How constant is Lambert scale?
A Lambert chart is not perfectly constant scale, but careful selection of its standard parallels can hold the scale error within a small operational limit.
The one-percent convention
For practical navigation, a chart may be treated as constant scale when distortion does not exceed about 1 percent. Distance measured with a ruler will then be within about 1 percent of the value obtained from the exact local scale. Always use the published chart information if greater precision is required.
Separation of the standard parallels
Scale is correct at the two standard parallels, contracted between them and expanded outside them. Increasing their separation increases the maximum scale error. Oxford gives the following guide for a Lambert sheet.
| Latitude difference between standard parallels | Approximate maximum scale error |
|---|---|
| 5⅓° | 0.1% |
| 16° | 1% |
| 23° | 2% |
| 28° | 3% |
| 32° | 4% |
Earth and chart convergence
The difference between Earth convergency and Lambert chart convergence explains both track relationships and the slight curvature of a great circle on the chart.
Two distinct angles
Earth convergency is the change in inclination between two meridians on the Earth. It is also the change in direction of a great-circle track between those meridians. Chart convergence is the angle between the same two meridians on the chart, or the change in direction of a straight chart line as it crosses them.
Half-angle relationships
The difference between a rhumb line and a great circle is the conversion angle, equal to half Earth convergency. The difference between a rhumb line and a straight line on the Lambert is half chart convergence.
| Comparison | Angle | Formula |
|---|---|---|
| Rhumb line to great circle | Conversion angle | ½ × change of longitude × sin mean latitude |
| Rhumb line to Lambert straight line | Half chart convergence | ½ × change of longitude × sin parallel of origin |
| Great-circle curvature relative to straight line | Difference of the two half angles | ½ × change of longitude × (sin mean latitude minus sin parallel of origin) |
Worked comparison
Take two meridians 20 degrees apart on a Lambert whose parallel of origin is 45 degrees north. Half chart convergence is ½ × 20 × sin 45 degrees = 7.071 degrees. At mean latitude 47 degrees, conversion angle is 7.314 degrees. At 45 degrees it is 7.071 degrees. At 43 degrees it is 6.820 degrees. Only at the parallel of origin are the two angles equal.
Great circles and rhumb lines
On a Lambert chart, a great circle is very nearly straight. A rhumb line has much greater curvature and bends towards the pole of projection.
Great-circle shape
At the parallel of origin, a great circle is represented by a straight line in the east and west sense. At another latitude it is a shallow curve concave to the parallel of origin. The curvature is normally so small that a straight line between two positions is accepted as the great-circle track for practical plotting.
Rhumb-line shape
Meridians are the straight rhumb-line exception. Every other rhumb line curves concave to the pole of projection, which means concave to the parallels of latitude. It lies farther from the straight chord than the great circle does.
“Great circles can be treated as straight lines for all practical purposes.”Oxford ATPL Book 10, chapter 22
“Great circle is closest to a straight line at the parallel of origin.”R.K. Bali, Air Navigation, chapter 3
Mid-meridian track relationships
A straight line, the corresponding great circle and the corresponding rhumb line are parallel at the mid-meridian of a Lambert route.
The mid-meridian rule
At the halfway meridian, the straight-line track is taken as the mean great-circle track and equals the rhumb-line direction for the route. At either endpoint, the straight line changes direction from that mean by half chart convergence.
Straightening a great circle
A great circle can be represented by a practical straight line by applying conversion angle at an endpoint. Find the rhumb-line track, then move from it towards the pole by the conversion angle to obtain the initial great-circle track. On Lambert, the plotted straight line already approximates that great circle, so the remaining difference is normally very small.
Worked route, 250 degrees at A
A straight line from A to B measures 250 degrees true at A. Chart convergence is 6 degrees in the Northern Hemisphere. Half chart convergence is 3 degrees, so the rhumb-line track from A to B is 247 degrees true. Its reciprocal is 067 degrees. At B, the straight-line bearing of A is 067 minus 3 = 064 degrees true.
Worked mid-meridian route
A straight route crosses the mid-meridian at 300 degrees true and measures 302 degrees true at A. The rhumb-line track is 300 degrees true. The two-degree endpoint difference is half chart convergence, so full chart convergence is 4 degrees. The straight-line bearing of A from B is approximately 118 degrees true.
| Place on route | Straight-line direction compared with rhumb line |
|---|---|
| First endpoint | Differs by half chart convergence |
| Mid-meridian | Parallel to the rhumb line and great circle |
| Second endpoint | Differs by half chart convergence in the opposite sense |
Plotting positions and measuring distance
Lambert chartwork uses the curved graticule directly. Plot positions by interpolation, measure direction at the relevant meridian and measure distance along a graduated meridian.
Plotting a position
- Find the two meridians enclosing the required longitude and interpolate the longitude fraction.
- Find the two parallels enclosing the required latitude and interpolate the latitude fraction.
- Mark the intersection carefully, allowing for the curvature of both parallels.
Measuring a bearing
Place the protractor or plotter against the meridian at the point where the direction is required. A straight route's angle to north changes as it crosses converging meridians, so a bearing measured against a distant meridian is not the bearing at the point of interest.
Measuring distance
Use a graduated meridian, because one minute of latitude represents one nautical mile. For a short leg, use the scale near the leg's mean latitude. If scale error on the sheet is within the accepted limit, a chart ruler may be used. A long leg crossing appreciably different scale should be divided into shorter sections.
| Quantity | Correct Lambert reference |
|---|---|
| Latitude | Graduated meridian |
| Longitude | Interpolation between enclosing meridians |
| Track or bearing | Meridian at the point where the direction applies |
| Distance | Latitude graduations on a meridian near the leg |
Plotting radio bearings
Radio waves follow great-circle paths. Lambert makes them nearly straight, but the reference meridian must match the place where the bearing was measured.
Bearings measured at the ground station
VDF and VOR bearings are referenced at the ground station. Correct magnetic bearing to true using variation at that station, then plot the true bearing from the station's meridian. No conversion angle is required because the radio path is already a great circle and is treated as straight on Lambert.
Bearings measured at the aircraft
ADF or NDB relative bearing, and an airborne weather-radar bearing, are measured at the aircraft. Add the aircraft's true heading to relative bearing to obtain the true great-circle bearing from the aircraft. When plotting the reciprocal from the NDB, draw a line through the NDB parallel to the aircraft's meridian and measure from that transferred reference. This automatically allows for chart convergence.
Worked ADF example
An aircraft heads 330 degrees true and observes an NDB at relative bearing 090 degrees. The true bearing from aircraft to NDB is 060 degrees, so the reciprocal position line is 240 degrees. Plot 240 degrees from a line through the NDB parallel to the aircraft's meridian. If chart convergence is 3 degrees and the bearing must instead be measured from the NDB meridian, use 243 degrees true.
| System | Where bearing is measured | Variation used | Plotting reference |
|---|---|---|---|
| VOR or VDF | Ground station | At the ground station | Station meridian |
| ADF or NDB | Aircraft | At the aircraft | Transferred aircraft meridian through NDB |
| Airborne weather radar | Aircraft | At the aircraft | Transferred aircraft meridian through plotted object |
Worked routes and chart comparison
The final skill is to combine scale, convergence and track relationships without borrowing a Mercator rule at the wrong moment.
Worked longitude from track change
A Lambert chart has n = 0.6. A straight line is 292 degrees true at A and 283 degrees true at B. Its direction has changed by 9 degrees, which equals chart convergence between the two meridians. Change of longitude is 9 divided by 0.6 = 15 degrees. If A is at 12 degrees east and B lies west of A, B is at 3 degrees west.
Worked longitude from rhumb and straight tracks
On a Lambert chart with n = 0.80, the initial straight-line track is 070 degrees true and the rhumb-line track is 082 degrees true. Their difference of 12 degrees is half chart convergence, so full chart convergence is 24 degrees. Change of longitude is 24 divided by 0.80 = 30 degrees. From 4 degrees west eastwards, the destination longitude is 26 degrees east.
| Property | Lambert conformal | Normal Mercator |
|---|---|---|
| Meridians | Straight and converging | Straight, parallel and equally spaced |
| Parallels | Concentric circular arcs | Straight parallel lines |
| Great circle | Nearly straight, shallow curve to parallel of origin | Curve concave to equator, except equator and meridians |
| Rhumb line | Curved towards pole, except meridians | Straight |
| Scale | Near constant between well-chosen standard parallels | Expands rapidly with secant latitude |
| Chart convergence | Longitude change × n | Zero |
| Typical use | Mid-latitude route charts and plotting | Low-latitude and rhumb-line plotting |
Advantages and disadvantages
Lambert permits practical straight-line great-circle plotting, direct plotting of ground-measured radio bearings and ruler measurement where scale error is small. Its non-rectangular graticule makes position plotting less simple than Mercator, rhumb lines are curved, and aircraft following a great circle needs continuously updated desired track rather than one unchanging compass direction.