science

Two-point equidistant projection

Two-point equidistant projection is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Two-point equidistant projection rather than just read about it. In short: The two-point equidistant projection or doubly equidistant projection is a map projection first described by Hans Maurer in 1919 and Charles Close in 1921. It is a generalization of the much simpler azimuthal equidistant projection.

Key takeaways

Two-point equidistant projection belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Two-point equidistant projection to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Two-point equidistant projection from memory before moving on to harder problems.

Reference excerpt

The two-point equidistant projection or doubly equidistant projection is a map projection first described by Hans Maurer in 1919 and Charles Close in 1921. It is a generalization of the much simpler azimuthal equidistant projection. In this two-point form, two locus points are chosen by the mapmaker to configure the projection. Distances from the two loci to any other point on the map are correct: that is, they scale to the distances of the same points on the sphere. The two-point equidistant projection maps a family of confocal spherical conics onto two families of planar ellipses and hyperbolas. The projection has been used for all maps of the Asian continent by the National Geographic Society atlases since 1959, though its purpose in that case was to reduce distortion throughout Asia rather than to measure from the two loci. The projection sometimes appears in maps of air routes. The Chamberlin trimetric projection is a logical extension of the two-point idea to three points, but the three-point case only yields a sort of minimum error for distances from the three loci, rather than yielding correct distances. Tobler extended this idea to arbitrarily large number of loci by using automated root-mean-square minimization techniques rather than using closed-form formulae. The projection can be generalized to an ellipsoid of revolution by using geodesic distance.

See also List of map projections Chamberlin trimetric projection 3D projection

References

Charles Close (1934). “A doubly equidistant projection of the sphere.” The Geographical Journal 83(2): 144-145. Charles Close (1947). Geographical By-ways: And Some Other Geographical Essays. E. Arnold. Waldo R. Tobler (1966). “Notes on two projections.” The Cartographic Journal 3(2). 87–89. François Reignier (1957). Les systèmes de projection et leurs applications a la géographie, a la cartographie, a la navigation, a la topométrie, etc... Institut géographique national.

Illustrations

Worked examples

Example 1 — a first encounter with Two-point equidistant projection

Start with the simplest possible case. Write down what Two-point equidistant projection claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Two-point equidistant projection before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Two-point equidistant projection ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Two-point equidistant projection

In research: Two-point equidistant projection appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Two-point equidistant projection in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Two-point equidistant projection is common in secondary-school and first-year university syllabi. It links to neighbouring topics Cartography stubs, Equidistant projections, so understanding it makes those chapters shorter.
In everyday life: Look for Two-point equidistant projection outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Two-point equidistant projection in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Two-point equidistant projection means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Two-point equidistant projection out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Two-point equidistant projection in simple terms?

Why does Two-point equidistant projection matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Two-point equidistant projection?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Two-point equidistant projection.