3.1 Course to Steer (CTS) Single-Hour Vector Triangles

3.1 Course to Steer (CTS) Single-Hour Vector Triangles

Navigating across tidal waters requires reconciling two independent frames of reference: water moving over the seabed (tidal stream) and the vessel moving through the water (propulsion). Aiming the bow directly at a distant waypoint ignores tidal drift, causing the yacht to trace a curved, inefficient path over the seabed. The single-hour Course to Steer vector triangle provides the geometric solution to maintain a straight ground track.

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Vector Conventions and Notation

Admiralty chartwork standards enforce distinct arrow notation for the triangle of velocities:

1. Ground Track (Track to Make Good): Marked with two arrowheads (>>). The planned path over the seabed from departure Waypoint A toward destination Waypoint B. Its speed is the Speed Over Ground (SOG).
2. Tidal Stream Vector: Marked with three arrowheads (>>>). The horizontal movement of water. Its direction is the Set (°True) and its magnitude is the Drift (hourly rate in knots).
3. Water Track (Course Through Water): Marked with one arrowhead (>). The vessel's direction of travel and Speed Through Water (STW), directed from the tidal-vector tip to intersect the ground track. Allowing for leeway converts this direction into the bow heading.

$\text{Water Track (1 arrow)} + \text{Tidal Stream (3 arrows)} = \text{Ground Track (2 arrows)}$

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Step-by-Step Vector Triangle Construction

To construct a 1-hour Course to Steer vector triangle on a nautical chart, execute five standardized steps:

Step 1: Draw the Ground Track

Plot departure Point A and destination Point B. Draw a straight line connecting them, measure its True direction, and mark it with two arrowheads pointing toward Point B.

Step 2: Lay Off the Tidal Stream Vector from Departure

From the tidal diamond or atlas for the passage hour, find the Set (°True) and Drift (knots). From Point A, draw a line in the direction of the Set, measuring 1 hour of tidal drift (e.g., 2.4 NM for 2.4 knots). Label the end Point C and mark with three arrowheads pointing from A to C.

Step 3: Swing the Boat Speed Arc

Set dividers to 1 hour of planned Speed Through Water (STW) on the latitude scale (e.g., 6.0 NM). Place the compass point at Point C (tide tip) and swing an arc intersecting the Ground Track at Point D.

Step 4: Draw the Water Track

Draw a straight line from Point C to Point D. Mark it with one arrowhead pointing from C to D. The direction of line CD is the True Water Track.

Step 5: Measure Speed Over Ground (SOG)

Measure the distance along the ground track from Point A to Point D. Distance AD equals Speed Over Ground (SOG) in knots. Dividing total distance AB by this SOG yields the Estimated Time of Arrival (ETA).

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Applying Wind Leeway Allowance

Leeway is downwind drift caused by wind pressure against the hull, sails, and topsides. To prevent leeway from pushing the yacht off the water track, the helmsman must steer into the wind:

• Wind on Port Side (Port Tack): Wind pushes the yacht to starboard. Steer upwind to port by subtracting leeway: $\text{True Heading} = \text{Water Track} - \text{Leeway}$.
• Wind on Starboard Side (Starboard Tack): Wind pushes the yacht to port. Steer upwind to starboard by adding leeway: $\text{True Heading} = \text{Water Track} + \text{Leeway}$.

Leeway depends on the vessel, wind and sea conditions. Use an appropriate estimate for the exercise or vessel, then check the resulting ground track underway.

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Converting True Course to Compass Course (CADET)

Convert True Heading to the steering compass heading by correcting for Magnetic Variation and Compass Deviation:

$\text{True (T)} \longleftrightarrow \text{Variation} \longleftrightarrow \text{Magnetic (M)} \longleftrightarrow \text{Deviation} \longleftrightarrow \text{Compass (C)}$

• Compass to True: Add East, Subtract West (CADET: Compass to True, Add East).
• True to Compass (Reversed): Add West, Subtract East.
• Remember: "Variation West, Magnetic Best; Variation East, Magnetic Least."

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Worked Numerical Example

Scenario Data

• Departure (A): Alpha Buoy (50°30.0'N, 001°40.0'W)
• Destination (B): Bravo Fairway Buoy (090° True, Distance 12.0 NM)
• Yacht STW: 6.0 knots
• Tidal Stream: Assume set 140°T at 2.4 knots remains constant for the entire passage, beginning at 1000 UT
• Wind: Northerly (from 000°), causing 5° leeway
• Variation: 2° West (charted)
• Deviation Card: On heading 069°(M), Deviation is 3° East

Solution Steps

1. Ground Track: Draw line 090°(T) from A toward B (2 arrows).
2. Tide Vector (AC): From A, lay off 2.4 NM on bearing 140°(T) to Point C (3 arrows).
3. Boat Speed Arc: From C, swing 6.0 NM arc across the 090° track to intersect at Point D.
4. Water Track (CD): Direction from C toward D measures 072°(T) (1 arrow).
5. Determine SOG: Distance AD measures 7.25 NM, giving SOG = 7.25 knots. $\text{ETA} = \frac{12.0\text{ NM}}{7.25\text{ knots}} = 1.655\text{ hr} \approx 1\text{ hr } 39\text{ min}$
6. Apply Leeway: Wind from 000° strikes port side. Steer 5° into wind (to port): $\text{True Heading} = 072^\circ\text{(T)} - 5^\circ = 067^\circ\text{(T)}$
7. Apply Variation: Variation is 2° West (Add West): $\text{Magnetic Heading} = 067^\circ\text{(T)} + 2^\circ\text{W} = 069^\circ\text{(M)}$
8. Apply Deviation: Deviation on 069°(M) is 3° East (Subtract East): $\text{Compass Course to Steer (CTS)} = 069^\circ\text{(M)} - 3^\circ\text{E} = 066^\circ\text{(C)}$

Order to Helmsman: "Steer 066° on the steering compass."