Marine Reference

Marine Navigation Formulas

A practical reference for the formulas, assumptions, and verification checks behind ToolPatch marine route, tide, under-keel-clearance, drift, fuel, anchoring, CPA/TCPA, COLREGs, and turning-circle calculators.

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How to use this reference

Treat each formula as a planning aid with named assumptions. The calculation tells you what the entered scenario implies; it does not certify that the voyage, transit, anchorage, or manoeuvre is safe.

Choose the planning constraint first: route, depth, current, fuel, anchoring, traffic, or manoeuvring.
Open the matching calculator and keep the entered units, references, and assumptions with the result.
Verify consequential decisions against official charts, tide/current sources, weather, local notices, vessel data, instruments, COLREGs, and bridge-team procedures.

Route, ETA, Set, And Drift

Use these checks when building a passage baseline before weather, traffic, depth, and local routing constraints are applied.

Calculator	Formula or method	Use when	Verify before relying
Great Circle vs Rhumb Line Calculator	Great-circle distance uses the haversine relation; rhumb-line distance and bearing use the Mercator latitude dpsi relation.	Compare shortest-path geometry with constant-bearing route planning before briefing a longer leg.	Check coordinates, route practicality, hazards, traffic schemes, weather routing, currents, and port approaches.
Drift Estimator	V_ground = V_vessel + V_current + V_wind_drift; drift offset = environmental vector * duration.	Estimate course over ground, speed over ground, and accumulated set over a short planning interval.	Use set as direction of motion, keep true/magnetic conventions consistent, and compare with GPS track and current data.
ETA Current/Wind Compensation Calculator	Effective speed = calm speed - wind penalty speed - current component; ETA = distance / effective speed.	Update an arrival estimate when current or weather reduces speed made good.	Recheck observed speed, changing current, sea state, speed limits, traffic, tide gates, and pilot windows.

Depth, Tide, And Clearance

Use these formulas when water depth, tide timing, draft, squat, and explicit margins control whether a transit is plausible.

Calculator	Formula or method	Use when	Verify before relying
Under-Keel Clearance Calculator	Available water = charted depth + tide height; UKC = available water - (draft + squat + wave allowance + safety margin).	Check a draft-limited channel, harbour entry, bar crossing, berth approach, or shallow-water transit.	Confirm chart datum, latest surveys, tide predictions, vessel draft, squat, wave conditions, pilot requirements, and local policy.
Tide Window Planner	Required tide = max((draft + safety margin) - charted depth, 0); the window is approximated with a 12.42-hour semidiurnal tide curve.	Estimate how much of a tide cycle may provide enough water around high tide.	Use official tide tables, local corrections, time zone, surge, river flow, bar conditions, and harbour guidance.

Fuel, Anchoring, And Operating Margins

Use these calculations when reserve policy, anchorage setup, and contingency margins need to be visible in the plan.

Calculator	Formula or method	Use when	Verify before relying
Fuel Endurance + Reserve Calculator	Reserve fuel = usable fuel * reserve percent; endurance = trip fuel / burn rate; range = endurance * speed.	Separate reserve fuel from trip-usable fuel before comparing range with route and contingency needs.	Check measured burn, speed, sea state, current, load, generator or hotel loads, weather detours, and diversion options.
Anchor Scope & Holding Calculator	Required rode = (depth + bow height) * scope ratio; holding confidence is a heuristic from scope, wind, and bottom type.	Review whether deployed rode meets an intended scope target and whether local holding assumptions deserve more caution.	Check tide, swing room, seabed, anchor type, wind, gusts, current reversal, nearby hazards, alarms, and watchkeeping.

Traffic And Manoeuvring

Use these estimates for training, bridge-team briefings, and scale awareness before relying on live instruments or vessel-specific data.

Calculator	Formula or method	Use when	Verify before relying
CPA/TCPA Risk Calculator	TCPA = - (r dot v) / \|v\|^2; CPA = \|r + v * TCPA\|, with risk bands from CPA and TCPA thresholds.	Triage a target when bearing, range, course, and speed suggest a possible close-quarters situation.	Recheck whenever either vessel changes motion; cross-check radar, AIS, visual bearings, COLREGs, traffic, and restricted water.
COLREGs Encounter Classifier	Encounter class is inferred from simplified course and relative-bearing geometry.	Discuss head-on, crossing, overtaking, or uncertain encounter geometry during training or debriefing.	Apply full COLREGs context, lookout, visibility, vessel status, local rules, and all available observations.
Turning Circle / Advance / Transfer Estimator	Turning radius is approximated from vessel length, speed, and rudder angle; tactical diameter = 2 * radius, advance = 2.6 * radius, transfer = 1.5 * radius.	Frame manoeuvre scale before pilot exchange, simulator discussion, or restricted-water briefing.	Replace estimates with pilot card, wheelhouse poster, sea-trial, simulator, or observed manoeuvring data before relying on them.

Common cross-checks

Keep references consistent

Do not mix chart datum with another tide datum, true bearings with magnetic bearings, wind-from direction with set-toward direction, or speed over ground with speed through water without converting intentionally.

Rerun when the situation changes

Marine calculations can change quickly when speed, tide, current, draft, weather, traffic, sensor data, or vessel intent changes.

Keep margin visible

A positive result can still be operationally weak if the margin is small or the input data is uncertain.

Use the calculator, not only the formula

The linked tools preserve labels, units, examples, limitations, and related workflow links that are easy to lose in a hand calculation.