m not a bomber FC like DBRB or good old TempelmanN, having learned primarily from my old CEO Cheeb Aman, I’ve enjoyed FCing the bombing experience once or twice myself. It’s important to mention this because an experienced bomber FC will be able to actually gauge the extent to which this analysis has a practical use, or even makes sense. I am a layman applying math to a topic I have little experience in.

On a sidenote, the passion and joy shown by Tempelman N in that video was and is an inspiration to me for who I want to be and for the attitude I want to represent in-game, and to a lesser extent, in real life. I’ve heard good and bad bits about the man, and he’s a flawed human like the rest of us, but in that video he is how I want to feel and how I want to make people feel about this fucking awesome game.

The Hypothesis

One of the easiest ways to perform a bombing run is to do so on a spawn point, i.e. a gate, cyno, or jump bridge. These are guaranteed points at which you know your opponent’s exact position, well ahead of time. Otherwise you have to be skilled in the art of using ad-hoc perches and bombing spots. For our purposes, we’re going to discuss simultaneous bombing (all bombs explode in the same server tick) rather than wave-bombing, and looking primarily at spawn points (gates, cynos, jump bridges). Today I want to discuss a theoretical application of geometry in regards to bomb positioning when done on spawn points. To do so I am going to make some fairly dangerous assumptions in order to make the maths prettier:

• Each wave of 7 bombs explodes at exactly the same point in space, and the minimum distance allowed between these two points is the bomb explosion optimal, 15km. In practice, you want to allow yourself a little wiggle room, but for the sake of the math we’re going to pretend we can bomb absolutely perfectly.
• Between the spawn time and the bombs launching and exploding, the ships will have moved a negligible amount. This can be focused with good timing, but practically speaking most fleet comps will be able to move some distance between spawning in system and noticing that they’re going to be bombed.
• Ships occupy finite points rather than spheres. The fact that they are spheres actually helps us as if any subset of their volume intersects with the bomb optimal radius, they still take damage.
• When we look at ship fitting statistics and damage output, we’re going to assume perfect skills.
• I have infinitely many bomber pilots to use. Again, this is a theoretical discussion, not a practical one.
• The server maths does not screw with any of the calculations.

Essentially, I am assuming that everyone can execute everything perfectly and that the server plays nice.

Firstly, let’s discuss the basics of bomb mechanics. After being launched, bombs will travel 30 km in the direction your Stealth bomber was facing when you launched it. The bomb will then explode, damaging each target with the same base damage (8k of the damage type associated with that bomber assuming all Vs and the correct bomb). This damage receives a multiplier of min(1, ship radius/explosion radius). Further, bombs do have HP and resistances to their own damage type, 7 bomb explosions do enough damage to kill a bomb, thus you can have maximum seven per wave that will intersect each other upon explosion, as the seventh and final bomb would destroy any further bombs. The explosion (signature) radius for bombs is 400m, so anything 400m or larger takes full damage, while anything smaller takes a reduced amount. Note that unlike missiles, the speed of the target has no impact on the damage it takes from a bomb.

Let’s use an example to be absolutely clear. Let’s look at this Svipul fit from C02.

Note here that we are not applying links, which would make this Svipul significantly tankier, we are in defensive mode, which does the same, and that the pilot has no implants. The Svipul is following the conventional wisdom of keeping his sig-blooming MWD offline, hoping to tank the damage and get on with the fleet, although he has forgotten to heat his hardeners. Furthermore, this is a pre-nerf Svipul.

Let us assume that the bomber FC was aware that this Svipul fit is going to be used and can analyse it in EFT or Pyfa before the fleet. Doing so reveals that the most effective bomb type to use is concussion, dealing kinetic damage. It’s resistances in the majority of its EHP (shield) are ties lowest in kinetic and thermal, and naturally the structure resistances are uniform. In armour, the kinetic resistance is lower than the thermal one, giving kinetic as our preferred bomb damage type.

Doing the math we have:

3672/(1-0.814) = 19,742 effective shield hit points
875/(1-0.562) = 1998 effective armour hit points
550/(1-0.565) = 1264 effective structure hit points

Summing these we get 23,004 effective kinetic hit points overall. But how many bombs do we need to kill this Svipul? To answer that we need to know how much damage a bomb does to one. Our sig is 59.7, so with all Vs our base bomb damage is 8000, which we multiply by 59.7/400 to get 1194 damage per bomb. 23,004/1194 = 19.3, meaning we need at least 20 bombs to kill the Svipul, which is one bomb short of three waves.

Before we continue much farther, we do need the values for the sizes of each gate. Sourced from the fantastic Seamus “Care-Bear Extraordinaire” Donohue, one of my favourite people on EVE Online, here they are:

Amarr Constellation-4646 m
Amarr System-4946 m
Minmatar System-4995 m
Gallente System-6754 m
Caldari System-7065 m
Caldari Constellation-8998 m
Gallente Constellation-9412 m
Gallente Region-10 km
Amarr Region-14 km
Caldari Region-15 km
Minmatar Region-24 km
Minmatar Constellation-25 km
Amarr Border-26 km
Caldari Border-28 km
Gallente Border-38 km
Minmatar Border-42 km
Smuggler Gate-not known to Seamus

Seamus in turn sourced these values from here.

It should be noted that I am also using similar zero-sphere theory/terminology from his excellent Fanfest presentation in 2014, which I highly recommend watching or at least understanding before continuing with this article. You can watch that here. Oops, I mean here.

TL;DR: everything in EVE is spherical and distances are calculated edge-to-edge, not center to center.

Okay, now let’s talk spawn distances. According to Professor Donohue, the center of your ship will spawn 12 to 14 kilometers from the edge (zero-sphere) of the stargate. This gives us the spawn distances from the center of the stargates by halving the above diameters and adding 12 to 14 kilometers.

Amarr Constellation-14323 – 16323 m
Amarr System-14473 – 16473 m
Minmatar System-14498 – 16498 m
Gallente System-15377 – 17377 m
Caldari System-15533 – 17533m
Caldari Constellation-16499 – 18499 m
Gallente Constellation-16706 – 18706 m
Gallente Region-17 – 19 km
Amarr Region-19 – 21 km
Caldari Region-19.5 – 21.5 km
Minmatar Region-24 – 26 km
Minmatar Constellation-24.5 – 26.5 km
Amarr Border-25.5 – 27.5 km
Caldari Border-26 – 28km
Gallente Border-31 – 33 km
Minmatar Border-33 – 35 km
Smuggler Gate-not known to Seamus

Note that aiming a single wave for the exact center of the gate will yield results only for the three smallest gate types, unless the response of the opposing fleet is to burn back to gate upon being bombed.

A small note before we continue. In the following diagrams I have set the distance between each cluster of bombs as EXACTLY 15km. This, of course, would render the bombing run useless as the bombs would detonate each other. The bombs should be launched as closely to these spots as is possible with the piloting of the stealth bombs. The natural, slight time difference between explosions will also help as they are only in range of each other (in this theoretical, perfect scenario) at the instant they explode. We’re effectively working with the mathematical “strictly greater than but not equal to” of those 15 km distances.

Orthogonal Bombing

The typical setup I have seen and used for simultaneous gate bombing is what I call orthogonal positioning, i.e. positioning at right angles. You have six squads positioned at right angles above, below, in front of, behind, left and right of the gate. This is a solid way of getting at least two waves of bombs to hit all of your targets. A 2D representation of how many waves hit what space is as follows:

Note that only four circles have been drawn here – the other two are in front of and behind the plane, but I have included their damage in their intersection points. The distance from the gate of the bomb explosion is approximately 10.6km (15/sqrt2 if you want to be precise) which means that if our fleet does not move, even on the smallest gates all ships will be hit by no more than one or two waves of bombs. We’re really not going to kill any Svipuls here unless they forget their hardeners. That said, this is an excellent focus of damage if the fleet to be bombed burns back to gate but does not make it in time.

For the sake of comparison to the following iteration of this idea, here is the same diagram, but this time we are bombing with only four waves in two dimensions.

The Hexabomb and other two-dimensional ideas

Squares are very inefficient at containing circles. Equilateral triangles of side length radius equal to the radius of the bomb optimal form a much more efficient lattice. As we continue through two-dimensions we’re effectively just adding or removing sides to our bombing ring, but when we move into three dimensions and into dual-ring bombing the triangle concept will become much more important.

Ideally, we want to pack as many bombs into as small as space as possible, with explosion radii overlapping each other as much as possible. If we bomb in this fashion our overlap zones are significantly larger, and focus less on the centre of our run and more on where the enemy spaceships will actually be. Let’s start with the basis of all the shapes I’m going to form for the rest of this discussion.

Now, if we did this on its own, it would be a worse outcome than the orthogonal bomb. There are fewer bombs and the blast is focused on the centre of the gate even more, instead of on where the spaceships will spawn. However, by copy/pasting six of these together, we get a much better outcome.

Each bomb explodes 15km from the center of the gate but for the central bomb. What we are really interested in is the region just inside the circle of exploding bombs, where each ship spawning is being consistently hit by three or four waves of bombs. For our current smallest gate, the spawn point is around the radius of the optimal if the central bomb, so we’re in good shape. However, if the gates are any larger we are in trouble.

We can make the focus of our bomb blasts a larger doughnut by adding more 15km sides to our constructed shape, increasing our ring of destruction’s distance from the gate. Here are examples of an Octabomb and a Dodecabomb:

Now, I am aware that we are entering the realm of pilot infeasibility: the Octabomb and Dodecabomb require 56 and 84 stealth bombers respectively. Do remember that we are developing theory here, practicality will come later.

Unfortunately, in this case we have lost the central bomb, meaning the majority of our overlapping regions have only two waves of damage (remember we are looking for the spawning ring to be just inside the polygon formed by our bomb explosion points). This however is solved by introducing an inner ring of bombs, as soon as the rings are large enough such that the inner ring of bombs do not overlap. This is first achievable in the Dodecabomb:

Note that we have now formed the Hexabomb inside the Dodecabomb, sans the central bomb:

We have returned to the point of guaranteeing that 3-4 waves of bombs will hit all our targets in a radius of between 15 and 30 km from the central point of the gate, which works well for all but the two largest gate types.

Unfortunately we now also require no less than 126 bombers, and we’re not even working in three dimensions yet. Given the assumption that the goal is to outright kill as many ships as possible, the solution is to take as many stealth bombers as possible (three waves minimum) and partially produce a segment Dodecabomb, guaranteeing that you take out a specific fraction of the opposing fleet. Functionally, we have produced a repeated set of equilateral triangles forming a ring that encloses the spawn sphere of the gate.

The Tetrahebomb and other three dimensional ideas

Now we move into three dimensions, but we are using exactly the same principle. Consider the Tetrahedron, or d4 if you’re a tabletop nerd like me.

If we consider each vertex of the die a bomb explosion, and have our side lengths as close to but not equal to 15, all points within the die (with a little extra volume outside peaking at the centre of the faces) will receive four waves of bomb damage.

If we expand to larger solids we can construct an Icosahedron by gluing several tetrahedrons together:

Again, this would require a prohibitive number of bombers, but a partial construction does guarantee you getting 4 waves of bombs on a given fraction of an enemy fleet.

Going beyond the normal Platonic solids, we can design rings of tetrahedra-based shapes of tetrahedrons glued to each other by the face. This allows us to create as great or as small a volume as we wish to bomb. The example below shows how two tetrahedra would be joined, but you would simply add more and more to fill the desired volume, depending on how many waves of bombers you wish to add. Interestingly, after the initial investment of 28 bombers (four waves), each additional tetrahedron only requires a single vertex – seven more pilots.

Importantly, if the enclosed space of a single Tetrahedra seems small, remember that this only represents the volume which experiences four waves. if we imagine extra tetrahedra joined to each face of the ones we have formed with our bombing points, this secondary set of volumes will all receive three waves of bomb damage. Still enough to kill our example Svipul.

As far as I have been able to reason, four is the upper limit of repeatable overlaps of bomb explosion optimals in three-dimensional space. You can do six as shown in the orthogonal bombing setup, but not over a continuous volume.

Potential Applications on Tranquility

I will reiterate once again that this has all been a theoretical discussion of bomb geometry, ignoring the realms of practicality in both pilots and worthwhile effort. The number and complexity of creating  the bookmarks required are extraordinary for all but the most basic applications of this theory.

That said, I do believe that small constructions of just a few tetrahedrons with appropriate bookmarks may be feasible for a practical application of bombing spawn points. Further, if we return to bombing waves, we can use these same structures to quickly do multiple sets of four waves of bombs over a given area. We can use this construction on the fly similarly to how we currently do it for orthogonal bombing, but with a more efficient area coverage.

For ad-hoc bombing, if we know which grid the fight is going to happen on (or a given set of grids depending on how many bookmarks you want to make), we place bookmarks in system far enough away from the grid in the same configuration as the vertices of the shape, allowing us to do ad hoc 4-wave strong simultaneous bombing runs. We must be careful such that the warp distances are not so short that placement on the grid will significantly alter formation – but not so far that the warp time means that by the time the bombers land and bomb, the opposing fleet is no longer in position (hyperspatial bombers ftw). The distance from each vertex to the centre of the structure is approximately 8.66km, the travel distance for each bomb is 30 km, so by simultaneously warping each of the four waves of our bomber to the enemy fleet at 40 km, we should have a pretty reasonable run going on.

If the new mega-grids prove too small for the formation to be non-negligibly altered, due to the current constraints of scanning, we would need four separate scanners warping each wing into position. Given the inability to pick out specific ships in a fleet of uniform hulls, it would probably be better to first warp a suicide ceptor to the opposing fleet, and then using that as a point for the fleet to warp to (this is of course assuming there are no bubbles in the way to catch and pull our bombers).

This is not going to result in some revolution in how we flying bombing fleets, merely an examination of ideal geometry for the task. Do you have any improvements or expansions of the the thoughts I’ve laid out here? Am I completely wrong due to an embarrassing and basic error? I’d love to hear your thoughts in the comments section!