Measure the approximation error

1. Evaluate angular and scalar residuals

After fitting, ask a concrete question: how far is the stored curve from the function that supplied its angles? The difference at a tested time is a residual. We will measure its size, then keep the largest size found across each segment and the series.

Suppose an invented source longitude is 359.99999° and the approximation is 0.00001°. Direct subtraction gives −359.99998°, but those directions are only 0.00002° apart across zero. Multiplying by 3600 converts this short distance to 0.072 arcsecond. For source declination −12° and approximation −11.99998°, ordinary subtraction and absolute value also give 0.00002°, or 0.072 arcsecond. These are differences between individual coordinates, not a 3D vector separation.

Declination uses absolute valueAbsolute valueClose The unsigned size of a number. Vertical bars mark this operation: |−0.00002| = 0.00002. It measures the size of a scalar difference without keeping its direction. : discard the sign of p − f, where p is the prediction and f is the supplied angle. Longitude requires the shorter distance around a circle. Reduce p − f into [0, 360), call that result r, and use 360 − r if r exceeds 180. In either case the residual is unsigned.

scalar residual = |p − f| × 3600 arcsec

longitude residual = min(r, 360 − r) × 3600 arcsec, r = (p − f) modulo 360

The builder checks 60 evenly spaced times per segment, including both endpoints. For an interval from 0 to 118 seconds, 60 points create 59 gaps, each 118/59 = 2 seconds long. More generally, point i is placed as follows, for i = 0 through 59.

tᵢ = lo + (hi − lo)i/59, i = 0…59

At each time, evaluate the source function f(t) and the fitted polynomial p(x), using x = 2(t − lo)/(hi − lo) − 1. The implementation uses Clenshaw’s recurrence to evaluate the coefficients; a later runtime lesson develops that calculation. Keep the largest converted residual over the 60 points, then the largest over all segments. Fitting minimized a sum of squares on its cosine grid; this measurement asks for a largest absolute difference on a different grid.

const circularResidualArcsec = (approx: number, source: number): number => {
  const r = ((approx - source) % 360 + 360) % 360;
  return Math.min(r, 360 - r) * 3600;
};
const scalarResidualArcsec = (approx: number, source: number): number =>
  Math.abs(approx - source) * 3600;

2. Measure adjoining-segment jumps

A curve may follow its source closely within a segment yet meet the next curve with a small step. To inspect that join, compare both predictions at their shared time. This checks how the curves meet each other, a different question from how either follows its source.

Take invented degree-1 longitude lists [359, 1] and [1.00002, 1]. The preceding curve ends at x = +1, giving 359 + 1 = 360°. The next begins at x = −1, giving 1.00002 − 1 = 0.00002°. Their short circular distance is 0.00002° = 0.072 arcsecond. Subtracting without wrapping would misleadingly suggest almost a full turn.

This difference at the join is called a seam jumpSeam jumpClose The distance between two segment predictions at their shared endpoint. This compares one curve with the next; a residual instead compares a curve with its source function. National Institute of Standards and Technology: Digital Library of Mathematical Functions — polynomial approximation. Both values describe the same time, although that time has a different normalized coordinate in each segment. For degree 1, p(x) = c₀ + c₁x, so its end is c₀ + c₁ and its start is c₀ − c₁.

left = p_previous(+1); right = p_next(−1)

jump = short circular or scalar endpoint distance × 3600

Use the same short circular distance for longitude and absolute difference for declination. Evaluate every consecutive pair, convert to arcseconds, and report the largest jump. Fewer than two segments leave no pairs, so the reported maximum remains zero. Unwrapped longitude values separated by a whole turn also have zero circular jump.

Zero jump does not tell us whether the curves follow their reference. Two adjoining longitude curves can agree perfectly with each other while both remaining 1° from the source. Their seam measurement is zero, while their source residual is large.

3. Try the fit ladder

When a curve misses its residual limit, there are two straightforward adjustments to try: give its polynomial more coefficients, or ask each polynomial to cover a shorter interval. The builder tries those adjustments in a fixed order and measures each result, rather than assuming an adjustment improves it.

For an initial 32-day, degree-10 specification, the attempts are 32/10, 32/12, 16/10, 16/12. Suppose their measured maxima begin 0.16″, 0.11″, 0.09″ against a 0.1″ limit. The third passes; the fourth is never run. The third attempt returns to the initial degree 10, even though the preceding attempt used degree 12.

The configured acceptance limit is a residual gateResidual gateClose The configured largest permitted measured residual for a row. In this builder equality passes, and the first passing ladder attempt is retained. It is a sampled workflow condition, not a continuous or independent accuracy guarantee. . Body longitude and declination use 0.1 arcsecond; NodeOmega longitude uses 1 arcsecond. At exactly the limit, the row passes. For NodeOmega beginning at 8 days/degree 10, residuals 1.1″ and then exactly 1.0″ select 8 days/degree 12. The remaining 4-day attempts are skipped. A body longitude residual of 1.0″ would fail its smaller gate.

attempts = (S,d), (S,d+2), (S/2,d), (S/2,d+2)

row passes when measured residual ≤ configured gate

Here S is the initial segment length in days and d is the initial degree. Each attempt refits the entire series; the builder stops at the first passing result rather than seeking the smallest possible residual. A shorter segment length can also change the series’ actual fitted end.

The comparison is a direct ≤ test, not a separate finite-data audit. The teaching gate function below takes a supplied measured maximum; it does not reproduce every way the current accumulator handles nonfinite source values. The ladder and thresholds are engineering choices in this repository, not a published astronomical accuracy standard.

const ladder = (days: number, degree: number): [number, number][] =>
  [[days, degree], [days, degree + 2], [days / 2, degree], [days / 2, degree + 2]];
const passesMeasuredResidual = (residualArcsec: number, gateArcsec: number): boolean =>
  residualArcsec <= gateArcsec;