2

Bear with me here, I'm neither a mathematician, nor an expert on any of these things, just interested in geometry. I'm interested in how much it really matters mathematically, if and how a triangle is degenerate and how one could find out if there will be a valid result for some formula just by looking at the degeneracy.

So first of all (and according to the definition of a triangle as a polygon with three edges and three vertices), there are different types of degenerate triangles:

  1. all three vertices in one point

  2. all three vertices on one line

    2.1 special case: all three vertices on one line and two of them in one point

  3. two right angles

Those are all the types of degeneracy I can think of right now.

So let's imagine you want to calculate the diameter of the circumscribed circle of a degenerate triangle of type 1. The result would still be valid, because if all the vertices are in one point, you can construct a circumscribed circle that passes through "all" those points, and then the diameter is just 0.

Doing the same for type 2 and 2.1 isn't hard either, the diameter would always be the distance between the two outermost vertices of this triangle on that line.

But doing it for type 3 is just, well, "impossible". When we have two right angles, one vertex is somewhere in infinity. So what's the diameter of the circumscribed circle now? Also infinity? Or "undefined"?

In this case, logically, or rather mathematically, there would be no real difference between a "normal" triangle and type 1, type 2 and type 2.1 degenerate triangles, but a significant difference between a normal triangle and a type 3 degenerate triangle.

And there are several calculations that would just "fail" with at least one type of degeneracy. My question here is, how can I know that beforehand? What types of degeneracy just make it "impossible" to calculate for example the area, or some other, more complicated stuff?

How can I know which formula will break with which type of degeneracy?

Bernard
  • 179,256
Nano Miratus
  • 143
  • 1
    What do you mean by degeneracy 3?. – Certainly not a dog Sep 20 '19 at 10:09
  • When you define a triangle just as a convex polygon with three vertices and three edges, it's technically valid to have two angles with 90°, like in this question: https://math.stackexchange.com/questions/1904183/is-a-triangle-with-two-right-angles-still-a-degenerate-triangle?rq=1

    But it is degenerate then, right? While I can see, how this degeneracy is completely different to the other types, you could still call that degenerate.

     – Nano Miratus Sep 20 '19 at 10:13
  • 1
    Yes, I understand what you mean. However I feel like defining a triangle as 3 vertices and 3 edges is kind of overdetermined, and perhaps that means there are more degeneracies that we haven’t thought of here (on this list). – Certainly not a dog Sep 20 '19 at 10:16
  • 2
    If vertices in infinity count as in (3),(which is questionable), then you get infinite for the diameter. But we can then also consider two (or all three) vertices in infinity.. – Berci Sep 20 '19 at 10:28
  • I completely agree. I just tried to "oversimplify" it in my question for that specific circumscribed circle example. And of course, there are probably more types of degeneracy. But my original question is about which type of degeneracy results in invalid, unexpected or infinite results. For example, putting in a type 2 triangle into Heron's formula gives us an expected result of 0 for the area. Which is great, but is any triangle formula that "safe"? Are there any formulae that rely on a triangle to not have all of it's points on one line? – Nano Miratus Sep 20 '19 at 10:34
  • 1
    How did you see the circumscribed circle about triangle of type 2? Can you possibly draw 1 circle passing through 3 different collinear points? – Fareed Abi Farraj Sep 20 '19 at 10:51
  • Oh whoops. You're right, it's only possible for type 1 and 2.1. But hey, that's exactly what I meant to "say" with my question, there are different types of degeneracy and some of them break some formulas. – Nano Miratus Sep 20 '19 at 11:00
  • 1
    @Fareed a straight line is a degenerate circle – Mark S. Sep 20 '19 at 11:00
  • @MarkS. I don't think a straight line is a degenerate circle, because a circle and it's higher dimensional buddies are all defined by "the set of all points that are at a given distance from a given point", and under those circumstances, the only one-dimensional circle you can construct is a point. A straight line can be a degenerate ellipse though. – Nano Miratus Sep 20 '19 at 11:04
  • 4
    @NanoMiratus: Sometimes, it's convenient to think of a circle as a "curve of constant curvature". In that context, a line —being a curve of curvature zero— counts as a (degenerate) circle. This way of thinking helps unify results such as the Descartes "Kissing Circles" Theorem. – Blue Sep 20 '19 at 11:46
  • Ugh, maths is weird. – Nano Miratus Mar 31 '20 at 14:14

0 Answers0