BC/Velocity-Band Transformation

This is how a bullet's measured ballistic coefficients at several velocities become a single drag curve the solver can fly. It matters because the alternative, picking one coefficient, is wrong at the ends of the speed range precisely where long-range shooting lives.

The problem with switching

The straightforward way to use several coefficients is to pick the one whose measured speed is nearest, and switch as the bullet accelerates. Do not do that. A coefficient that jumps produces a drag curve with a step in it, and a step in the drag curve is a step in the trajectory and in its derivative. The integrator would have to cope with a discontinuous right-hand side, and the solver's smoothness assumptions would no longer hold exactly where the bullet is moving fastest.

So the measurements are treated as samples of a smooth underlying function rather than as a lookup table. The bullet flies the smooth curve, and the measurements are merely where that curve is pinned.

What the caller supplies

A list of velocity and coefficient pairs, ordered by increasing velocity. Each velocity becomes a Mach number through the local speed of sound, so the same measurements work at different altitudes and temperatures. Each coefficient is normalized to the same atmosphere convention the library uses internally, so the numbers mean what they mean everywhere else in lob.

The measurements must be well-formed: strictly increasing velocity, positive and finite coefficients, and velocities inside the range the reference drag curve covers. Violations are rejected at build time rather than producing a curve nobody intended. Tables are not sorted for you, because silently reordering a caller's data hides a mistake rather than fixing it.

Building the curve

Two splines are constructed and then combined.

The retardation curve is the inverse of the bullet's coefficient against Mach, fitted through the measured pairs. Because the operation lob needs is a division of the reference curve, it is convenient to interpolate the reciprocal, and doing so keeps the arithmetic local to the spline.

The reference drag curve is the selected standard curve on the same knots the library always uses, so nothing about the representation changes when bands are supplied.

The bullet's curve is their product:

Cd(M) = S(M) * Cd_std(M)

Rather than evaluating two splines per step, the product is merged once, at build time, into a single spline. The merge preserves the derivative as well as the value at every knot, using the product rule, because the integrator's accuracy depends on the curve being smooth in its derivative as well as its height. The result is indistinguishable, to the solver, from any other drag curve it has ever been handed.

Why this interpolation

The fitted curve is a shape-preserving interpolant rather than a natural cubic spline. It passes exactly through each measured pair, which is what "the manufacturer measured this" demands, and between measurements it does not overshoot the way a natural spline does. Overshoot would invent drag that nobody measured, in the region between data points where the bullet is actually flying.

Outside the measured range the curve is held flat, at the value of the nearest measurement. A bullet leaves the muzzle slower than any published band and arrives slower than the fastest band, so extrapolation is not an edge case, it is most of the flight. Holding the edge value is the assumption that needs the fewest additional facts. A caller who knows better can supply additional bands near the ends and define the behavior explicitly.

The constant case

If every supplied coefficient is the same, the retardation curve is flat and the whole construction collapses to dividing the reference curve by one number, which is exactly what the single-coefficient path does. The tests assert the two paths agree, so a caller who happened to measure one bullet at several speeds gets the same answer as a caller who trusted a single number for it.

Limitations

Bands and a custom drag table are mutually exclusive. Supplying both discards whichever came first; lob does not blend a measured table with measured coefficients.
Extrapolation outside the measured range is flat, by choice, as described above.
The atmosphere convention applies to every band uniformly. There is no per-band atmosphere reference, which is not a limitation anyone has yet needed.
Coefficients are only as good as the velocities they were measured at. The transformation is exact at the measurements and interpolative between them, which is the best that any method can do with the data given.