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
    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

1

On this figure :

triangulation of annulus

What you want is to have $c$ = $r_2 - d$. This leads you to the equation $\frac{r_2}{r_1} = \cos{\theta} + \sin{\theta}$. We assume here that $\theta > 0$.

Use the trigonometry formulas to convert the cosine and sine as functions of $t = \tan{\frac{\theta}{2}}$, and this gives you t as the root of a degree $2$ polynomial : $ \begin{aligned} \frac{1+2t-t^2}{1+t^2} &= \frac{r_2}{r_1} \\ - \left(1+\frac{r_2}{r_1}\right)t^2 +2t +\left(1-\frac{r_2}{r_1}\right) &= 0\\ \end{aligned} $

Solving this you get $ \begin{aligned} \Delta &= 4\left(2-\frac{r_2^2}{r_1^2}\right)\\ t &= \frac{ 1\pm \sqrt{2-\frac{r_2^2}{r_1^2}} }{ \left(1+\frac{r_2}{r_1}\right) } \end {aligned} $ and we assumed $\theta$ positive.

Using $\arctan$, and rounding to find the best integer $k$ (since the number of samples is integral) such that $\theta = \frac{\pi}{k}$ should give you the angle $\theta$ to use as half of the sampling frequency for both circles. Then both circles should be sampled using twice this angle frequency, but shifted with respect to one another with an angle $\theta$.

  • 0
    Since the base edge rotates a bit when moving from one triangle to the next, the angle $b$ will be (slightly) exceeding $90$ degrees in this set-up; in situation 3 therefore, $k$ will increase. Note furthermore that in general the triangles won't fit perfectly (i.e. the last one won't match up with the first), so that an exact angle of $90$ degrees won't always be possible.2012-10-20