What do we call these two parts of the s-curve?

I'm translating a text where a 4-parameter dose-response log curve is fitted to experimentally obtained data. The text says:

The obtained dose-response curve (S-shaped) should contain points on the upper and lower plateaus and at least 4 points in the linear region and bending regions.

As I understand, there should be data points at the very top and the very bottom of the curve, and 4 data points belonging to the range that includes the "linear region" (where the curve goes at a slant linearly) and the two "bends".

Do they call them "bending regions", "bends" or "inflecting regions", "inflections" or something else? • In differential calculus, an inflection point, point of inflection, flex, or inflection (inflexion (British English)) is a point on a curve at which the curve changes from being concave (concave downward) to convex (concave upward), or vice versa. But don't believe that bit about Brits spelling it inflexion - I'd say the author of the Wikipedia page pretty much made that up on the spot. Whatever - it's "domain-specific", and easily googled. – FumbleFingers Feb 19 '17 at 15:48
• @FumbleFingers - so the inflection point is right in the middle on my figure? So I guessed, and this prompted me to ask the question. – CowperKettle Feb 19 '17 at 15:51
• Personally, I might say that (taken in isolation) the first curve on your example was "exponential", and the second was "converging". In theory the bit in the middle has no "length" at all. But it's a bit of a hot issue in futurology these days, because most people instinctively assume that right now we're somehow at that point, and that future increases (in the pace of technological innovation, for example) will either continue at the existing rate or flatten off. Human minds aren't good at dealing with exponentiation. – FumbleFingers Feb 19 '17 at 16:11
• An inflection point is a point where the curvature changes direction, so it would be in the middle here. – fixer1234 Feb 19 '17 at 20:11
• I think this should be asked on ELU, since it is of restricted usage, and not likely something the average English learner would encounter. Maybe even on Math SE. – user3169 Feb 20 '17 at 2:21

These are the "knees" of the curve.

An exponential curve is self-similar, but it still has a "knee". The "knee of the curve" (of an exponential) is where it visually appears to change from being approximately horizontal to approximately vertical.

Mathematically, an S-curve is very similar to a pair of exponential curves joined at the S-curve's midpoint. Thus, it makes sense to call the transitions between the middle "linear" portion and the tails "knees".

• Very interesting, and I'm surprised to say I've never come across this definition until now! – Phylyp Mar 20 '17 at 19:51
• I've heard the term "knee" used very commonly in relation to the current-voltage curves for diodes, which have a similar shape. – LMS Mar 20 '17 at 20:24

The curve you've displayed there is the representation of a Sigmoid function or Logistic function. In fact, the picture you've used seems to be very similar to that used for the logistic function on Wikipedia. :-)

The points on that curve are:

• The inflection point - this is at the very centre of the picture, where the blue line crosses '0.5' (i.e. at the coordinates (0, 0.5) ). This inflection point represents the point where the concave curve on the left becomes a convex curve on the right (concave/convex assuming it's viewed from the top down)
• The critical point/stationary point - these are the points indicated by the pink arrows. In layman's terms this is where each of the curves turn (or in automotive terms, it is the apex of a corner).