The polar stereographic projection
The polar stereographic is a true perspective projection designed around a pole. It gives a useful, almost constant-scale chart for high-latitude and polar navigation.
How it is constructed
A plane touches the reduced Earth at the pole being charted. For a North Polar chart, the projection point is the South Pole. Straight rays from that opposite pole pass through the graticule and meet the tangent plane. The result can show one hemisphere or, if extended, more than one hemisphere.
| North Polar chart element | Location |
|---|---|
| Plane of projection | Tangent at the North Pole |
| Projection point | South Pole, diametrically opposite |
| Meridians | Straight lines radiating from the North Pole |
| Parallels | Concentric circles centred on the North Pole |
Graticule, scale and conformality
The chart is conformal and its scale is correct at the pole. Both scale and the spacing of parallels increase with distance from the pole.
Co-latitude
Co-latitude is 90 degrees minus latitude. At 70 degrees north the co-latitude is 20 degrees. The distance along a meridian from the North Pole to 70 degrees north is 20 × 60 = 1,200 NM.
Worked scale checks
At 78 degrees north, co-latitude is 12 degrees and half co-latitude is 6 degrees. Sec² 6 degrees is about 1.011, so a polar scale of 1:1,000,000 becomes approximately 1:989,000. Scale remains within about 1 percent of polar scale between 90 and 78 degrees north. Between 78 and 70 degrees north, the error grows from about 1 to 3 percent.
| Latitude band | Scale behaviour | Operational reading |
|---|---|---|
| 90° to 78° | Within about 1% of polar scale | Practically constant scale |
| 78° to 70° | About 1% to 3% expansion | Use local scale with care |
| Below 70° | Expansion increases | Projection becomes less attractive |
Why it is conformal
Meridians and parallels cross at 90 degrees, and scale expands equally in every direction at a point. Local angles and small shapes are therefore correct. Large shapes and areas become increasingly distorted away from the pole.
Convergence and route shapes
Every meridian reaches the pole as a straight radial line, so polar stereographic chart convergence has a factor of exactly one.
Where convergence is correct
Chart convergence between two selected meridians is constant across the chart. It equals their longitude difference everywhere, but matches Earth convergency exactly only at the pole. Earth convergency decreases with decreasing latitude.
Great circles and rhumb lines
Both are curves concave to the pole of projection, except that meridians are straight. A rhumb line has the greater curvature. Great-circle curvature becomes very small near the pole, so at latitudes above about 70 degrees a straight line is normally taken as the great-circle route for practical plotting.
Worked route at 75 north
Between A at 75 north, 60 west and B at 75 north, 60 east, chart convergence is 120 degrees. The triangle formed with the pole is isosceles, so each base angle is 30 degrees. The initial straight-line track from A to B is 030 degrees true and the final track is 150 degrees true.
Why grid navigation is needed
At high latitude, true north changes direction rapidly as the aircraft crosses meridians. A constant true heading therefore does not remain aligned with one straight chart route.
The true-north problem
All meridians meet at the pole. Moving east or west across high latitudes rotates the local direction of true north. A straight near-great-circle route on the chart consequently has a true track that changes continuously. Frequent true-heading corrections become large and awkward close to the pole.
A fixed chart datum
Grid navigation replaces changing local true north with a set of parallel grid-north lines. One chosen datum meridian defines Grid North. Every grid line is parallel to that datum, so the direction of a straight chart route measured from Grid North remains constant.
| Direction datum | Behaviour across high latitude |
|---|---|
| True North | Rotates as each converging meridian is crossed |
| Magnetic North | May change rapidly and become weak near a magnetic pole |
| Grid North | Parallel throughout the chart and gives constant grid direction |
Grid direction
A track angle measured clockwise from Grid North is a grid track. A heading referenced to Grid North is a grid heading. Holding a constant grid heading, after allowing for drift, keeps the aircraft aligned with the straight near-great-circle chart route.
Grid convergence and conversions
Grid convergence is the angle between Grid North and local True North. It is determined by the chart convergence between the datum meridian and the aircraft's meridian.
East and west convergence
Convergence is east when True North lies east of Grid North, and west when True North lies west of Grid North. For conversion, the Oxford mnemonic is: convergence east, true least; convergence west, true best.
Worked conversions
A grid direction of 105 degrees with 20 degrees west convergence gives a true direction of 125 degrees. The same grid direction with 20 degrees east convergence gives 085 degrees true. Grid 090 degrees, convergence 10 degrees west, variation 8 degrees west and deviation 2 degrees east produces true 100, magnetic 108 and compass 106 degrees.
Sign pattern by hemisphere
For a standard North Polar grid aligned with Greenwich, 45 west gives 45 east convergence and 45 east gives 45 west convergence. In the standard South Polar grid, convergence has the same east or west name as longitude.
Grivation and magnetic steering
When a magnetic compass is usable, grivation combines variation and grid convergence into one correction between magnetic and grid direction.
Algebraic addition
Treat east as one sign and west as the opposite sign. Convergence 17 west plus variation 4 east gives grivation 13 west. Convergence 11 east plus variation 4 east gives 15 east. Convergence 14 east plus variation 4 west gives 10 east.
| Convergence | Variation | Grivation |
|---|---|---|
| 17° W | 4° E | 13° W |
| 11° E | 4° E | 15° E |
| 14° E | 4° W | 10° E |
The grid MORU check
MORU is used as a quick sense check for the magnetic and grid relationship. The reliable arithmetic rule is more important: from Grid to Magnetic, subtract east grivation and add west grivation. Reverse the operation from Magnetic to Grid. Thus grid 090 with grivation 20 east gives magnetic 070.
Isogrivs
Lines joining places of equal grivation are isogrivs. A pilot steering by magnetic compass can change magnetic heading as successive isogrivs are crossed, maintaining the required grid heading in the same way that changing variation is applied to hold a true heading.
Compass and gyro grid steering
Grid direction can be steered with a magnetic compass where the field is reliable, or with a directional gyro where magnetic direction is unusable.
Magnetic compass method
Convert the required grid heading to magnetic heading using grivation, then apply deviation for the compass indication. Update the magnetic heading when crossing isogrivs. This method becomes unreliable where horizontal magnetic field strength is weak or variation changes too rapidly.
Gyro method
Align the directional gyro initially with Grid North. Correct Earth rate, which is 15 × sin latitude degrees per hour. Do not apply the normal transport-wander correction, because the purpose is to retain the fixed grid reference rather than follow changing local true north.
| Gyro effect | Grid-steering treatment |
|---|---|
| Real drift | Monitor and correct; normally small with a good gyro |
| Earth rate | Correct at 15 × sin latitude degrees per hour |
| Transport wander | Do not correct in the normal way for grid operation |
| Residual transport wander | A small projection-related correction may remain |
Why constant grid heading works
True track changes along the near-great-circle route because the meridians converge. Grid North does not rotate across the chart. The same straight route therefore keeps one grid direction, making manual high-latitude steering manageable.
Polar grids and worked checks
A standard polar grid fixes Grid North to the Greenwich meridian and turns the chart's full 360 degrees of meridians into a simple direction-reference system.
North Polar worked track
At 45 west on a standard North Polar grid, convergence is 45 east. A constant grid track of 090 degrees converts to true track 045 degrees. At 45 east, convergence is 45 west, so the same grid track becomes 135 degrees true. Grid direction stays constant while true direction changes by 90 degrees.
South Polar worked track
On a standard South Polar grid, convergence carries the same east or west name as longitude. For grid track 070 degrees, at 45 west add 45 degrees to obtain 115 degrees true; at 45 east subtract 45 degrees to obtain 025 degrees true.
Finding the datum meridian
At 45 north, 110 west, a grid track of 132 degrees and true track of 082 degrees differ by 50 degrees east convergence. In the Northern Hemisphere west of the datum gives east convergence, so the aircraft is 50 degrees west of the datum. The datum is therefore 60 degrees west.
| Polar property | Rule |
|---|---|
| Projection | Perspective plane, tangent at pole, projected from opposite pole |
| Scale | Correct at pole, expands as sec² half co-latitude |
| Graticule | Radial straight meridians and concentric circular parallels |
| Chart convergence | Equals longitude change, n = 1 |
| Great circle | Nearly straight above about 70 degrees latitude |
| Grid direction | Measured from a fixed parallel Grid North |
| Grivation | Variation plus convergence, algebraically |