computer science

Minimum bounding box algorithms

Minimum bounding box algorithms is a computer 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 Minimum bounding box algorithms rather than just read about it. In short: In computational geometry, the smallest enclosing box problem is that of finding the oriented minimum bounding box enclosing a set of points. It is a type of bounding volume. "Smallest" may refer to volume, area, perimeter, etc. of the box.

Key takeaways

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

Reference excerpt

In computational geometry, the smallest enclosing box problem is that of finding the oriented minimum bounding box enclosing a set of points. It is a type of bounding volume. "Smallest" may refer to volume, area, perimeter, etc. of the box. It is sufficient to find the smallest enclosing box for the convex hull of the objects in question. It is straightforward to find the smallest enclosing box that has sides parallel to the coordinate axes; the difficult part of the problem is to determine the orientation of the box.

Two dimensions For the convex polygon, a linear time algorithm for the minimum-area enclosing rectangle is known. It is based on the observation that a side of a minimum-area enclosing box must be colinear with a side of the convex polygon. It is possible to enumerate boxes of this kind in linear time with the approach called rotating calipers by Godfried Toussaint in 1983. The same approach is applicable for finding the minimum-perimeter enclosing rectangle. A C++ implementation of the algorithm that is robust against floating point errors is available.

Three dimensions In 1985, Joseph O'Rourke published a cubic-time algorithm to find the minimum-volume enclosing box of a 3-dimensional point set. O'Rourke's approach uses a 3-dimensional rotating calipers technique, and is based on lemmas characterizing the minimum enclosing box:

There must exist two neighbouring faces of the smallest-volume enclosing box which both contain an edge of the convex hull of the point set. This criterion is satisfied by a single convex hull edge collinear with an edge of the box, or by two distinct hull edges lying in adjacent box faces. The other four faces need only contain a point of the convex hull. Again, the points which they contain need not be distinct: a single hull point lying in the corner of the box already satisfies three of these four criteria. It follows in the most general case where no convex hull vertices lie in edges of the minimal enclosing box, that at least 8 convex hull points must lie within faces of the box: two endpoints of each of the two edges, and four more points, one for each of the remaining four box faces. Conversely, if the convex hull consists of 7 or fewer vertices, at least one of them must lie within an edge of the hull's minimal enclosing box. It is also possible to approximate the minimum bounding box volume, to within any constant factor greater than one, in linear time. The algorithm for doing this involves finding an approximation to the diameter of the point set, and using a box oriented towards this diameter as an initial approximation to the minimum volume bounding box. Then, this initial bounding box is partitioned into a grid of smaller cubes, and grid points near the boundary of the convex hull of the input are used as a coreset, a small set of points whose optimum bounding box approximates the optimum bounding box of the original input. Finally, O'Rourke's algorithm is applied to find the exact optimum bounding box of this coreset. A Matlab implementation of the algorithm is available.

The minimal enclosing box of the regular tetrahedron is a cube with side length 1 2 {\displaystyle {\frac {1}{\sqrt {2}}}} that of the tetrahedron; for instance, a regular tetrahedron with side length 2 {\displaystyle {\sqrt {2}}} fits into a unit cube, with the tetrahedron's vertices lying at the vertices (0,0,0), (0,1,1), (1,0,1) and (1,1,0) of the unit cube.

See also Smallest enclosing ball Minimum bounding rectangle

References

Worked examples

Example 1 — a first encounter with Minimum bounding box algorithms

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

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

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

Frequently asked questions

What is Minimum bounding box algorithms in simple terms?

Why does Minimum bounding box algorithms matter?

Because it connects several computer 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 Minimum bounding box algorithms?

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 algorithms.