BC Bands Worked Example

A concrete example of BC/Velocity-Band Transformation, using coefficients that are also used by the test suite so the result is checkable.

The measurements

A bullet measured at three velocities, each with a G7-referenced coefficient that rises as velocity increases (from 0.20 at 2000 fps to 0.40 at 3000 fps).

fps  ->  BC 0.20
fps  ->  BC 0.30
fps  ->  BC 0.40

Read the low coefficient at low speed correctly. A coefficient is a divisor, so the low-speed measurement means more drag than the reference bullet, not less. The bullet is losing more speed early and saving it late, which is what a low-drag superbullet does.

The resulting curve

Below the slowest band the curve is the reference divided by the first coefficient. Between bands it is the smooth interpolation. Above the fastest band it is the reference divided by the last coefficient.

Reference drag curve against the curve transformed from velocity bands

The visible features are the ones the transformation promises: the curve is smooth everywhere, with no kinks at the measured Mach numbers, and outside the measured range it runs parallel to the reference curve, shifted by the edge coefficient.

Building the context

#include <array>
#include <cassert>
#include <cstdint>
 
#include "lob/lob.hpp"
 
const float kFps[] = {2000.0F, 2500.0F, 3000.0F};
const float kBcs[] = {0.20F, 0.30F, 0.40F};
 
const lob::Context kCtx =
    lob::Builder()
        .BCAtmosphere(lob::AtmosphereReferenceT::kIcao)
        .BCDragFunction(lob::DragFunctionT::kG7)
        .DiameterInch(0.308)
        .MassGrains(168.0)
        .InitialVelocityFps(3100)
        .ZeroDistanceYds(100.0)
        .BCVelocityBands(kFps, kBcs, 3)
        .Build();
assert(kCtx.error == lob::ErrorT::kNone);
 
std::array<uint32_t, 4> kRanges = {300U, 600U, 900U, 1200U};
std::array<lob::Output, 4> kOutputs = {};
lob::Solve(kCtx, kRanges, kOutputs.data(), kRanges.size());

Note that the reference curve here is G7 rather than G1. The reference is independent of the transformation, and any of the standard curves can be transformed.

What changes relative to a single coefficient

The comparison worth making is against using the middle coefficient for the whole flight. That produces a proportionally scaled curve, which is the wrong shape rather than merely the wrong size.

Because the bands say the bullet is draggier than the reference at low speed and slipperier at high speed, the trajectory drops less early and more late than the single-coefficient baseline. At short range the bands-based solution is the flatter one; at long range it is the steeper one. A single coefficient that happens to match at one range is simultaneously optimistic at another, and the bands remove that trade-off rather than relocating it.

This is the entire reason to supply bands. Not a better constant, a better shape.

BC Bands Worked Example

The measurements

The resulting curve

Building the context

What changes relative to a single coefficient

Related