` = "Rhumb Line Azimuth"`

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The Rhumb Line Azimuth calculator computes the azimuth angle for a rhumb line between to latitude / longitude coordinates.

INSTRUCTIONS: Choose units and enter the following:

• (LT1) Latitude One
• (LN1) Longitude One
• (LT2) Latitude Two
• (LN2) Longitude Two

Rhumb Line Azimuth (Az): The azimuth is returned in both decimal degrees and degrees, minutes, seconds.

The Math / Science

In sailing, azimuth refers to the angle between a fixed reference direction (typically true north) and the direction of an object or point of interest on the horizon, measured clockwise around the observer’s horizon. This measurement is usually given in degrees, from 0° to 360°.

Heading, on the other hand, is the direction in which the vessel's bow is pointed, also measured in degrees relative to true or magnetic north. It indicates the direction of travel but does not necessarily correspond to the actual path due to wind, current, or drift.

Difference between Azimuth and Heading:

• Azimuth: Measures the direction of a specific object or location relative to true north from the observer's position.
• Heading: The direction the boat is currently pointed, relative to true or magnetic north.

So, while the azimuth represents the angle between north and the object you're observing, the heading refers to where your boat is actually pointing or moving. These two angles can differ, especially if you're navigating towards a distant point that's not directly ahead.

A rhumb line, also known as a loxodrome, is a path on the surface of a sphere (such as the Earth) that crosses all meridians of longitude at a constant angle. Unlike a great circle route, which is the shortest path between two points on the globe, a rhumb line maintains a constant bearing (direction) and is easier to navigate using a compass.

Key Characteristics of a Rhumb Line:

1. Constant Bearing: A rhumb line crosses each meridian at the same angle, making it a line of constant compass direction.
2. Ease of Navigation: Because the bearing remains constant, it's easier to follow using traditional navigation techniques, such as a compass, without continuously adjusting the course.
3. Non-Shortest Path: On a spherical surface, a rhumb line is not the shortest distance between two points. The shortest path is a segment of a great circle.
4. Spiraling towards the Poles: On a Mercator projection map, which is commonly used for navigation, rhumb lines appear as straight lines. However, on a spherical surface, rhumb lines spiral towards the poles.

Practical Usage:

• Marine and Air Navigation: Rhumb lines are particularly useful in marine and air navigation for courses that need to be followed over long distances without changing direction.
• Mercator Projection: On a Mercator projection map, rhumb lines are represented as straight lines, which simplifies the navigation process.

Example:

If a ship were to sail from London to New York City along a rhumb line, it would maintain a constant compass direction throughout the journey, even though this path is longer than the great circle route.

Algorithm 

The rhumb line (loxodrome) azimuth between two points on the globe, you can follow these steps:

1. Convert Latitude and Longitude to Radians:
   • ϕ₁ and λ₁​ are the latitude and longitude of the first point in radians.
   • ϕ₂ and λ_2​ are the latitude and longitude of the second point in radians.
2. Calculate the Change in Longitude:
   • Δλ=λ₂−λ₁
3. Calculate the Mean Latitude:
   • ϕ_m=(ϕ₁+ϕ₂)/2​​
4. Calculate the Change in Latitude:
   • Δϕ=ϕ2−ϕ1
5. Calculate the Meridional Parts for Both Latitudes:
   • The meridional part, M(ϕ), can be calculated using the formula:
     • M(ϕ)=ln⁡(tan⁡(π/4+ϕ/2))
   • Calculate M(ϕ1) and M(ϕ2):
     • M(ϕ1)=ln⁡(tan⁡(π/4+ϕ1/2))
     • M(ϕ2)=ln⁡(tan⁡(π/4+ϕ2/2))
6. Calculate the Change in Meridional Parts:
   • ΔM=M(ϕ2)−M(ϕ1)
7. Calculate the Rhumb Line Azimuth (bearing):
   • The azimuth α\alphaα can be calculated using the formula:
     • α = arctan⁡( Δλ / ΔM )
8. Adjust the Azimuth:
   • Ensure the azimuth is in the correct quadrant by adjusting for the signs of Δλ\Delta \lambdaΔλ and ΔM\Delta MΔM.
   • If Δλ>0 and ΔM>0, the azimuth α\alphaα is in the first quadrant.
   • If Δλ>0 and ΔM<0, the azimuth α\alphaα is in the fourth quadrant.
   • If Δλ<0 and ΔM>0, the azimuth α\alphaα is in the second quadrant.
   • If Δλ<0 and ΔM<0, the azimuth α\alphaα is in the third quadrant.

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Sailing and Navigation Calculators:

• Bruce Number: Computes the ratio of the square root of sail area to the cube root of displacement of a sail boat
• Sail Area / Displacement Ratio: Compute SA/D ratio for sail boats.
• Displacement-Length Ratio: Computes a metric to describe how heavy a boat is in relation to its waterline.
• Capsize Screening Formula: Computes a metric to describe a boat's stability in rough seas.
• Vessel Displacement Volume: Computes the cubic feet displacement associated with the weight displacement.
• Weight of Seawater: Computes the weight and mass of a volume of seawater.
• Time to Sink: Compute the time it takes for a breach (water in) to surpass the displacement of the boat.
• Haversine Distance (decimal degrees): Compute the Distance Between two Points on the Earth (Great Circle Arc)
• Haversine Distance (Degrees, Minutes and Seconds): Compute the Distance Between two Points on a Sphere (Great Circle Arc) using degrees, minutes and seconds verses decimal degrees.
• Rhumb Line Azimuth: Computes the azimuth heading one can navigate for a path that crosses all meridians of longitude at a constant angle.
• Rhumb Line Distance: Computes the distance traveled between two points on the globe if traveled via a rhumb line.
• Correction Angle: navigation equation that compensates for air or water currents.
• Decimal Degrees: Compute decimal degree angles from degrees, minutes and seconds,[CLICK HERE].
• Nautical Travel Cost: Computes the total cost to travel a distance based on the cost rate and distance traveled.
• Travel Time between Coordinates: Compute the time to travel between two location (latitudes and longitudes) based on a average speed
• Distance to Sea Level Horizon: Computes the distance to the horizon from a specified height using a spherical model the mean spherical radius of the Earth

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Navigation Calculators:

• Haversine Distance (decimal degrees): Compute the Distance Between two Points on the Earth (Great Circle Arc)
• Haversine Distance (Degrees, Minutes and Seconds): Compute the Distance Between two Points on a Sphere (Great Circle Arc) using degrees, minutes and seconds verses decimal degrees.
• Rhumb Line Azimuth: Computes the azimuth heading one can navigate for a path that crosses all meridians of longitude at a constant angle.
• Rhumb Line Distance: Computes the distance traveled between two points on the globe if traveled via a rhumb line.
• Travel Time between Coordinates: Compute the time to travel between two location (latitudes and longitudes) based on a average speed.
• Navigation Speed: Computes the required average speed needed to go between coordinates on a great circle arc within a given amount of time.
• Distance to Sea Level Horizon: Computes the distance to the sea horizon at an altitude
• Correction Angle: navigation equation that compensates for air or water currents.
• Distance to Sea Level Horizon: Compute the distance to the horizon based on a height above the orb using a spherical Earth model.
• Decimal Degrees: Compute decimal degree angles from degrees, minutes and seconds,[CLICK HERE].