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.
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,[1] and is found more commonly by the 1940s,[2] being common enough to draw criticism.[3] The unabridged Webster's Dictionary (1971 edition) gives definition 3h of knee as:
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: