science

Loximuthal projection

Loximuthal 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 Loximuthal projection rather than just read about it. In short: In cartography, the loximuthal projection is a map projection introduced by Karl Siemon in 1935, and independently in 1966 by Waldo R. Tobler, who named it.

Key takeaways

Loximuthal 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 Loximuthal projection to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Loximuthal projection from memory before moving on to harder problems.

Reference excerpt

In cartography, the loximuthal projection is a map projection introduced by Karl Siemon in 1935, and independently in 1966 by Waldo R. Tobler, who named it. It is characterized by the fact that loxodromes (rhumb lines) from one chosen central point (the intersection of the central meridian and central latitude) are shown straight lines, correct in azimuth from the center, and are "true to scale" in the sense that distances measured along such lines are proportional to lengths of the corresponding rhumb lines on the surface of the Earth. It is neither an equal-area projection nor conformal.

Description

A loxodrome on the surface of the Earth is a curve of constant bearing: it meets every parallel of latitude at the same angle. Suppose its bearing is θ north of east, so, for example, due east is θ = 0; due north is θ = a right angle; due west is θ = a half circle. The loxodrome's whole length as it goes from the south pole to the north pole is fairly routinely seen to be πR csc θ where R is the radius of the Earth (in particular if the loxodrome goes straight east, it circles the Earth infinitely many times without getting closer to either pole, so its length is ∞. Let a loxodrome pass through the point whose longitude and latitude are both 0; call this the "central point". Suppose one starts at the central point and travels a certain distance in a certain direction along this loxodrome and arrives at geographic location . Let f(p) be the point in the (x, y)-plane reached by going that same distance in that same direction from the origin (0, 0). Thus f(p) ∈ R × [−⁠πR/2⁠, ⁠πR/2⁠]. That point f(p) is the image of p on the map. More than one loxodrome goes from the central point to p, but there is a unique shortest one: the one that does not cross the 180° meridian on its way from the central point to p. If one were to include loxodromes crossing the 180° meridian, one would get infinitely many images of the whole Earth, occupying the entire strip R × [−⁠πR/2⁠, ⁠πR/2⁠]. Using only the unique shortest loxodrome from the central point to each point p gives only one copy, occupying a sort of oval.

Illustrations

Worked examples

Example 1 — a first encounter with Loximuthal projection

Start with the simplest possible case. Write down what Loximuthal 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 Loximuthal 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 Loximuthal 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 Loximuthal projection

In research: Loximuthal 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 Loximuthal 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: Loximuthal projection is common in secondary-school and first-year university syllabi. It links to neighbouring topics Map projections, so understanding it makes those chapters shorter.
In everyday life: Look for Loximuthal 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 Loximuthal projection in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Loximuthal 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 Loximuthal projection out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Loximuthal projection in simple terms?

In cartography, the loximuthal projection is a map projection introduced by Karl Siemon in 1935, and independently in 1966 by Waldo R. Tobler, who named it.

Why does Loximuthal 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 Loximuthal 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 Loximuthal projection.