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Lambert azimuthal equal-area projection

Lambert azimuthal equal-area 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 Lambert azimuthal equal-area projection rather than just read about it. In short: The Lambert azimuthal equal-area projection is a particular mapping from a sphere to a disk. It accurately represents area in all regions of the sphere, but it does not accurately represent angles.

Lambert azimuthal equal-area projection — main illustration
Lambert azimuthal equal-area projection — illustration

Key takeaways

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

Reference excerpt

The Lambert azimuthal equal-area projection is a particular mapping from a sphere to a disk. It accurately represents area in all regions of the sphere, but it does not accurately represent angles. It is named for the Swiss mathematician Johann Heinrich Lambert, who announced it in 1772. "Zenithal" being synonymous with "azimuthal", the projection is also known as the Lambert zenithal equal-area projection. The Lambert azimuthal projection is used as a map projection in cartography. For example, the National Atlas of the US uses a Lambert azimuthal equal-area projection to display information in the online Map Maker application, and the European Environment Agency recommends its usage for European mapping for statistical analysis and display. It is also used in scientific disciplines such as geology for plotting the orientations of lines in three-dimensional space. This plotting is aided by a special kind of graph paper called a Schmidt net.

Definition

To define the Lambert azimuthal projection, imagine a plane set tangent to the sphere at some point S on the sphere. Let P be any point on the sphere other than the antipode of S. Let d be the distance between S and P in three-dimensional space (not the distance along the sphere surface). Then the projection sends P to a point P′ on the plane that is a distance d from S. To make this more precise, there is a unique circle centered at S, passing through P, and perpendicular to the plane. It intersects the plane in two points; let P′ be the one that is closer to P. This is the projected point. See the figure. The antipode of S is excluded from the projection because the required circle is not unique. The case of S is degenerate; S is projected to itself, along a circle of radius 0. Explicit formulas are required for carrying out the projection on a computer. Consider the projection centered at ⁠ S = ( 0 , 0 , − 1 ) {\displaystyle S=(0,0,-1)} ⁠ on the unit sphere, which is the set of points ⁠ ( x , y , z ) {\displaystyle (x,y,z)} ⁠ in three-dimensional space R 3 {\displaystyle \mathbf {R} ^{3}} such that ⁠ x 2 + y 2 + z 2 = 1 {\displaystyle x^{2}+y^{2}+z^{2}=1} ⁠. In Cartesian coordinates ⁠ ( x , y , z ) {\displaystyle (x,y,z)} ⁠ on the sphere and ⁠ ( X , Y ) {\displaystyle (X,Y)} ⁠ on the plane, the projection and its inverse are then described by

… excerpt ends here. Continue reading the full article.

Illustrations

Lambert azimuthal equal-area projection: Lambert azimuthal equal-area projection of the world. The center is 0° N 0° E. The antipode is 0° N 180° E, near Kiribati in the Pacific Ocean. That point is represented by the entire circular boundary of the map, and the ocean around that point appears along the entire boundary.
Lambert azimuthal equal-area projection of the world. The center is 0° N 0° E. The antipode is 0° N 180° E, near Kiribati in the Pacific Ocean. That point is represented by the entire circular boundary of the map, and the ocean around that point appears along the entire boundary.
Lambert azimuthal equal-area projection: The Lambert azimuthal equal-area projection with Tissot's indicatrix of deformation.
The Lambert azimuthal equal-area projection with Tissot's indicatrix of deformation.
Lambert azimuthal equal-area projection: A cross sectional view of the sphere and a plane tangent to it at S. Each point on the sphere (except the antipode) is projected to the plane along a circular arc centered at the point of tangency between the sphere and plane.
A cross sectional view of the sphere and a plane tangent to it at S. Each point on the sphere (except the antipode) is projected to the plane along a circular arc centered at the point of tangency between the sphere and plane.
Lambert azimuthal equal-area projection illustration
Lambert azimuthal equal-area projection: Animation of a Lambert projection. Each grid cell maintains its area throughout the transformation. In this animation, points on the equator remain always on the z = 0 {\displaystyle z=0} plane.
Animation of a Lambert projection. Each grid cell maintains its area throughout the transformation. In this animation, points on the equator remain always on the z = 0 {\displaystyle z=0} plane.

Worked examples

Example 1 — a first encounter with Lambert azimuthal equal-area projection

Start with the simplest possible case. Write down what Lambert azimuthal equal-area 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 Lambert azimuthal equal-area 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 Lambert azimuthal equal-area 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 Lambert azimuthal equal-area projection

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

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

Frequently asked questions

What is Lambert azimuthal equal-area projection in simple terms?

The Lambert azimuthal equal-area projection is a particular mapping from a sphere to a disk. It accurately represents area in all regions of the sphere, but it does not accurately represent angles.

Why does Lambert azimuthal equal-area 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 Lambert azimuthal equal-area 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 Lambert azimuthal equal-area projection.

Tags

  • Equal-area projections

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