Add through-water motion and current vectors; inspect east/north components, COG and SOG with an explicit zero-leeway assumption.
navigation-sphere-vector/1.0.0
Current and ground-motion vector workspace
Add a vessel’s motion through water to the motion of that water over the ground. This forward-only teaching model assumes zero leeway, so heading equals the direction of through-water motion.
Method and symbols
Heading H is the direction the bow points. Through-water velocity describes translation relative to the water. They coincide here only because leeway and sideslip are set to zero.
Current set S is the direction TOWARD which the water moves over the ground. Current drift D is its speed. It does not use the meteorological “wind from” convention.
Ground velocity is v⃗g = v⃗w + v⃗c. Its magnitude is speed over ground (SOG); its direction is course over ground (COG). The model does not predict a future position.
All directions are true, measured clockwise from north. East and north are positive orthogonal axes; 000° = north, 090° = east. 360° is accepted as 000°. Speeds are knots: 1 kn = 1852/3600 m/s exactly. Signed COG − heading is clockwise-positive.
Add like components: Eg = 10 + 0 = 10 kn; Ng = 0 + 2 = 2 kn. A speed component is a speed, not a displacement.
SOG = √(10² + 2²) = √104 ≈ 10.198039 kn. COG = atan2(10, 2) ≈ 078.690068° true. The COG is 11.309932° to the left of heading in this example.
Convert speed: 10.198039 × 1852/3600 ≈ 5.246325 m/s. Current changes ground motion; it has not changed the specified heading or the 10 kn water speed.
Eg = 10 + (0) = 10 kn
Ng = 0 + (2) = 2 kn
SOG = √[(10)² + (2)²] = 10.198 kn
1 · Through water2 · Current3 · Over ground
Velocity diagram, not a track plot. Arrows show direction; zero vectors have no direction.
Velocity components in the local east/north frame
Quantity
Value
Unit
Through water · East
10
kn
Through water · North
0
kn
Current · East
0
kn
Current · North
2
kn
Over ground · East
10
kn
Over ground · North
2
kn
SOG
10.198
kn
SOG
5.24632
m/s
COG (true)
78.6901
° true
Signed COG − heading
-11.3099
°
Interpretation and limits
Constant vectors in one local horizontal plane. No wind/leeway, tides varying in space or time, acceleration, turning, sensor errors or Earth curvature.
At zero ground speed COG is undefined. A resultant no larger than 64 × machine epsilon × (V + D) is treated as numerical zero; raw components and the threshold are included in the CSV.
If V = 0, heading still describes bow orientation but does not define a water-motion direction. If D = 0, current set has no effect.
100 kn per input is an educational validation bound, not a vessel rating or a claim about plausible currents. There is no inverse course-to-steer solver, collision decision or operational steering advice.
Sources, method and version
Vector method: NOAA educational vector addition and NOAA current terminology. This page’s examples, implementation, bounds and tests are original; the NOAA graphic’s approximate numerical example is not reused.