mathematics

Minimum bounding box

Minimum bounding box is a mathematics 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 Minimum bounding box rather than just read about it. In short: In geometry, the minimum bounding box or smallest bounding box (also known as the minimum enclosing box or smallest enclosing box) for a point set S in N dimensions is the box with the smallest measure (area, volume, or hypervolume in higher dimensions) within which all the points lie. When other kinds of measure are used, the minimum box is usually called accordingly, e.g., "minimum-perimeter bounding box".

Key takeaways

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

Reference excerpt

In geometry, the minimum bounding box or smallest bounding box (also known as the minimum enclosing box or smallest enclosing box) for a point set S in N dimensions is the box with the smallest measure (area, volume, or hypervolume in higher dimensions) within which all the points lie. When other kinds of measure are used, the minimum box is usually called accordingly, e.g., "minimum-perimeter bounding box". The minimum bounding box of a point set is the same as the minimum bounding box of its convex hull, a fact which may be used heuristically to speed up computation. In the two-dimensional case it is called the minimum bounding rectangle.

Axis-aligned minimum bounding box The axis-aligned minimum bounding box (or AABB) for a given point set is its minimum bounding box subject to the constraint that the edges of the box are parallel to the (Cartesian) coordinate axes. It is the Cartesian product of N intervals each of which is defined by the minimal and maximal value of the corresponding coordinate for the points in S. Axis-aligned minimal bounding boxes are used as an approximate location of an object in question and as a very simple descriptor of its shape. For example, in computational geometry and its applications when it is required to find intersections in the set of objects, the initial check is the intersections between their MBBs. Since it is usually a much less expensive operation than the check of the actual intersection (because it only requires comparisons of coordinates), it allows quickly excluding checks of the pairs that are far apart.

Arbitrarily oriented minimum bounding box The arbitrarily oriented minimum bounding box is the minimum bounding box, calculated subject to no constraints as to the orientation of the result. Minimum bounding box algorithms based on the rotating calipers method can be used to find the minimum-area or minimum-perimeter bounding box of a two-dimensional convex polygon in linear time, and of a three-dimensional point set in the time it takes to construct its convex hull followed by a linear-time computation. A three-dimensional rotating calipers algorithm can find the minimum-volume arbitrarily-oriented bounding box of a three-dimensional point set in cubic time. Matlab implementations of the latter as well as the optimal compromise between accuracy and CPU time are available.

Object-oriented minimum bounding box In the case where an object has its own local coordinate system, it can be useful to store a bounding box relative to these axes, which requires no transformation as the object's own transformation changes.

Digital image processing In digital image processing, the bounding box is merely the coordinates of the rectangular border that fully encloses a digital image when it is placed over a page, a canvas, a screen or other similar bidimensional background.

See also Bounding sphere Bounding volume Minimum bounding rectangle Darboux integral

References

Illustrations

Minimum bounding box: A sphere enclosed by its axis-aligned minimum bounding box (in 3 dimensions) — A sphere enclosed by its axis-aligned minimum bounding box (in 3 dimensions)

Worked examples

Example 1 — a first encounter with Minimum bounding box

Start with the simplest possible case. Write down what Minimum bounding box claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, 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 Minimum bounding box 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 Minimum bounding box 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 Minimum bounding box

In research: Minimum bounding box appears in mathematics 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 Minimum bounding box 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: Minimum bounding box is common in secondary-school and first-year university syllabi. It links to neighbouring topics Geometric algorithms, Geometry, so understanding it makes those chapters shorter.
In everyday life: Look for Minimum bounding box 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 Minimum bounding box in 20 minutes

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

Frequently asked questions

What is Minimum bounding box in simple terms?

Why does Minimum bounding box matter?

Because it connects several mathematics 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 Minimum bounding box?

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 Minimum bounding box.