Points on a cirlce to a line segment

Jun 2014
2
0
USA
Hello,

Let say I have a circle with a known circumference that has arbitrary points on it around the circle. I want to convert/unravel that circle to a line segment and know where the arbitrary points on the circle reside on the line segment.

So for example, if a circle with a circumference of 10 is centered at (0,0) and has four total points that I care about every 90 degrees starting at 0 degrees. Then the circle is converted to a line segment along the X axis, what would the coordinates of the points that were on the circle now be on the line segment? I'm looking for a repeatable formula.

Thanks in advance for any help.
 
Jun 2014
2
0
USA
Duh. So simple. I was overcomplicating it. Thanks!
 

Prove It

MHF Helper
Aug 2008
12,897
5,001
Because I tend to not like having to remember a lot of formulas, I remember that arclength is a proportion of the circumference, where the proportion is determined by the angle. The whole circle sweeps out an angle of $\displaystyle \begin{align*} 2\pi \end{align*}$ radians, so the proportion of the circle is $\displaystyle \begin{align*} \frac{\theta}{2\pi} \end{align*}$. Thus the arclength is this proportion of the circumference:

$\displaystyle \begin{align*} l &= \frac{\theta}{2\pi} \cdot C \\ &= \frac{\theta}{2\pi } \cdot 2\pi \, r \\ &= \theta\,r \end{align*}$