Racing Line Comparison
Same car, same corner, same gates — which path is quicker, and why?
Tyre grip μg = 10.8 m/s². Engine and brake limits are further capped by the
| Metric | Line A | Line B | B − A |
|---|---|---|---|
| Path distance to finish (m) | 192.32 | 192.26 | −0.05 |
| Estimated section time (s) | 7.287 | 7.114 | −0.173 |
| Time at corner exit gate (s) | 3.894 | 4.032 | +0.138 |
| Minimum speed (mph) | 48.9 | 46.4 | −2.5 |
| Exit speed at exit gate (mph) | 52.5 | 52.2 | −0.3 |
| Full throttle from (m) | 127.1 | 132.1 | +5.0 |
| Speed at finish (mph) | 73.2 | 79.8 | +6.6 |
| Max lateral acceleration (g) | 1.10 | 1.10 | ±0.00 |
| Max grip utilization (%) | 100.0 | 100.0 | ±0.0 |
| Minimum radius (m) | 44.2 | 39.9 | −4.4 |
| Entry sector (s) | 2.545 | 2.681 | +0.136 |
| Corner sector (s) | 1.349 | 1.351 | +0.002 |
| Straight sector (s) | 3.394 | 3.082 | −0.312 |
- Line A (Geometric Apex) is 0.05 m longer, so it must travel further for the same gates.
- Line A (Geometric Apex) carries 2.5 mph more minimum speed (tightest radius 44.2 m for Line A (Geometric Apex) vs 39.9 m for Line B (Late Apex)).
- Line A (Geometric Apex) crosses the corner exit gate 0.3 mph faster; that speed advantage is carried down the exit straight.
- Line B (Late Apex) is 0.138 s behind at the corner exit gate but wins back 0.312 s on the 90 m straight.
- Break-even straight ≈ 40 m: with a shorter straight Line A (Geometric Apex) wins, with a longer one Line B (Late Apex)'s exit speed pays off and it wins.
- Line A (Geometric Apex) gains 0.136 s in the entry sector (start → corner entry gate).
- Line A (Geometric Apex) gains 0.002 s in the corner sector (entry gate → exit gate).
- Line B (Late Apex) gains 0.312 s in the straight sector (exit gate → finish).
- Net effect: the straight sector decides it for Line B (Late Apex), even though it has the lower minimum speed — the slowest point alone does not set the section time.
Physical principle
Section time is distance divided by speed, summed metre by metre. A line can win by being shorter, by carrying more speed through the tightest bend, or by starting the exit straight faster — and those goals conflict. The solver finds the fastest speed profile each line allows, then integrates time.
Solver & equations
v ≤ √(μg/|κ|)a_long² + (v²κ)² ≤ (μg)²v²ᵢ₊₁ = v²ᵢ + 2·a·Δst = Σ 2Δs / (vᵢ + vᵢ₊₁)Forward pass: accelerate with min(engine, remaining grip). Backward pass: brake with min(brakes, remaining grip). Δs ≈ 0.25 m.
Your results
Line B (Late Apex) is faster by 0.173 s. The sector-by-sector breakdown under “Why the times differ” shows where it is won.
Fair-comparison rules: both lines start at the same point at 75 mph, finish at the same point, and use the same vehicle. The finish gate is 90 m after the corner exit gate and has no speed constraint: each car is timed at full acceleration through it, so exit speed keeps paying off for the whole straight. A longer straight makes exit speed worth more; a shorter one makes the time to the corner exit matter more. For these two lines the break-even straight is about 40 m.
Try this
- Set Line A to Geometric Apex and Line B to Late Apex, then drag the straight from 40 m to 600 m. Watch the time-gap chart: where does the late apex catch up, and does it win?
- Compare a shorter racing line against a longer but smoother one. Which produces the lower estimated section time?
- Lower max acceleration to 2 m/s² (a low-power car), then raise it to 10 m/s². Does the winner change? Why does exit speed matter more for one car than the other?
- Set both lines to the same preset — the times must match exactly. Then nudge one apex by a metre.
- Raise the entry speed until a warning appears. Which line runs out of braking distance first?
Model: point mass, flat dry track, constant μ, friction-circle combined grip, constant engine and brake limits (no power curve, aerodynamic drag or downforce, so the value of exit speed on long straights is overstated), no load transfer, tyre temperature, wear or suspension dynamics. Times are theoretical estimates for learning, not predictions of a real car or advice for real driving.