mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...

, where a knee point is the optimum point for some decision, for example when there is an

increasing function In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of orde ...

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 economics, diminishing returns are the decrease in marginal (incremental) output of a production process as the amount of a single factor of production is incrementally increased, holding all other factors of production equal ( ceteris paribu ...

. In

heuristic A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate, ...

use, the term may be used informally, and a knee point identified visually, but in more formal use an explicit

objective function In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cos ...

is used, and depends on the particular optimization problem. A knee may also be defined purely geometrically, in terms of the

curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canonic ...

or the

second derivative In calculus, the second derivative, or the second order derivative, of a function is the derivative of the derivative of . Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, ...

Definitions

The knee of a curve can be defined as a

vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...

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 ''Webster's Dictionary'' is any of the English language dictionaries edited in the early 19th century by American lexicographer Noah Webster (1758–1843), as well as numerous related or unrelated dictionaries that have adopted the Webster's n ...

'' (1971 edition) gives definition 3h of ''knee'' as:

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 , who criticize:

Applications

* Elbow method *

Maximum power point tracking Maximum power point tracking (MPPT) or sometimes just power point tracking (PPT), is a technique used with variable power sources to maximize energy extraction as conditions vary. The technique is most commonly used with photovoltaic (PV) solar sy ...

References

* * {{refend Curvature (mathematics) Mathematical optimization Operations research