This is about mathematical terminology in English.

A simple curving line can have a top peak and a bottom valley. What is the correct term to use for the top/bottom point?

In my native tongue, we call it a toppunkt, literally meaning top-most point (although it counts for both peaks and valleys). When looking it up I have found the word vertex, which Wikipedia agrees with. Is this really the correct term?

Oxford Dictionaries doesn't seem to agree but only defines vertex as opposite angular points in a geometrical shape.

Vertex literally means turning point in Latin, so it does make sense. But from simple google searches, the word vertex doesn't seem to be used very often and the term turning point actually seems to be used as well.

I would appreciate help with sorting out the terminology here.

Note that I am aware that such a point is called a stationary point in 2D (and higher dimensions). I am specifically asking to the 1D case, though, where there is a special term.

  • 1
    This is very confusing. What do you mean by "a simple curve"? What branch of mathematics or what application of it? Graphs? Celestial navigation? Cartesian geometry? Topology? – Lambie May 12 at 16:50

This is quite technical mathematical language, the vertex, or apex is the point at which the curvature is a minimum or a maximum. For a vertical parbola, the vertex occurs at the maximum point. For a car driver, the apex of a turn is the point at which the steering wheel is turned the most.

However, it is more common to speak of

  • a "stationary point" (zero gradient, applies to 1D as well higher dimensions),
  • a "turning point" (a stationary point that isn't an inflexion) or
  • a "(local) maximum/minimum" (which will be at a stationary point, if the curve is differentiable)

"Turning point" is the most useful term for the top or bottom of a curve.

Find the turning points of the curve y = x³ - x, and for each point, determine if it is a maximum or minimum.

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  • Thank you for the answer. So, you would advise me to use the term turning point in technical work rather than vertex or apex? Or is it a "colloquial" term? Regarding your first point, I am aware of the term stationary point which my native language also uses - but of some reason we never use that term in the 1-dimensional case although it strictly is correct. Not even in technical literature. Instead we have the term toppunkt to use for 1D. I am assuming that this is also the case in English - but please correct me if wrong - and I am thus looking for the translation of toppunkt. – Steeven May 14 '19 at 8:21
  • If you're talking about a sine wave or similar the usual terms would be "peak" and "trough." Turning point is used as a technical term, along with (local) maxima and (local) minima. – Showsni May 14 '19 at 15:29

In mathematics, the vertex is a point at which two lines meet and form an angle. That could refer to the highest peak of a triangle, but it could equally refer to any of the other two angles. The names for any part of a geometric shape usually remain the same regardless of its orientation.

The name for the highest point or top of a shape or object is the apex.

The bottom, or lowest point of a shape is normally just called the base.

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