Checking spline error between samples put 82% of tracks outside their 10 m tolerance

Fit a cubic to a GPS track parametrised by time and nothing constrains the curve between two samples. The fitter's residual is evaluated where samples exist ^[5]. When the recorder drops out for a minute the curve crosses that gap on its own, and the post reports it drawing loops ^[8]. Douglas-Peucker cannot do that, because its simplified polyline satisfies the tolerance along the whole line by construction ^[9]. The correlations locate the cause: error tracks the longest gap between samples at r = 0.69, and spatial jumps at only 0.22 ^[7].

The author, who publishes as artsensiva, wrote the lesson as "verify a method's guarantee where the method is used, not where it's convenient to check" ^[10]. Two of the four comparisons described end by finding a bug in his own fitter ^[5][23].

Splitting at gaps over 30 s and at physically impossible speeds turned 200 files into 585 trips, about 2.9 per file ^[11][1]. With error checked on a dense grid and knots added where the tolerance broke, all 585 passed ^[12]. Recall went to 0.972 against Douglas-Peucker's 0.996, intervals not overlapping ^[13]. In miss rates that is 2.8% against 0.4%, so the corrected spline still returns about seven times as many wrong nearest neighbours ^[2]. The dense-grid fix bought roughly a tenfold cut in misses, from 29.3% to 2.8% ^[4]. For those recall figures to say anything about another store, the queries have to be Frechet nearest neighbour at a 10 m tolerance on tracks sampled densely enough that a 30 s split is the right cut ^[3][11].

The compactness test was set up against the harder opponent. Because the spline holds accuracy at every point in time, the fair baseline is the time-aware DP+SED, synchronized Euclidean distance ^[14]. The spline lost to that and to plain DP at every tolerance ^[15]. The narrowing gap, 3.4x at 50 m down to 1.4x at 2 m, is what approximation theory predicts for a polyline against a cubic on a smooth curve ^[15][16]. That pointed at clean, high-precision data as the place a spline might win, so the next run used synthetic roads made of straights, arcs and clothoids ^[17]. With zero noise the spline came out 22% more compact at a 1 m tolerance ^[18].

That 22% came from the sweep whose feasibility counts were suspicious. In every cell the share of tracks where a method could hit the tolerance at all was identical for spline and polyline: 3/30, 25/30, 24/30, 26/30 ^[19]. A 20 cm tolerance was unreachable almost everywhere even with no noise ^[20]. For a polyline that follows from the chord error L^2/8R, which the post puts at 1.7 m for 20 m/s on a 30 m radius ^[21]. Invert it and the chord is 20 m, so those samples sit about a second apart ^[3]. A cubic through exact samples should have tracked the arc to centimetres ^[22]. It could not, because the earlier fix tethered it to the straight segments joining the samples, and on clean data that tether obliged it to reproduce the polyline's error ^[23].

The second lesson in the post reads: "a fix that's correct in one experiment becomes a constraint in the next. Matching numbers across methods that should behave differently are almost always a signal." ^[24] The fifth experiment answers it with least squares straight on x(t) and y(t), no tethering, plus an oracle spline fitted to the true curve with no noise ^[25]. The published text breaks off mid-sentence there, before those numbers and before it identifies the by-product the author says was worth the trip: "along the road a different result appeared, and that one was probably worth the whole exercise" ^[27][26]. The code, the per-milestone reports and the decision history are public, under DOI 10.5281/zenodo.22850469 ^[2].