Tower Defense Geometry

Click to place a tower. The number on it is how much path its circle covers — which is exactly what its damage is proportional to. Move around a corner and watch it change. Everything runs in your browser; nothing is uploaded.

Damage is a length

A tower shoots whatever is inside its radius, so the damage it deals to one passing enemy is its fire rate times the time that enemy spends in range. At a constant speed that time is the length of path inside the circle, divided by the speed. Everything else cancels — where along the path the coverage sits, which direction the enemy came from, none of it matters.

So "where should I build?" is really "which circle covers the most path?", and that has an exact answer.

A straight caps at two radii; a corner reaches four

On a straight path the best any circle can do is sit on the line and cover a diameter — 2.00r, searched over the whole plane, with no placement doing better. A corner lets one circle cover both arms at once, and the sharper the turn the more it gets:

A hairpin corner is worth just under twice a straight stretch. For a single corner 4r is the ceiling, because two arms is all a corner has — though a whole path can do better than that, which is the next section.

The last column is the part worth taking away. Building on the corner itself is a mistake. Sitting exactly on the vertex covers a radius along each arm — 2r, which is precisely what a straight stretch gives, so the corner has bought you nothing. The gain comes from standing back from it, where the circle cuts a chord out of each arm instead of a radius. At a 45° turn the vertex gets you 54% of what is available; at the sharpest turns it falls to 50%.

The formula, and the edge it has

For a turn of a right angle or sharper there is a closed form, and it is exact rather than close. Stand one radius back along the bisector and the circle cuts a chord of 2r·cos(θ/2) from each arm, so the coverage is

4r · cos(θ / 2)

That is 2.83r at a right angle, 3.70r at 45°, and it climbs towards 4r as the turn tightens. Every row at 90° or below in the table above matches it to under half a per cent, with the best placement sitting at exactly one radius.

Above a right angle it stops being true, and not gently. The arms are far enough apart that the vertex cuts the chords short before they finish, so the formula runs low: 1.5% at 100°, 13% at 120°, 50% at 150°. At a straight 180° it predicts zero coverage when the honest answer is 2r. It is a genuine result on one side of a right angle and nonsense on the other, which is why the code returns it with a flag rather than as a general rule.

The "best spot" column shows why. Below a right angle it sits at exactly one radius, every time. Above it, the optimum creeps in towards the vertex — 0.84r at 100°, 0.58r at 120°, 0.27r at 150° — until at a straight join the best place is on the line itself and the corner has stopped being a corner.

A whole path can beat four radii, because a corner is not the only thing in range

Four radii really is the most a single corner can give — two arms is all it has. But that is a fact about corners, not about paths. Where a route doubles back on itself, a third stretch of it comes inside the same circle:

The last column is the whole explanation. The hairpin path reaches 4.55r — past the four-radius corner ceiling — because the best spot catches three separate stretches of the route at once, not two arms of one corner. The straight path catches one, which is exactly why it caps at two radii.

The straight path is the weakest to defend and it is not close — there is simply nowhere on it that a tower can do more than cover its own diameter. That is the whole reason tower defense maps are drawn with switchbacks.