StarCraft surround and choke calculator

The short version

Your army's damage is not what you paid for. It is what is in range at the moment the shooting starts.
A column fires one unit at a time. A line fires as many as the ground lets you spread. An arc fires all of them.
The choke does not slow you down. It decides how wide you are allowed to be, and everything behind that width is supply you are paying for and not using.
Range is the multiplier. Melee units get almost nothing from a line and everything from a surround. Ranged units are already half surrounded by their own range.
Drag the army in the field below. Gold means firing.

How much of your army is firing

Pick a shape, then drag the army around the enemy. Gold means that unit is in range. The number is the share of what you paid for that is actually shooting.

2of 12

Firing

17%

Flux

The share of your damage that is reaching him.

96px

Reach

1 tiles of range, plus his body.

10units

Paying, not shooting

Supply you brought and did not use.

Column, 12 units, range 1: 2 of them are within reach when the fight starts, so you are firing 17% of the damage you paid for.

Shape	Firing	Flux	What it costs you
Column	2 of 12	17%	Single file. Everybody behind the front is paying rent.
Line abreast	8 of 12	67%	As wide as the gap lets you be, and then it folds into ranks.
Arc	9 of 12	75%	Around the front of him. The queue becomes a row.
Full surround	12 of 12	100%	All the way round, which needs open ground and a slower enemy.

What flux is

Lesson 4 defines flux as an integral: take your damage output across the surface you are attacking through, and normalise it by the damage you have.

\Phi = \frac{\int D(u)\cdot d\vec{A}}{\sum D(u)}

(1)

That is exactly right and completely unusable at the keyboard, so the lab computes the thing the definition is a definition of: of the units you brought, how many can reach him.

You can also write the arc itself, as the circle you stand on:

y = \sqrt{2Rx - x^2}

(2)

which is the same circle of radius $R$ written as a height above the line of approach. The practical version is shorter: stand at range, all the way round.

Why a column is so much worse than it looks

A column is what you get when you press attack move through a narrow gap and walk away. It looks efficient, because every unit is moving. But the second unit is a rank behind the first, and a rank is one unit width, so at melee range it is already out of the fight.

Set the shape to Column and the range to 1 in the lab. Two units out of twelve. Then set it to Arc. The other ten did not get better weapons. They got a seat.

Why the arc stops helping for ranged units

An arc seats units around a circle of radius equal to your reach, and the number of seats on it is the circumference over your spacing: about 2 pi R / s. For a melee unit that circle is small, so the seats run out and the extras go into a second ring that cannot reach. For a dragoon at range 4 the circle is four times bigger, so there are four times as many seats and you are firing everything before the shape matters.

This is the whole reason melee armies care about surrounds and ranged armies care about concave spreads: they are the same fix applied to a circle of a different size.

Homework 4: what a bad surround costs

The class set a surround as homework: an arc of length 60 against a body 20 wide, with the attacker's estimate of the distance wrong by 10. The question was what the error costs.

It costs two different things, and the second one is the one that loses the fight.

Coverage. You stand at radius $R$ , which is half his body plus your reach. An arc of length $L$ wraps $L / 2 \pi R$ of him. Ten pixels further out and the same 60 of arc wraps less, because the circle you are standing on is bigger. In the numbers the class gave, a 38% wrap becomes a 27% wrap.
Time. Every unit on that arc is now ten pixels beyond its own range. At the moment you thought the fight had started, nobody is firing at all. Flux is zero until you close, and closing is a second and a half of him shooting and you not.

Press the surround preset in the lab and drag the army ten pixels out to see both effects at once.

What this lab assumes

Assumption	Value	Why, and how sure we are
The enemy is a body, not a point	64 pixel radius	About a dozen units bunched up. The number matters: surrounding one zergling and surrounding an army are not the same problem.
Unit spacing	24 pixels	Roughly a zealot's width. Real formations are looser, so treat every count here as the good case.
Contact	units press to touch	They walk in until they touch his body or queue behind a friend, which is what attack move does.
The choke	caps the width only	It does not slow anybody down here. Real chokes also break your formation on the way through, which makes the column result worse and never better.
Geometry only	no fight is run	These are positions, not a fight. Use them to count who can shoot, then take the fight to the Battle Lab.

Flux Lab