Track progress s

The shared coordinate for every comparison is the distance s (metres) along a fixed closed centerline, measured from the start/finish gate in the direction of travel. It is not the racing line: a lap's lateral offset from the centerline is a measured quantity, and two laps taking different lines are aligned at physically identical places.

Centerline construction

1. Valid GPS points of the racing laps are ordered by a provisional progress parameter (unwrapped polar angle about the track centroid, which works for ovals).
2. They are binned into 180 angular bins; the median x, y of each bin forms a closed polyline (empty bins are filled by circular interpolation, then a circular median over ±1 bin is applied). Off-line driving is left out when it matters. The line most passes took is found first: every pass of the car through an angle (interpolated between consecutive samples at most 10 m apart, crossings fewer than 25 samples apart counting once) gives one vote, so a slow infield road — which leaves several times the samples of a fast lap — cannot outvote the laps. Samples more than 15 m across the track from that line are dropped, but only if that changes the centerline's length by more than 2 %; otherwise the centerline is kept exactly as built from every sample, so a healthy session's gate and lap times do not move. (A four-lap stint at La Crosse, Oct 2026, with the infield road driven at ~25 mph before and after, gave a 1066 m centerline for an 826 m track and every lap was refused; without the road it is 826 m.)
3. The polyline is smoothed with a least-squares Fourier series of the closed curve: the smallest number of harmonics (from 4 up to 32) whose RMS residual to the bin medians is at most 1 m.
4. The curve is resampled at a uniform arc-length spacing of 1 m: $C(s)$, $s \in [0, L)$, with heading $\psi_c(s) = \operatorname{atan2}(y', x')$ and signed curvature $\kappa = (x'y'' - y'x'')/(x'^2 + y'^2)^{3/2}$ from central differences.
5. Steps 1–3 always produce a counter-clockwise curve (increasing polar angle). For a track driven clockwise (see Lap detection, step 2) the curve is reversed before resampling, so s, the heading and the curvature follow the car: curvature is positive in left turns and negative in right turns.
6. s = 0 is placed at the gate; direction is the direction of travel.

The centerline is versioned (its hash is an input of every downstream result). Segment boundaries are stored as fractions of $L$ so they survive a rebuild.

Continuity-constrained projection

Independent nearest-neighbour projection can snap a noisy point to the opposite straight of a narrow oval or to a parallel pit lane. Instead, each lap is projected sequentially with a prediction window:

• Predicted progress: $s_{pred} = s_{prev} + v_i\,\Delta t$.
• Search window: $[s_{pred} - \max(5\,\mathrm{m},\; 0.5 v\Delta t + 3\sigma_s),\; s_{pred} + \max(10\,\mathrm{m},\; 1.5 v\Delta t + 3\sigma_s)]$ with $\sigma_s$ = 3 m.
• Among the centerline points in the window the one with the smallest perpendicular distance $|n|$ is chosen, provided $|n| \le$ 30 m (the limit $n_{max}$) and $s \ge s_{prev} -$ 2 m.
• Confidence: $c = \mathrm{clip}(1 - |n|/n_{max}) \cdot \mathrm{clip}(1 - |s - s_{pred}|/\text{window})$.
• If no candidate qualifies the sample is marked lost, $s = s_{pred}$, confidence 0. More than 5 consecutive lost samples invalidates the lap (GPS projection lost).
• The gate crossing pins $(t_{start}, 0)$ and $(t_{end}, L)$ so s unwraps continuously through the gate.

The signed lateral offset $n$ is positive to the left of the direction of travel.

From samples to a grid

t(s) for a lap is a monotone piecewise-cubic interpolant (PCHIP) through the running maximum of the projected s values, anchored at $(0, 0)$ and $(L, T)$, evaluated on the 1 m grid. PCHIP preserves monotonicity and cannot overshoot. Speed, path accelerations, lateral offset and projection confidence are linearly interpolated onto the same grid. Every chart, the map cursor and the readout use these grid values directly; the browser never interpolates between them.

Verified

• Real recordings: no lost projection samples and mean projection confidence 0.85–0.96 on every racing lap.
• Synthetic: a 40 m-wide oval with 3 m GPS noise never projects to the wrong straight; a pit lane 8 m outside the front stretch creates no lap; a 20 m infield excursion is lost for ≤ 5 samples and the lap survives; crossing the gate with ±3 m noise unwraps s without a discontinuity.
• Centerline arc length on a synthetic ellipse within 0.1 % of the analytic value; curvature sign tested.