The ‘knee of a curve’ – useful clue but incomplete support

This paper describes the concept of the ‘knee of a curve’ as used to illuminate trade spaces for resource investment decisions. The knee is generally considered to be the point beyond which the curve flattens out, meaning the return on investment is diminishing. Although the concept of the knee seems quite intuitive, it can be ambiguous when applied to generic cost-performance functions. This paper presents two distinct mathematical interpretations of the knee using Calculus and Analytic Geometry: first, the point of maximum curvature; and second, a derivation using the area between the curve and the curve's upper bound. This paper concludes that the knee can provide useful clues in the Analysis of Alternatives, but is by no means conclusive.