Navigation Course - Route and Distance Calculation

Introduction

• Topic: Rhumb lines at short distances
• Objective: calculate the course, route, and nautical miles between two points.

Coordinate Transformation

Starting and Ending Point

• Point A: φ of A = 32°12' N; λ of A = 3°24' E
• Point B: φ of B = 36°54' N; λ of B = -2°36' W

Conversion of Coordinates

• North and South become + or - respectively.
• East and West become + or - respectively.

Calculation of Latitude Difference (Δφ)

• Δφ = φ of B - φ of A
• Example: 36°54' - 32°12' = 4°42' (or 282 minutes)
• Direction: North

Calculation of Longitude Difference (Δλ)

• Δλ = λ of B - λ of A
• Example: -2°36' - 3°24' = -6° (or -360 minutes)
• Direction: West

Calculation of Mean Latitude (φ mean)

• φ mean = (φ of A + φ of B) / 2
• Example: (32°12' + 36°54') / 2 = 34°33'

Calculation of True Course

Formula

• tan(χ) = (Δλ * cos(φ mean)) / Δφ
• χ = arctan(calculated value)
• True Course = 360° - χ
• Example: χ = 46°43', so True Course = 313°57'*

Calculation of Route in Nautical Miles

Formula

• m = Δφ / cos(χ)
• Example: 282 minutes / cos(313°57') = 409.14 NM

Calculation of Estimated Time of Arrival (ETA)

• Process: subtract 24 hours to calculate the correct day and time.
• Example: Departure on the 22nd at 21:35, ETA = 24th at 00:52 (after 2 days).

Second Exercise (Known Point and Route)

• Similar calculation but starting from a known point and a given course.
• Speed: 18.9 kn, Route: 390 NM.
• Calculation of the ETA and final coordinates.

Calculation of Latitude Difference (Δφ) for the Second Point

• Δφ = M * cos(χ)
• Result: -5°53'28"
• Newly calculated latitude: 31°00.5'*

Calculation of Longitude Difference (Δλ) for the Second Point

• Δλ = M * sin(χ) / cos(φ mean)
• Result: 3°18.7'
• Newly calculated longitude: 0°42.7' E*

Common Errors

• Confusion between degrees and minutes.
• Errors in switching between degrees and minutes during calculations.

Conclusion

• Completed the calculation of routes and distances using transformed coordinates, calculator, and trigonometric formulas.
• Correct interpretation of signs and conversion between degrees and minutes are fundamental for accuracy.