ashfordeOU Table Interpolation (cross-cutting/numerics/interpolation)
Use when the task is estimating a value between the tabulated points of an aerospace data table: linear interpolation on one segment, piecewise linear interpolation over the whole table, the natural cubic spline through the data points, and boundary extrapolation.
Domain quick reference
- Linear interpolation on one segment: y = y0 + (y1 - y0) * (x - x0) / (x1 - x0). The result lies on the straight line through (x0, y0) and (x1, y1); at x = x0 it returns y0 and at x = x1 it returns y1. The same formula extends the line beyond the segment when x is outside [x0, x1].
- Piecewise linear table interpolation: locate the bracketing segment with a binary search, apply the linear formula on that segment only. The result is continuous and monotone between knots: it never overshoots the neighboring table values.
- Natural cubic spline: the piecewise cubic that passes through every knot, is twice differentiable at the knots, and has zero second derivative at both ends (m0 = m[n-1] = 0). The interior second derivatives m[1..n-2] solve a tridiagonal linear system, solved with the Thomas algorithm.
- Spline segment evaluation on [x[i], x[i+1]] with h = x[i+1] - x[i]: S(x) = m[i] * (x[i+1] - x)^3 / (6 h) + m[i+1] * (x - x[i])^3 / (6 h) + (y[i] / h - m[i] * h / 6) * (x[i+1] - x) + (y[i+1] / h - m[i+1] * h / 6) * (x - x[i]).
- Extrapolation: linear tables extend with the end segment slope; the spline extends with the end segment polynomial. Both require the explicit extrapolate flag, and both degrade quickly far from the table, the spline faster than the line.
- Table requirements: at least 2 points, xs and ys of equal length, every value finite, xs strictly increasing. Tables of lift coefficient versus angle of attack, drag polar points, and atmosphere profiles are the classic aerospace use.
- Method choice: linear interpolation preserves monotonicity and never oscillates; the natural spline is smoother but can overshoot between knots, and it does not reproduce a quadratic function exactly because its end second derivatives are forced to zero.
- Sanity checks: the interpolant reproduces the table exactly at the knots; linear and spline results agree at the knots and stay close in the interior for well-behaved tables.
Workflow
- Validate the table with validate_table: at least 2 points, equal lengths, finite values, strictly increasing xs. A sorted table is the precondition for the binary search.
- Choose the method: piecewise linear when monotonicity matters or the data are noisy; the natural cubic spline when a smooth curve through the knots is wanted.
- For one segment, interpolate with linear_interpolate(x, x0, y0, x1, y1); for a whole table use interpolate_linear(xs, ys, x).
- For the spline, either call interpolate_cubic(xs, ys, x) directly or split the steps: natural_cubic_spline_coefficients to get the second derivatives, then cubic_spline_evaluate(xs, ys, m, x).
- Handle the boundaries: pass extrapolate=True only when a value beyond the table ends is genuinely needed, and prefer linear extrapolation over spline extrapolation far outside the table.
- Verify the result: the value at any knot equals the table value, and for interior points the linear and spline results should agree closely on smooth data before the lookup result is used.
Pitfalls
- Querying outside the table without the extrapolate flag: both interpolate_linear and cubic_spline_evaluate raise ValueError; pass extrapolate=True deliberately, never to silence the error.
- Feeding an unsorted table: the binary search returns the wrong segment and the value is silently wrong; validate_table raises on non-increasing xs.
- Trusting spline overshoot: the natural spline can exceed the local table range between knots, especially near steep steps; linear interpolation never does. Check the spline value against the neighboring knots before using it.
- Expecting the natural spline to reproduce smooth functions: it reproduces the knots exactly but a quadratic sampled at 4 points gives interior second derivatives of 2.4, not 2; do not demand exactness away from the knots.
- Extrapolating far beyond the table: a cubic continues to grow or fall steeply; linear extension with the end slope is the safer boundary behavior for design margins.
- Degenerate segments: a repeated x value divides by zero in the linear formula; validate_table rejects repeated abscissas.
- Confusing this leaf with finite-difference-derivatives: derivatives of tabulated data belong to that leaf, interpolated values between tabulated points belong here.
- Confusing interpolation with least-squares-regression: regression fits a model to scattered data, interpolation passes exactly through the given points.
Behavior contract (gate 3)
The interpolation math is exercised by the gate 3 contract test: scripts/test_interpolation.py against scripts/interpolation_logic.py (stdlib unittest, offline, 42 cases). Run:
python3 scripts/test_interpolation.py
Compliance
- NACA Report 824 (Summary of Airfoil Data) is US government work, public domain, and the pack anchor for tabulated aerodynamic data per standards-map.yaml; it is referenced, not reproduced. Table interpolation is generic numerical methodology, summary and formulas only.
- compliance: STANDARDS-REF, gated: false.