Swept Path Analysis: A Visual Guide to How Vehicles Turn

When a truck rounds a corner, the rear wheels don't follow the front wheels. They cut inside the curve. The space the vehicle sweeps through is wider than the vehicle is, and bigger than the radius of either wheel alone. Swept path analysis is the geometric tool civil engineers use to predict that swept area exactly, so a fire truck reaches a hydrant, a delivery van clears a bollard, and a refuse vehicle backs out of a dead end without climbing the kerb.

This guide explains the geometry from first principles, walks through a worked pen-and-paper example, and finishes with what to consider beyond the textbook model. No specific country's regulations are required; the underlying physics applies everywhere.

TL;DR. In a steady-state turn, a rigid vehicle's swept envelope is bounded by just two reference paths: the outer-front corner (outer edge) and the inner-rear wheel (inner edge: wheel, not corner). A third path, the outer-rear corner, defines the tail swing at the start of every turn: a brief outboard swing of the rear caused by the rear overhang, regardless of how fast the steering input is applied. All three can be computed from five numbers: length, width, wheelbase, front overhang, rear overhang. Worked example below.

What is a swept path?

A swept path (also called vehicle tracking or off-tracking analysis) is the area a vehicle physically occupies while turning. It is the union of every position the vehicle body passes through during the manoeuvre.

You need it because:

• An intersection, a fire-truck turning bay, a parking-garage ramp or an alleyway is only as useful as the vehicles that can fit through it.
• The geometric envelope is bigger than the bare vehicle width, and the difference matters: a typical lorry occupies a footprint about 1.5× its own width during a sharp turn.
• Without the envelope, you can't verify clearance from kerbs, walls, columns, parked cars, or fire hydrants.

The geometry: Ackermann steering, in plain words

Rudolph Ackermann's 1817 patent describes how a steered vehicle's wheels move during a turn. Its central insight:

During a turn, every wheel of the vehicle traces a circle around a single shared centre point, the instantaneous centre of rotation (ICR).

The ICR sits on the extension of the rear axle (because the rear wheels don't steer, their rolling direction is fixed perpendicular to the axle). The front wheels are angled exactly so their perpendicular lines also pass through this same ICR. That alignment (front wheels turning to slightly different angles so their lines meet at one point) is what we call Ackermann geometry.

The consequence: every point on the vehicle traces its own circle around the ICR. The largest circle is traced by the outer-front corner of the body. Because of the front overhang, the corner sits further from the ICR than the outer-front wheel, and it defines the outer edge of the swept envelope. The smallest circle is traced by the inner-rear wheel at the axle, which defines the inner edge. The body sweeps the annular region between these two paths.

The math, if you want it. With the ICR on the extension of the rear axle, let R be the distance from ICR to the centre of the rear axle (the rear-axle radius). Then the inner-rear-wheel radius is R − w/2 and the outer-front-corner radius is √[(R + w/2)² + (wb + fo)²], where w is vehicle width, wb is wheelbase, and fo is front overhang. Knowing any one of these radii pins down all the others.

The reference paths

Three reference paths cover the envelope. Two bound it during the steady-state portion of the turn; one more, the outer-rear corner, defines a brief outboard sweep at the start of every manoeuvre.

Steady-state bounds

In pure circular motion these two paths alone fully describe where the vehicle is. Everything else on the body sits inside the annulus between them.

Tail swing

The single most common mistake, even in some published work, is to assume the inner rear corner of the body defines the inner edge of the envelope. It doesn't. The corner is wider than the wheel, so its path lies outboard of the wheel's path. The wheel, not the corner, sets the inner clearance.

The five numbers you need

For a non-articulated vehicle (passenger car, van, two-axle truck, bus), five dimensions fully describe the geometry of a constant-radius turn:

The minimum turning radius (the smallest circle the vehicle can drive) is fixed by the maximum steering angle of the front wheels, which is a property of the vehicle itself. Most manufacturer datasheets list it directly as the outer turning radius (front-outer-corner) or kerb-to-kerb turning circle.

Worked example: a typical passenger car

Let's compute the four reference paths for a standard passenger car making its tightest right-hand turn. Dimensions roughly matching a real-world VW Golf or BMW 3-series:

The outer turning radius of 5.60 m is the path of the outer front corner, measured from the ICR. We work backwards from there.

Step 1: Find the rear-axle radius

The outer front corner sits at distance (wb + fo) = 3.60 m forward of the rear axle, and w/2 = 0.925 m outboard of the rear-axle centre. By Pythagoras:

R_out² = (R_rear + w/2)² + (wb + fo)²

5.60² = (R_rear + 0.925)² + 3.60²

31.36 = (R_rear + 0.925)² + 12.96

(R_rear + 0.925)² = 18.40

R_rear + 0.925 = 4.29

R_rear = 3.36 m

So the centre of the rear axle traces a circle of radius 3.36 m around the ICR.

Step 2: Inner rear wheel

The inner rear wheel is half a vehicle width inboard of the rear-axle centre:

R_inner_wheel = R_rear − w/2 = 3.36 − 0.925 = 2.44 m

Step 3: Outer rear corner

The outer rear corner sits ro = 1.10 m behind the rear axle and w/2 = 0.925 m outboard. Its distance from the ICR is:

R_outer_rear = √[(R_rear + w/2)² + ro²] = √[4.29² + 1.10²] = 4.43 m

Step 4: Inner front corner

The inner front corner sits (wb + fo) = 3.60 m forward of the rear axle and w/2 = 0.925 m inboard. Its distance from the ICR is:

R_inner_front = √[(R_rear − w/2)² + (wb + fo)²] = √[2.44² + 3.60²] = 4.35 m

Result

The swept envelope is the annular region between the inner-rear-wheel circle (2.44 m) and the outer-front-corner circle (5.60 m). For this passenger car, that's a band 3.16 m wide, roughly 1.7× the vehicle's own width.

Note that the outer-rear corner (4.43 m) and the inner-front corner (4.35 m) both sit inside this annulus in steady-state circular motion. The outer-rear corner becomes limiting at the start of the turn (tail swing); the inner-front corner is interior throughout for this passenger-car geometry.

Why we used a 90° turn assumption. This calculation gives you the radii of the full circles each reference traces. A real intersection turn typically uses only an arc of those circles (the steering angle is held constant briefly, then released). Software tracks the actual arc swept; the radii here tell you how wide it gets at maximum input.

Real-world considerations beyond the textbook model

The Ackermann low-speed model is exact in principle, but real vehicles deviate from it for three reasons. Most regulatory checks ignore these because they're designed around low-speed manoeuvres, but it's worth knowing where the model breaks.

1. Heading changes are not instantaneous

The model assumes the steering input changes the front-wheel angle immediately. In reality, the driver turns the steering wheel over half a second to two seconds, and during that time the actual swept path is a transition curve, not a circular arc. For passenger cars at planning scales this is negligible; the transition is visible only at sub-metre resolution. For long articulated vehicles (semi-trailers, B-doubles), the transition matters and software typically models it explicitly.

2. At higher speed, the rear wheels track outboard

Above roughly 30 km/h, tyre slip becomes significant. The rear wheels develop a small slip angle (typically 1–3°) that pushes the rear of the vehicle outboard of its kinematic path. This widens the outer envelope, sometimes by 0.5 m for a lorry. For driveways, parking, fire access and most urban planning, low-speed kinematics is correct. For motorway entry/exit ramps, racetracks and crash reconstruction, you need a dynamic model.

3. Articulated vehicles add a second envelope

A semi-trailer is two rigid bodies connected by a hinge. Each follows its own Ackermann geometry around the same ICR, but the trailer's rear axle traces a smaller circle than the towing vehicle's. This is the famous off-tracking of a long lorry: the trailer cuts the corner. Articulated buses, B-trains, and double-trailer combinations stack this effect; the rearmost trailer can cut the corner by several metres on a sharp turn.

There's a second subtlety unique to articulated geometry. The rigid-vehicle shortcut of "track only the body corners and connect the dots" no longer gives the exact envelope, because the trailer's hinge point itself moves along the tractor's path while the trailer rotates around it. At certain phases of the turn the inner edge of the trailer's swept area is defined not by the trailer's corner but by a point along the trailer's side, typically near the rear axle group. Production tools like AutoTURN, Vehicle Tracking and AutoPATH handle this by sampling many points along the entire vehicle perimeter at every timestep and taking the union, rather than relying on four corners. The four-corner approximation is accurate enough for first-order checks but can underestimate the true envelope on sharp turns.

For these, hand calculation is impractical and software is essentially required.

From paper to PathSweeper

The numbers above show that even a single passenger car needs five inputs and four output radii. A real planning task is rarely a single constant-radius circle; it's a sequence of turns, of varying radii, possibly with reverse manoeuvres, against a real site plan with kerbs, parked cars, columns and trees. Doing all of that by hand is a project in itself.

PathSweeper computes all of this in real time as you draw the trajectory, against your uploaded site plan. The same underlying geometry, but in seconds rather than an afternoon, including articulated vehicles, reverse manoeuvres, and corner-by-corner clearance reporting.

Country-specific design vehicles

The geometry is universal, but the vehicles you check against are usually mandated by national or regional standards. Each country specifies a list of design vehicles with concrete dimensions you must use for regulated submissions:

• Germany: RBSV 2020 design vehicles

Each of those guides lists what's free to use and where the licensed publications fit in.

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