// RESEARCH · MATH AA IA · INTERACTIVE MODEL

The search spiral.

A maritime rescue drone has to search the most likely water first, while the water itself drifts, the wind pushes it sideways, and the battery counts down. This is the model from my Math IA — an Archimedean spiral re-parametrised so the drone holds a constant airspeed — running live. Drag sliders, pick a scenario, scrub the timeline.

The math ↓

STANDBY On the pad

T+00:00

HDG000

BANK 0° PITCH 0° CRAB 0°

100%

reserveRTH needsearch budget

Flight left: —
RTH needs: —
Search left: —
Lands with: —
Still to cover: —

AS: —
GS: —
TRK: —
POD: —
ALT: —
WIND H/X: —
HOME: —
DATUM: —
SEARCHED: —

Drag ⌂ home or ◇ LKP to move them · scroll to zoom · drag to pan · double-click to reset

// SAME DRONE · SAME INTERCEPT · SAME BATTERY

How the IA spiral stacks up

Probability of detection over the search — the higher and earlier, the better

Distance from the datum while searching — likeliest water first means a steady climb (IA Fig. 5)

IA spiral

Naive

Ground

Square

Sector

Pattern	Search score	Order	POD at bingo	50% POD	Targets

Every pattern starts at the IA spiral's intercept and leaves at its own bingo (elapsed + RTH ≥ usable endurance). POD is scored in the water's frame against a Gaussian of spread σ around the drifting datum. The three spirals are the same equation solved in three frames: the water's (IA), the ground's (GPS-locked, ignores current) and the air's (ignores both).

00:00 / 00:00

Standby Intercept Spiral search Return to home

Scenarios

Contacts

// THE MODEL, IN FIVE LINES

From a spiral to a flight plan

An Archimedean spiral, pitched by the camera

Arms 2Rs apart leave no gaps and no overlap, so the pitch is fixed by the scan radius.

r (θ) = r_{0} + \frac{R_{s}}{π} θ

Centred on a drifting datum

The survivor moves with the water, so the spiral does too.

P (t) = C_{0} + v_{c} t + r (θ) [\begin{matrix} \cos θ \\ \sin θ \end{matrix}]

Hold the airspeed constant

The drone's speed through the air is fixed, whatever the wind and current are doing.

‖ θ^{'} w (θ) + v_{eff} ‖ = U, v_{eff} = v_{c} - v_{wind}

…which is a quadratic in θ′

Take the + root. Because |v_eff| < U makes AC < 0, √Δ > |B|: it is always the positive one.

θ^{'} = \frac{- B + \sqrt{B^{2} - 4 A C}}{2 A}

A = r^{2} + b^{2}, B = 2 w \cdot v_{eff}, C = {‖ v_{eff} ‖}^{2} - U^{2}

Straight legs: intercept & return

Out to the moving spiral start S(t) and back home, each at airspeed U. Smallest positive τ of:

‖ D + v_{rel} τ ‖ = U τ \Rightarrow τ = \frac{D \cdot v_{rel} + \sqrt{{(D \cdot v_{rel})}^{2} + (U^{2} - {‖ v_{rel} ‖}^{2}) {‖ D ‖}^{2}}}{U^{2} - {‖ v_{rel} ‖}^{2}}

The battery decides when the spiral ends

Rather than solving T = (t₁ − t_delay) + (t₂ − t₁) + (t₃ − t₂) for θ(t₂) after the fact, the simulation checks it live: the drone leaves the spiral the instant elapsed flight time + RTH time from here reaches the usable endurance. That's exactly an autopilot's battery “bingo” — and it lands with the reserve you set, to the second.

Refinements since the submitted paper

Straight legs. The paper wrote the intercept and return times as a vector divided by a vector, (C₀ + v_ct₁ − H)/(v_air + v_wind). Here each leg is the interception quadratic above (closed form), which also gives the heading to hold.
Aim point. The intercept targets the spiral's first point C(t₁) + (r₀, 0), not the datum itself — it only matters when r₀ ≠ 0.
Root choice. Choosing the + root isn't just “realistic”: whenever |v_eff| < U it is provably the only positive root. When |v_eff| ≥ U no constant-airspeed spiral exists — the model now says so.
Autonomous ODE. θ′ depends on θ alone, so t(θ) is a plain integral ∫dθ/θ′(θ). No special ODE solver is needed; RK4 is used here only for even time-sampling.
Appendix code. The Streamlit prototype integrated θ′ = U/√(r² + b²) (the still-air formula) with the current bolted on afterwards, so its airspeed wasn't actually constant; its spiral was also centred v_c·t₁ behind the datum. This page integrates the full quadratic.
w(θ). In §4.2 the line w(θ) = r′(θ)² + r(θ)·θ′ is a typo for the vector w(θ) = [r′cosθ − r sinθ, r′sinθ + r cosθ], with ‖w‖² = r² + r′². A(θ) was already right.