mathematics

Hidden line

Hidden line 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 Hidden line rather than just read about it. In short: In mathematics, a hidden line is a geometric edge line that is not visible from an observer's view of a solid shape or object. A common practice is to draw the visible edges as solid lines and the hidden lines as dotted lines, dashed lines, or thinner lines than the visible lines.

Key takeaways

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

Reference excerpt

In mathematics, a hidden line is a geometric edge line that is not visible from an observer's view of a solid shape or object. A common practice is to draw the visible edges as solid lines and the hidden lines as dotted lines, dashed lines, or thinner lines than the visible lines.

Hidden lines add geometric information about the unseen sides of an object. They are used to help a person visualize drawings of geometric objects in three-dimensional space.

A three-dimensional object drawn with solid visible and hidden lines is a wire-frame model of the object.

Geometry objects without hidden lines When a student is first introduced to a geometric cube, it's easier to start with something they are familiar such as a common box which is a cube without hidden lines. Drawing an object without hidden lines, matches an observer's view of the object. Omitting hidden lines makes an object easier to draw.

To draw higher dimensional cubes, hypercubes, without hidden lines, make the faces opaque. Then, the hidden lines are no longer visible, they are removed from the observer's view. This works with n-cubes in three-dimensional space.

Hidden lines in technical drawings Hidden lines represent edges of a physical object that are not visible from a specific viewpoint. Visible lines are viewable edges from a specified viewpoint. In a technical drawing hidden lines are either included or not included, depending on the viewer’s requirement. "Hidden lines are used in engineering drawings to represent features that cannot be seen in a particular view but are necessary to fully define the part or assembly. Hidden lines are represented by a series of short dashes, evenly spaced, with the first dash in contact with the visible line from which it starts and the last dash in contact with the visible line at which it ends." In a computer drawing application, the hidden line option may be referred to as a wire frame with hidden edges. The wireframe hidden edges would be dashed lines.

Removing hidden lines is important in computer design and graphics applications. There are algorithmic solutions to remove hidden lines or partially hidden lines during an object's rendering. "The general problem of efficiently and meaningfully displaying three-dimensional objects in two dimensions is central to computer graphics."

See also Hidden-surface determination Technical drawing Perspective Drawing of Surfaces with Hidden Line Removal The arXiv article is written in Spanish. PDF English translation.

References

External links What are the advantages and disadvantages of using dashed or dotted hidden lines?

Illustrations

Worked examples

Example 1 — a first encounter with Hidden line

Start with the simplest possible case. Write down what Hidden line 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 Hidden line 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 Hidden line 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 Hidden line

In research: Hidden line 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 Hidden line 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: Hidden line is common in secondary-school and first-year university syllabi. It links to neighbouring topics Elementary geometry, Graphical projections, Multi-dimensional geometry, so understanding it makes those chapters shorter.
In everyday life: Look for Hidden line 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 Hidden line in 20 minutes

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

Frequently asked questions

What is Hidden line in simple terms?

Why does Hidden line 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 Hidden line?

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 Hidden line.