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Secants/tangents/passants are defined as lines intersecting/touching/not touching a given circle. I wonder what the equivalent terminology for a hexagon is? There are four cases:

  1. Intersecting the hexagon
  2. Touching the hexagon along an entire side
  3. Touching the hexagon in a corner point
  4. Not touching the hexagon at all

What would a proper terminology be for those terms?

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TL;DR

Here is the language I would probably use for each of your four objects:

  1. "A line intersecting the hexagon in exactly two distinct points"
  2. "An extended side of the hexagon"
  3. "A line intersecting the hexagon at only one point"
    or
    "a line through a vertex, not intersecting any other point of the hexagon"
    or
    "a line through a vertex, not passing through the interior of the hexagon"
  4. "A line that does not intersect the hexagon"

The detailed answer

Proper terminology can be helpful. For example, when we apply differential calculus to the geometry of graphs in a Cartesian plane, we sometimes say things such as, "The derivative of $y=\sqrt{1-x^2}$ is the slope of the tangent to the unit circle at $(x,y).$" If we had to say "the line that intersects the unit circle at exactly one point" instead of "tangent to the unit circle," we would waste a lot of time thinking about what that means every time we made a statement like this. It would be even worse when we were dealing with $y=x^3$ or $y=\sin x$ instead of $y=\sqrt{1-x^2},$ because many of the tangents to those two graphs are not lines that intersect the curve at exactly one point, so we would need to keep repeating an even more cumbersome definition.

But proper terminology is a matter of what you need to use it for. People make definitions of words because it's easier to write one word than to write out the full definition every time you need to use it; also because a word that is often used will develop associations around it that will help people get oriented to what is being said in ways that much longer phrases do not.

Sometimes people have defined things that were useful to them, but that are not used much any more. For example, in trigonometry, there are six functions of an angle (sine, cosine, tangent, cotangent, secant, and cosecant) that are frequently used, but many others have been defined that you hardly see any more. Here is a link to a figure that shows five of these lesser-known functions, and even this figure leaves out definitions such as sagitta (a synonym for versine) and haversine (which actually is still mentioned sometimes in the context of spherical trigonometry because of a useful and well-known formula).

The term passant appears to be one of those that is not generally known (at least in the English-language literature--one can find passante in German-language sources). The Wikipedia article "Circle" defines passant, but on the talk page for that article there is some controversy about that. The only cited English-language uses of it there (which were the the only ones I have found) were in a couple of on-line articles by Prof. G. Eric Morehouse at the University of Wyoming, who seems to be using it in finite geometries in order to denote a line that has no intersection with some given figure; this definition is useful to him because he is able to count the (finite) number of passants of each of various figures in his geometry.

So to get a good answer to your question about terminology relevant to a hexagon, the first thing to discuss is why you need such terminology; what are you going to use it for?

If you have a need for a term equivalent to passant, the correct word seems likely just to be passant, in the sense used by Prof. Morehouse. (Even then, if you are writing something you intend other people to read, you should write the definition of passant before you use it, since it's such an uncommon term; and you should also ask yourself whether it was worth the trouble to do so or whether you can make do without that word altogether.)

The infinite straight line that contains one side of a hexagon is an extended side of the hexagon. If you are treating the hexagon as a closed curve and using derivatives to find the direction of the curve at any point, then you could consider the tangent to the hexagon at any point except a vertex to be the same as the extended side through that point. There would not be a tangent at a vertex since the direction is undefined there.

A hexagon has several diagonals, which are segments that begin and end at non-adjacent vertices of the hexagon. If you used the term extended diagonal to refer to a line that intersected a hexagon at two non-adjacent vertices, I think you would be correctly understood by people who know geometry.

If there's some reason you need a term for an arbitrary line passing through the interior of a simple hexagon, but not necessarily through vertices, you might be able to call it a secant, though I do not think I have ever seen such a term used. (Instead it seems more common to write something like, "A line intersects the hexagon at point $G$ along side $AB$ and at point $H$ along side $BC.$")

By the way, as you might guess from the "TL;DR" portion of this answer, I disagree with the way you used your terms in defining the secant and tangent of a circle. The standard mathematical definition of intersect is "to have at least one point in common," and a circle is defined as a set of points all at the same distance from a given point; that is, the points of the circle are on its circumference, not inside it. Hence standard definitions of secant and tangent are like the ones given in this source:

A secant is a line that intersects the circle in two different points and a tangent is a line that intersects the circle in exactly one point, called the point of tangency.

Strictly speaking, your case 1 of a line "intersecting the hexagon" would also apply to all the lines mentioned in cases 2 and 3; but I inferred from the context of your question that you meant these cases to be disjoint.

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  • $\begingroup$ Thank you for your reply, got the confidence I needed! I need these terms to be used in a scientific paper in which a hexagonal grid appears. As I will need those terms quite often, your suggestions 1. - 4. are indeed unpractical as they are just too long. Therefore, I will define my own notation as you suggest and bear your hints in mind. Thanks again for the detailed explanation! $\endgroup$ – BJPrim May 9 '17 at 17:34
  • $\begingroup$ For the purpose you've described, yes, it makes sense to define passant and any other specialized terms that will make your paper easier to write and easier to read. $\endgroup$ – David K May 9 '17 at 17:43

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