Rate of Turn Indicator (ROTI) Explained: ROT Formula and Wheel-Over Point.

Introduction

The rate of turn indicator (ROTI) shows the instantaneous rate of change of the ship’s heading. It is connected to the steering repeater, which feeds it at a rate of 60 to 200 pulses per minute.

The rate of turn (ROT) is measured in degrees per minute (°/min). As per the IMO standard, the equipment should be graduated with not less than 30° on either side, at an interval of 1°.

The ROTI is extensively useful while altering a course, especially in confined areas, congested waters and narrow channels.

Calculating the Rate of Turn

ROT can be calculated with this formula:

ROT = V / R

• V is the velocity of the ship
• R is the radius of the arc

Use V in knots and R in nautical miles to get ROT in degrees per minute. For example, at 12 knots on a turn of radius 1.5 nautical miles, ROT ≈ 12 / 1.5 = 8°/min.

Proof of ROT = V / R

Let the ship sail the arc DC of radius R while its heading changes by θ degrees. DC is the distance of the arc the ship sails along while altering course.

Step 1: Arc length. An arc equals the angle in radians multiplied by the radius. Since θ is in degrees and 1 radian = 57.3°:

DC = (θ / 57.3) × R

Step 2: Time taken. From the standard formula, distance = speed × time, so the time to sail the arc is:

t = DC / V = (θ × R) / (57.3 × V) (in hours)

Step 3: Convert to minutes. ROT is measured per minute, so multiply by 60. The factor 60 / 57.3 is close to 1, so it is taken as 1:

t = (θ × R × 60) / (57.3 × V) ≈ (θ × R) / V (in minutes)

Step 4: Rate of turn. ROT is the change of heading divided by the time taken:

ROT = θ / t = θ / (θ × R / V) = V / R

Hence ROT can be calculated by V / R. The exact value is 57.3 V / (60 R), or about 0.955 V / R, so V / R slightly overestimates the true rate, by about 5%.

Two Ways of Altering Course

• Altering with a constant radius turn
• Altering with a constant ROT

Constant Radius Turn

While altering the vessel, the ship’s speed drops, which proportionally affects the ROT. Hence more rudder movement has to be given to keep the ship moving along the arc of the circle.

Constant ROT

The speed is assumed to be constant, so the average speed is taken for the calculation. With the radius also constant, this gives a constant ROT.

Wheel-Over Point (WOP)

The radius of the arc determines the wheel-over point, along with the ship’s characteristics (the F distance).

The wheel-over point is the point where the rudder order is to be given to start the course alteration. The ship cannot take a sharp turn because of its moment of inertia, so the wheel order is given a few miles before the new course.

F Distance

The F distance is the distance covered between the wheel-over point and the point where the ship starts to turn. It depends on many factors such as the ship’s length, beam, speed, displacement and trim.

Calculating the WOP

The WOP can be found with two formulas:

1. Estimated WOP: R × tan (θ / 2) = x
2. Close to accurate: R (1 − cos θ) + F sin θ = T

Method 1 (estimated WOP):

• Measure x using R × tan (θ / 2).
• Cut x on the new course and on the current course.
• Apply the F distance from the mark to get the WOP.

Method 2 (close to accurate): T is the perpendicular distance between the new course and the WOP. From the WOP, apply the F distance, measure BD and cut BC. DC is the arc.

For example, with θ = 60° and R = 0.5 nautical miles, x = 0.5 × tan 30° ≈ 0.29 nautical miles. With an F distance of 0.1 nautical miles, T = 0.5 × (1 − cos 60°) + 0.1 × sin 60° ≈ 0.25 + 0.09 = 0.34 nautical miles.

Proof of x = R tan (θ / 2)

The turning arc touches the current course at D and the new course at C. Let O be the centre of the arc, and B the point where the two courses meet.

1. OD is at right angles to the current course, and OC is at right angles to the new course, so the angle DOC equals the course alteration θ.
2. By symmetry, OB cuts the angle DOC in half, so the angle DOB = θ / 2.
3. Triangle ODB has a right angle at D, with OD = R and BD = x.

tan (θ / 2) = BD / OD = x / R, so x = R × tan (θ / 2)

Proof of T = R(1 − cos θ) + F sin θ

T is split into two parts.

Part 1: the arc. The new course is at distance R from the centre O, because OC is at right angles to it. The point D lies at angle θ from OC, so along that direction D is only R cos θ from O. The distance of D from the new course is therefore:

R − R cos θ = R(1 − cos θ)

Part 2: the F distance. From the WOP to D the ship runs straight for F along the old course, which makes angle θ with the new course. The part of F measured at right angles to the new course is:

F sin θ

Adding the two parts gives the distance of the WOP from the new course:

T = R(1 − cos θ) + F sin θ

Limitations

• Use speed over ground, not speed through water.
• Achieve the required ROT quickly. Once it is achieved, bring the rudder to midship. As the ship turns its speed drops, and so does the ROT, so it would later be difficult to turn the ship further.
• In a constant radius turn, monitor the ground speed. As the ship’s speed drops, the rate of turn can be reduced correspondingly.

Conclusion

The rate of turn indicator turns a change of heading into a number the officer can plan with. Knowing that ROT = V / R, and how the radius and the F distance fix the wheel-over point, lets you start a turn at the right place and keep the ship on its planned track, especially in confined and congested waters.