Why a conical chart?
Lambert's conformal conic was designed to give navigators a near constant-scale chart on which a straight line is a practical approximation to a great-circle route.
The need behind the projection
The Mercator is valuable because rhumb lines are straight, but its scale changes rapidly with latitude and most great circles are curved. Modern navigation systems can follow great-circle routes, so a chart that shows them almost straight and keeps scale nearly constant is more useful for long mid-latitude flight planning.
The simple conic starting point
Place a cone over a reduced Earth with its apex on the extended polar axis. In the simple conic, the cone touches the Earth along one parallel of latitude. Light rays imagined from the Earth's centre project the graticule onto the inside of the cone. The cone is then cut along one generator and opened into a flat sector.
| Construction feature | Simple conic result |
|---|---|
| Surface | A cone with its axis aligned to the Earth's spin axis |
| Contact | One parallel of tangency |
| Scale | Correct on the parallel of tangency, expanding away from it |
| Meridians | Straight lines converging at the apex |
| Parallels | Concentric arcs centred on the apex |
The cone constant and convergence
The opening of the cone fixes how strongly chart meridians converge. The controlling number is the sine of the parallel of origin.
The angle of the cone
In an axial cross-section of a simple conic, the angle at the apex is twice the latitude of tangency. A tangency latitude of 45 degrees gives a 90 degree apex angle; 60 degrees gives 120 degrees. When the cone is opened, the amount of chart sector occupied by 360 degrees of longitude is controlled by sine latitude.
Worked convergence
For a parallel of origin of 45 degrees, n = sin 45 degrees = 0.7071. Across 100 degrees of longitude, chart convergence is 100 × 0.7071 = 70.71 degrees. Across 10 degrees it is 7.071 degrees. The same constant applies everywhere on that Lambert chart.
Relation to Earth convergency
Earth convergency is change of longitude multiplied by sine mean latitude. Lambert chart convergence uses sine parallel of origin instead. They are equal along the parallel of origin. Elsewhere Earth convergency changes with latitude, while Lambert chart convergence between two chosen straight meridians remains constant.
“The sine of the parallel of origin is called the constant of the cone.”Oxford ATPL Book 10, chapter 21
“Lambert's convergence equals change in longitude multiplied by sine latitude of origin.”R.K. Bali, Air Navigation, chapter 3
From simple conic to Lambert
The simple conic changes scale too quickly and is not conformal. Lambert modified the projection to reduce scale error and preserve angles.
Two standard parallels
Instead of touching the reduced Earth along one parallel, the Lambert cone conceptually cuts through it. The two circles of intersection become the standard parallels. Scale is exactly correct on both. The former tangency parallel becomes the parallel of origin and lies halfway between the standard parallels.
What Lambert kept
The cone angle does not change, so the value of n and the chart convergence do not change. The parallel of origin still defines the cone constant. The standard parallels define where scale is correct. These are different jobs and must not be confused.
| Reference | Role on a Lambert chart |
|---|---|
| Parallel of origin | Mathematical basis, halfway between standard parallels, defines n and minimum scale |
| Lower standard parallel | Scale exactly correct |
| Upper standard parallel | Scale exactly correct |
| Cone constant n | Controls chart convergence |
Why the chart is non-perspective
Once the cone is brought inside the reduced Earth, further mathematical adjustment is required to make the scale in all directions equal at each point. The final Lambert projection is therefore non-perspective. It cannot be constructed solely by straight light rays from one physical projection point.
Scale and the one-sixth rule
Lambert scale is exactly correct at two standard parallels, slightly contracted between them and expanded outside them.
The complete scale pattern
- Outside the standard parallels, scale is greater than correct and expands farther away.
- At either standard parallel, scale is exactly correct.
- Between the standard parallels, scale is contracted.
- At the parallel of origin, halfway between them, scale is least.
The one-sixth rule
To minimise the maximum scale variation across a chart sheet, place the upper standard parallel about one sixth of the sheet height down from the top and the lower standard parallel about one sixth up from the bottom. The parallel of origin lies halfway between them. The rule spreads the small contraction inside the standard parallels against the expansion near the sheet edges.
| Example sheet | Latitude | Scale state |
|---|---|---|
| Upper edge | about 58° N | Slightly expanded |
| Upper standard parallel | 55° N | Correct |
| Parallel of origin | 45° N | Least, contracted |
| Lower standard parallel | 35° N | Correct |
| Lower edge | about 32° N | Slightly expanded |
Conformality and the graticule
Lambert's mathematical correction makes the chart conformal while retaining the characteristic fan-shaped graticule of a conic projection.
Orthomorphic construction
Conformal and orthomorphic describe the same practical property. Meridians and parallels cross at right angles, and scale at a point is the same in every direction. Small shapes and local bearings are therefore represented correctly.
Recognising the graticule
Meridians are straight lines radiating from the pole of projection. Parallels are arcs of concentric circles centred on that pole. The geographic pole is normally outside the chart sheet. Spacing between parallels is nearly uniform over an operational chart, but is adjusted mathematically to preserve conformality and the designed scale pattern.
| Property | Lambert result | Meaning |
|---|---|---|
| Construction | Conic, non-perspective | Mathematical modification is essential |
| Conformality | Yes | Local angles and small shapes are correct |
| Meridians | Straight and converging | They radiate from the off-sheet pole |
| Parallels | Concentric circular arcs | They cross meridians at 90 degrees |
| Chart convergence | Constant for a given longitude difference | It does not vary with latitude on one chart |
A limited preview of route shape
A straight line on a Lambert chart is a very close approximation to a great circle for practical plotting. Most rhumb lines are curved. Their detailed shapes, conversion-angle corrections and plotting method belong to Chapter 12.
Use, limitations and exam checks
Lambert charts are most useful over broad mid-latitude regions where near constant scale, conformality and nearly straight great-circle routes simplify planning and plotting.
Operational uses
The projection is widely suited to aeronautical route charts, area charts and meteorological charts in temperate latitudes. It gives better scale control than a Mercator across a wide east and west extent and allows distance to be measured with a ruler when the published chart's scale error is acceptably small.
Advantages and limits
| Advantage | Related limitation |
|---|---|
| Nearly constant scale over the designed sheet | Scale is not mathematically constant everywhere |
| Conformal, so local bearings are correct | Area is not preserved exactly |
| Straight line closely approximates a great circle | Rhumb lines are generally curved |
| Useful over middle latitudes | A different projection is preferred close to a pole or for an equatorial world chart |
Worked constant check
Suppose the standard parallels are 41 degrees 20 minutes north and 11 degrees 40 minutes north. Their midpoint is 26 degrees 30 minutes north, which is the parallel of origin. The constant of the cone is sin 26 degrees 30 minutes, approximately 0.446. A 20 degree longitude difference therefore gives about 8.92 degrees of chart convergence.
Summary sequence
- Find the parallel of origin, normally halfway between the standard parallels.
- Calculate n as sine parallel of origin.
- Multiply longitude difference by n for chart convergence.
- Remember that scale is correct at both standard parallels and least at the parallel of origin.
- Recognise the conic graticule and its best use in mid-latitude aviation.