From a globe to a flat chart
A chart projection transfers the Earth's curved graticule to a flat surface. Because a sphere cannot be opened flat without stretching, tearing or compressing it, every projection preserves some properties and sacrifices others.
Chart projections in principle
The historical picture uses a transparent reduced Earth, a light source and a receiving surface. Rays carry meridians, parallels and geographical detail to a plane, cylinder or cone. Modern charts are usually constructed mathematically, but the light-source model remains useful for understanding their geometry.
The reduced Earth
The reduced Earth is a scale model of the globe on which projection is based. For a 1:1,000,000 chart, imagine a globe reduced to one millionth of the Earth's linear dimensions. The projection surface touches or cuts this model at selected lines or points.
| Term | Meaning | Consequence |
|---|---|---|
| Projection | Transfer of the graticule to a chart surface | Creates a usable flat representation |
| Reduced Earth | Scale model used as the geometric basis | Its scale becomes the reference scale |
| Point or line of tangency | Where the surface touches the reduced Earth | Scale is commonly correct there |
| Secant projection | Projection surface cuts the reduced Earth | Produces two standard lines or a standard circle |
Perspective and non-perspective charts
A perspective or geometric projection can be produced directly by rays from a projection point. A non-perspective projection is built or modified mathematically. Many operational charts are non-perspective because mathematical adjustment gives the navigation properties required.
The three developable surfaces
Plane, cylinder and cone are called developable surfaces because each can be laid flat without stretching its own material after the graticule has been placed on it.
Azimuthal or plane projection
A flat plane touches the reduced Earth at one point. When tangent at a pole, meridians radiate as straight lines from the pole and parallels appear as concentric circles. Scale is correct at the point of tangency and changes away from it.
Cylindrical projection
A cylinder surrounds the reduced Earth. In the normal arrangement it touches at the equator. When opened, meridians and parallels form a rectangular graticule: meridians are parallel straight lines and parallels are straight lines across them. The simple perspective version distorts shape because scale does not change equally in both directions.
Conical projection
A cone is placed over the reduced Earth and normally touches along a parallel. After the cone is slit and opened, meridians are straight lines converging towards the cone apex, while parallels are arcs of circles centred on that apex. A secant cone cuts the Earth along two standard parallels.
| Family | Typical contact | Meridians | Parallels |
|---|---|---|---|
| Azimuthal or plane | One point | Straight, radiating from the centre in a polar case | Concentric circles |
| Cylindrical | Equator in the normal tangent case | Parallel straight lines | Parallel straight lines |
| Conical | One parallel, or two if secant | Straight, converging towards the apex | Arcs centred on the apex |
Properties of an ideal chart
An ideal chart would preserve directions, distances, shapes and areas while also making routes easy to plot. A flat chart cannot achieve all these aims everywhere.
Representation of the Earth's surface
| Desired property | Why it helps | Reality on a flat chart |
|---|---|---|
| Correct angles | Measured bearings match directions on Earth | Achievable locally on a conformal chart |
| Constant and correct scale | One ruler scale works everywhere | Impossible over a large flat sheet |
| Correct shapes | Features retain recognisable form | Small shapes can be good; large regions distort |
| Correct areas | Equal Earth areas appear equal | Possible on equal-area maps, but not together with every navigation property |
Navigation requirements
| Desired property | Operational value |
|---|---|
| Rhumb lines straight | A constant-direction track plots as one straight line |
| Great circles straight | The shortest route plots directly |
| Latitude and longitude easy to plot | Positions can be transferred accurately |
| Adjacent sheets fit correctly | A route can continue across chart boundaries |
| Worldwide coverage | One family can support broad route planning |
What cannot be perfect
Constant correct scale is possible only on the globe. A flat chart can make scale correct along selected lines or nearly constant over a limited region. The true shape of a large spherical area also cannot be preserved perfectly on a plane. Small areas can be represented accurately enough for recognition and navigation.
Scale can never be constant and correct.Oxford ATPL Book 10, chapter 17
A projection may be made accurate along selected lines, but distortion grows elsewhere.R.K. Bali, Air Navigation, chapter 3
Orthomorphism and conformality
Orthomorphic and conformal mean the same thing in this syllabus: angles on the Earth are represented by the same angles on the chart, so bearings measured on the chart are locally correct.
Condition 1, right-angle graticule
Meridians and parallels meet at 90 degrees on the Earth, so they must also cross at 90 degrees on a conformal chart. If that right angle is distorted, every other local bearing is distorted with it.
Condition 2, equal scale in every direction
At a given point, scale must be the same in every direction, or change at the same rate in every direction. If the north and south scale changes without an equal east and west change, a square becomes a rectangle and its diagonal no longer keeps the correct direction.
Why bearings matter most
A pilot needs a line measured on the chart to represent the correct direction on Earth. Exact area is rarely needed for navigation, while a bearing error directly affects the intended track. This is why aeronautical navigation charts are selected from conformal projection families.
How projections are constructed
Projection names may describe the receiving surface, the position of that surface, the projection point and whether mathematical correction has been applied.
Geometric perspective construction
In a true perspective projection, rays from a point project the reduced Earth onto the surface. The source may be at the Earth's centre, at the surface opposite the point of tangency, or effectively at infinity. These arrangements generate different spacing and distortion.
Geometric but mathematically modified
A geometric result may be adjusted mathematically to improve a navigation property. The normal Mercator is the standard example: the simple cylindrical graticule is modified so scale changes equally in the north and south and east and west directions, making the chart conformal.
Entirely mathematical construction
Some projections are generated from equations without a literal light-source construction. They can still be understood by comparison with a plane, cylinder or cone. Lambert's conformal conic is treated as a mathematically corrected conical projection.
| Construction | Defining feature | Typical example |
|---|---|---|
| Perspective or geometric | Direct projection by rays | Polar stereographic concept |
| Geometric, mathematically modified | Projected graticule is adjusted | Normal Mercator |
| Mathematical | Coordinates are generated by formula | Lambert conformal conic treatment |
Tangent and secant surfaces
A tangent surface touches the reduced Earth along one point, one parallel or one great circle, depending on its shape and orientation. A secant surface cuts the reduced Earth and usually creates two standard lines. Scale is correct where the conceptual surfaces intersect.
Comparing projection families
The quickest way to identify an unfamiliar chart is to read its graticule. Meridian and parallel shapes reveal the developable surface and suggest where scale is most accurate.
| Feature | Plane family | Cylindrical family | Conical family |
|---|---|---|---|
| Typical use region | Around the point of tangency, often polar | Around the tangent great circle, often low latitude in the normal case | A band around one or two standard parallels |
| Meridians | Radiate from the centre in a polar chart | Parallel straight lines in a normal chart | Straight lines converging towards the apex |
| Parallels | Concentric circles in a polar chart | Parallel straight lines | Arcs centred on the apex |
| Correct scale | At a point or standard circle | Along a tangent or secant great circle | Along one or two standard parallels |
| Main compromise | Distortion increases away from the centre | Distortion increases away from the contact line | Distortion increases away from the standard parallel band |
Graticule recognition
- Parallel straight meridians and straight parallels suggest a normal cylindrical family.
- Meridians radiating from a central pole with circular parallels suggest a polar plane family.
- Meridians converging towards a remote apex with curved parallels suggest a conical family.
Choosing the useful compromise
No projection is simply best. A polar route favours a chart centred on a pole. A low-latitude constant-direction route may favour a normal Mercator. Mid-latitude route charts often use a conformal conic projection. The operational region should lie near the chart's standard point, line or parallels where distortion is controlled.
Exam checks
- Conformal means correct local angles, not correct areas.
- A straight line is not automatically a great circle or a rhumb line; that depends on projection.
- Constant correct scale over the entire flat chart is impossible.
- The graticule is the network of meridians and parallels.
- Adjacent sheets and worldwide coverage are desirable, but they do not define conformality.