Here Comes Geometry

I had an interesting challenge for this laser cutter project: I needed to cut identical slots, following the path of a polygon, into (what would eventually be) a circular piece of MDF.  I wanted to make sure that everything stayed consistent, so my first impulse was to use the polygon tool in Illustrator, though that turned out to be more trouble than it was worth.  While I was able to construct a perfect 21-sided polygon that was the size I needed, there was no way to mark out the slots I needed to cut on the individual sides, and do so consistently.  Instead, I decided to take a bottom-up approach and construct a jig that would allow me to draw out my pattern by hand.

I started by using the line segment tool to create a horizontal side.

Next, I used the rectangle tool to construct a box of the appropriate size.

Using the Direct Select tool, I clicked on the center of the rectangle, then pressed the Command key and aligned it with the center of the line segment (pressing Command allows you to snap the anchor point to another when moving it).

Then, copy and paste the object.

And rotate.  Using basic geometry, I determined the angle between sides by using the following formula: [180(n-2)] / n, where n equals the number of sides.  I decided on a hexagon for the demonstration, since calculations are much easier (and angles much more precise) than a 21-sided figure.

It was necessary for me to also include measurements for the shape's circumcircle (the circle which encloses a given polygon and shares its radius), so I was able to use basic trigonometry to calculate its radius using the following formula: s / [2 sin (180/n)], where s equals the length of a side and n the number of sides.  Once established (roughly 3" in this case), it was a matter of drawing the appropriate line segment, then rotating it into position (again using Command while moving the segment to snap it to the intersection of the existing sides.

From this point, I was able to create my jig for the laser cutter.  I constructed a shape to enclose the entire object, then deleted all but the rectangles and the end points of the radius.  This device could then be used with a compass, and would allow me to both draw the circle I needed while placing the slots at regular intervals.