2
$\begingroup$

First of all - I am sorry if it is the wrong forum or if this is a very trivial question. I am not a mathematician nor a trigonometry genius - and therefor I would ask a simple answer that someone like me could understand (and not just fancy formulas if possible)

Giving a circle with a known radius ($r$) and another circle with an offset of ($t$) I would like to fill the "gap" ($t$) with non-overlapping triangles with the closest possible angle to $45^\circ$.

  • 1 - How can I know how many triangles will enter the space ?
  • 2 - How can I calculate their exact angles (a) (b) ?
  • 3 - assuming I want an EXACT angle of (b) = $90^\circ$ (and not approximation) - how can I know the number of triangles and also calculate the "left-over" ??

enter image description here

UPDATE I: as per comment : a visual example of wanted result. enter image description here

  • 0
    You seem to have drawn the triangles with all their corners on the two circles, such that the ones with their long edge on the inner circle (e.g. the yellow edge) extend beyond the gap (the mathematical term for which is "annulus" by the way) whereas the ones with their long edge on the outer circle are entirely inside the gap -- is that what you want? Also, angle $a$ seems to be approximately $360-45=315$ degrees, not $90$ -- isn't it $b$ that you want to be exactly $90$ degrees?2012-10-19
  • 0
    yes, it is b that I want 90 (sorry) .. and about the yellow line - what I need is actually to fill the gap. but because it is a circle, I do not know how (if ever) it is possible to calculate that angle. hence I made the yellow line..2012-10-19
  • 0
    I don't understand how you're using "fill". Are you saying that you want the triangle with the yellow edge to be entirely inside the gap but you don't know how to do that, or do you just want to *cover* the gap and don't mind if parts of the triangles extend beyond the gap? If by "fill" you mean that you want to cover the entire gap with triangles that don't extend beyond the gap, then that's impossible since the gap is curved and the triangles aren't. (Or perhaps you're thinking of generalized triangles that are allowed to have curved edges?)2012-10-19
  • 0
    @joriki - please see my update. as for your other comment about that being impossible - what I am doing is a real-world project in which I do not need the "base" of the triangles to be precise (it is done manually) - but I do need the angles in order to execute that project. this is the reason for the straight yellow line . I know that without it it would not be a triangle . but for me, it is also ok to know the calculation as if it were a triangle (even if it is going outside the bounds of the circle. so your definition of "generalized triangles that are allowed curved edges" is on the spot2012-10-19
  • 0
    Please clarify the question itself; people shouldn't have to read through the comments to find that it's actually $b$ and not $a$ that's supposed to be $90$ degrees.2012-10-19

1 Answers 1