Cubic Hermite Splines

Every drag curve in lob is stored as a piecewise cubic Hermite spline fitted with shape-preserving tangents. That one choice is what makes the solver fast, and the shape-preserving part is what makes it safe.

Why a spline at all

A drag curve is a function the integrator evaluates hundreds of times per trajectory. Whatever representation is chosen has to be cheap to evaluate and cheap to locate a point on. A lookup table means a search and a branch on every evaluation. The spline inverts that: the curve is reduced ahead of time to a fixed set of knots and a small block of coefficients, and evaluating it is arithmetic with no searching at all.

The second requirement is less obvious. The integrator's accuracy depends on the curve being smooth, not merely continuous. A curve with a kink in it puts a kink in the trajectory, and an integrator taking fixed steps cannot resolve a kink exactly however small the step becomes. So the representation has to preserve smoothness.

What the representation is

The curve is divided into intervals at the knots. Within each interval the curve is a cubic, described in local coordinates by the value and slope at the interval's two endpoints. Specifying value and slope at both ends of every interval is enough to pin down a cubic, which is why this is called a Hermite spline: it interpolates the values it is given and lets the slopes decide the curvature in between.

What makes lob's version special is how those slopes are chosen.

The tangent rule

The naive choice for the slope at a knot is an average of the data either side of it. That produces a smooth curve, and it also produces curves that swing wildly between closely spaced data points: a natural cubic spline famously overshoots, inventing values higher than anything in the data. For a drag curve that is not a cosmetic problem. An invented peak in drag between two measured points is a peak the projectile actually flies through.

lob fits tangents with the Fritsch-Carlson rules instead, and the decision tree is the whole design:

At the ends of the curve, the tangent is the slope of the adjacent span.
Inside, if the data slopes one way on the left and the other way on the right, the knot is a local extreme and the tangent is zero. A horizontal tangent at a peak or trough cannot overshoot, because the curve flattens there instead of turning sharply.
Everywhere else, the tangent is a weighted average of the two neighboring slopes, taken as a harmonic mean and weighted toward the closer neighbor.

The result is that a curve passing through monotonic data stays monotonic, and a curve through a peak does not overshoot it. Every drag curve in lob, from the compiled-in standard curves through to a custom table and a velocity-band transformation, is fitted with these same rules.

Knots

The knots are a fixed, deliberately non-uniform set spanning the Mach range the reference curves cover, stored alongside the coefficient tables so the standard curves need no construction at runtime at all.

They are clustered where the curves are hardest to represent, which is through the transonic region where drag turns sharply, and spread out where the curve is smooth and easy to follow. The placement was chosen by measuring, not by taste: candidate knot sets were scored against the full reference tables, and the set that shipped is the one that kept the worst-case deviation across every standard curve within the accuracy budget the solver is validated to. Since the knots are fixed, any incoming table can be resampled onto them without re-deciding where they should go, so a caller's custom data lands on the same footing as the standard curves.

Evaluation

Evaluation holds a view over the knot and coefficient arrays along with the index of the interval it last used. A new Mach number is resolved by moving that index, forward or backward, until the Mach number falls inside the interval it points at.

This is worth a second look, because it is not a binary search. The projectile's position progresses along the curve in one direction, so the cached index walks the knots the same way the projectile does and finding the right interval costs a single comparison in the overwhelming majority of steps. Mach may vary during flight (not necessarily moving monotonically along the Mach axis in all phases), so movement along the curve is the relevant invariant for the cached index. The rare search, such as the transonic hunt during angle finding, is where the index has to move more.

Once the interval is known, evaluating is a polynomial in local coordinates using that interval's four coefficients, and the derivative is another. Nothing is searched, nothing is allocated, and the whole curve is a small contiguous block of numbers.

Why not linear interpolation

Linear interpolation between samples is simpler still, and lob keeps an implementation of it around for comparison. It loses on smoothness. A linear segment is continuous but its slope jumps at every knot, so the curve is not differentiable there, and the derivative the solver uses does not exist at those points. The Hermite form is continuously differentiable everywhere, which is the property the integrator is actually relying on.

What backs it

The spline tests check the tangent rules against their definitions, that a curve passes through its input values with the intended slopes, that interval location finds the right segment in both directions, that the cached index behaves when the search reverses, and that the derivative matches the curve.

Two of them check the budget rather than the machinery. One confirms the fixed knots keep every standard curve within its accuracy budget across the whole Mach range. The other measures what that budget is worth: it perturbs the drag coefficient across the transonic band and measures the effect on the resulting trajectory, showing the trajectory impact of the residual curve error is small enough to sit below the solver's own noise. That is a claim about impact, not about curve shape, which is the quantity that actually matters to a caller.