15/08/2026
⚓ LESSON 136 — GREAT CIRCLE INTERMEDIATE POINTS
Turning a Great Circle Into Practical Waypoints
A Great Circle is the shortest path between two positions on a sphere.
But a vessel cannot simply steer one constant course from departure to destination because the Great Circle course normally changes continuously.
So how do navigators make the route practical?
📍 By calculating intermediate points along the Great Circle.
🧭 WHAT ARE INTERMEDIATE POINTS?
Intermediate points are selected positions lying on the same Great Circle between departure and destination.
They may be calculated at selected:
Longitudes → find the corresponding latitude
or
Latitudes → find the corresponding longitude
These positions can then be plotted as waypoints.
🌍 WHY DO WE NEED THEM?
Instead of trying to follow a continuously changing theoretical course, the navigator can represent the Great Circle with a series of manageable route legs:
Departure → WP1 → WP2 → WP3 → WP4 → Destination
The more appropriately spaced the waypoints are, the closer the practical route approximates the intended Great Circle.
⚠️ However, waypoint spacing must also consider navigational safety and operational requirements—not geometry alone.
📐 USING THE VERTEX
Once the Great Circle vertex is known, intermediate points can be calculated using the relationship between the vertex and another point on the same Great Circle.
A useful Great Circle relationship is:
tan Lat = tan Lat V × cos DLoᵥ₋ₚ
Where:
Lat = latitude of the intermediate point
Lat V = latitude of the vertex
DLoᵥ₋ₚ = longitude difference between the vertex and the intermediate point
🧮 EXAMPLE
From Lesson 135:
⭐ Vertex ≈ 40°39′N, 072°15′W
Suppose we want the Great Circle latitude at:
Longitude = 090°00′W
Difference of longitude from the vertex:
DLo = 90°00′ − 72°15′
= 17°45′
Then:
tan Lat = tan 40°39′ × cos 17°45′
This gives approximately:
📍 Intermediate Point ≈ 39°16′N, 090°00′W
That point lies on the same theoretical Great Circle.
🗺️ BUILDING THE ROUTE
Repeat the calculation for several selected longitudes:
120°W → Departure
110°W → WP1
100°W → WP2
90°W → WP3
80°W → WP4
72°15′W → Vertex
60°W → Destination
The calculated latitudes will trace the Great Circle toward the vertex and then away from it.
⚠️ IMPORTANT
Calculated intermediate points are geometrical positions.
Before using them in an actual passage plan, the navigator must check:
🌊 Weather and sea conditions
🗺️ Charts and navigational hazards
📏 UKC and safety margins
🚢 Traffic and routing systems
🧊 Ice or seasonal restrictions
🚫 No-go areas and company requirements
A mathematically shortest route is not automatically a safe navigational route.
❌ COMMON MISTAKES
❌ Connecting departure and destination with one constant-course assumption.
❌ Using the wrong DLo from the vertex.
❌ Mixing E/W longitude signs.
❌ Rounding intermediate calculations too early.
❌ Assuming calculated GC points can automatically be entered into the passage plan without safety assessment.
🧠 REMEMBER
Great Circle mathematics gives the theoretical route.
Intermediate points make that route practical to plot.
Passage planning makes it safe to navigate.
⚓ MARITIME JOURNEY PH REMINDER
Calculate → Plot → Assess → Verify → Navigate
