ArticleslgStudy

mathematics

Knee of a curve

Knee of a curve 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 Knee of a curve rather than just read about it. In short: In mathematics, a knee of a curve (or elbow of a curve) is a point where the curve visibly bends, specifically from high slope to low slope (flat or close to flat), or in the other direction. This is particularly used in optimization, where a knee point is the optimum point for some decision, for example when there is an increasing function and a trade-off between the benefit (vertical y axis) and the cost (horizont…

Knee of a curve — main illustration
Knee of a curve — illustration

Key takeaways

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

Reference excerpt

In mathematics, a knee of a curve (or elbow of a curve) is a point where the curve visibly bends, specifically from high slope to low slope (flat or close to flat), or in the other direction. This is particularly used in optimization, where a knee point is the optimum point for some decision, for example when there is an increasing function and a trade-off between the benefit (vertical y axis) and the cost (horizontal x axis): the knee is where the benefit is no longer increasing rapidly, and is no longer worth the cost of further increases – a cutoff point of diminishing returns. In heuristic use, the term may be used informally, and a knee point identified visually, but in more formal use an explicit objective function is used, and depends on the particular optimization problem. A knee may also be defined purely geometrically, in terms of the curvature or the second derivative.

Definitions The knee of a curve can be defined as a vertex of the graph. This corresponds with the graphical intuition (it is where the curvature has a maximum), but depends on the choice of scale. The term "knee" as applied to curves dates at least to the 1910s, and is found more commonly by the 1940s, being common enough to draw criticism. The unabridged Webster's Dictionary (1971 edition) gives definition 3h of knee as:

an abrupt change in direction in a curve (as on a graph); esp one approaching a right angle in shape.

Criticism Graphical notions of a "knee" of a curve, based on curvature, are criticized due to their dependence on the coordinate scale: different choices of scale result in different points being the "knee". This criticism dates at least to the 1940s, being found in Worthing & Geffner (1943, Preface), who criticize:

references to the significance of a so-called knee of a curve when the location of the knee was a function of the chosen coordinate scales

Detection methods

Kneedle algorithm The Kneedle algorithm detects the best balanced tradeoff based on the mathematical curvature concept, which is defined and well studied for continuous functions. Alternatively, the kneepointDetection() function from the SamSPECTRAL R package can be used to find the knee point, where is a "phase change" in the data, by fitting two lines using linear regression.

Applications Elbow method Maximum power point tracking

References

Illustrations

Knee of a curve: Explained variance. The "elbow" is indicated by the red circle. The number of clusters chosen should therefore be 4.
Explained variance. The "elbow" is indicated by the red circle. The number of clusters chosen should therefore be 4.
Knee of a curve: Photovoltaic solar cell I-V curves where a line intersects the knee of the curves where the maximum power transfer point is located.
Photovoltaic solar cell I-V curves where a line intersects the knee of the curves where the maximum power transfer point is located.

Worked examples

Example 1 — a first encounter with Knee of a curve

Start with the simplest possible case. Write down what Knee of a curve 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 Knee of a curve 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 Knee of a curve 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 Knee of a curve

In research
Knee of a curve 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 Knee of a curve 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
Knee of a curve is common in secondary-school and first-year university syllabi. It links to neighbouring topics Curvature (mathematics), Mathematical optimization, Operations research, so understanding it makes those chapters shorter.
In everyday life
Look for Knee of a curve 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.

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Knee of a curve in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Knee of a curve 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 Knee of a curve out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Knee of a curve in simple terms?

In mathematics, a knee of a curve (or elbow of a curve) is a point where the curve visibly bends, specifically from high slope to low slope (flat or close to flat), or in the other direction. This is particularly used in optimization, where a knee point is the optimum point for some decision, for e…

Why does Knee of a curve 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 Knee of a curve?

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 Knee of a curve.

Tags

  • Curvature (mathematics)
  • Mathematical optimization
  • Operations research

Keep exploring