Geometry Inspired Jigs and Tricks

In the process of woodworking, metalworking and general making, there's sometimes a need for a way to measure and mark specific geometric locations - center points, equal division etc. If you manufacture with computer aided procedures (cnc milling, laser cutting..), it's not really a problem, but if you're usually working with hand tools, then it gets trickier to mark and measure everything accurately.

In this instructable (which is my first, by the way, so any feedback is welcome), I'll present a few jigs and tricks that I use to makes life a little easier. What's cool about this tricks, is that they use simple geometric principles, and I'll try to explain the math behind every jig or trick.

And if you find these tricks useful, I would really appreciate a vote for this instructable.

How many times did you try to eye-ball the center of a round object? I found it to be one of the more frustrating things to find accurately. Despite the difficulty to find it, the math behind the center point of a circle is simple: it's the point from which the distance to any point in the outer perimeter of the circle is equal.

The jig that will be presented here, relies on a basic geometric principle: the angle bisector of 2 lines that are tangent to a circle, must intersect the center of the circle. This happens because the 2 tangent lines, if coming out of the same point, must be equal, and because they are symmetric in relation to the circle, their angle bisector, which is in fact their ''symmetry line'', must coincide with the circle's ''symmetry line'', which of course is the diameter.

So I designed a simple jig, which could be scaled up or down (STEP files attached) according to the size of the circle of which you want to find the center. It basically mimics the principle shown above, as the jig 'wings' represents the tangent lines and the center portion represents the angle bisector. Please note that there are 2 different designs for the middle part of the jigs, for your preference. I personally find the 'slot' type much more convenient to use, since it doesn't require your pencil to be very sharp. However, the second type is much more rigid and robust. So feel free to try them both!

In order to use the jig, you simply press the jig against the circle, and then draw a line across the middle part of the jig. Then you need to press the jig to the circle from another direction (doesn't matter), and draw another line. the intersection of the 2 lines we've just drawn is the exact center of the circle.

If you don't have access to a 3D printer or can't make the jig shown earlier, there's a simple graphic method to find the center of circle, using another geometric principle. Theoretically, because all of the possible diameters of a circle intersect in a single point (which is the center point), it's enough to draw 2 intersecting diameters in order to find the center point. But how can we draw a diameter without knowing where's the center point? well, there's a theorem that says that In a circle, any inscribed angle that subtends on a diameter, is a right angle. So we can use this theorem in reverse order, and by drawing a right inscribed angle, find the diameter that it subtends on.

So the steps are:

1. Draw a right inscribed angle using a straight edge ruler.

2. Draw the chord that connect its intersections with the circle (which is in fact a diameter).

3. Repeat the process 1 or 2 times more, with different angle locations.

4. Find the circle center, which is the intersection point of the diameters you've just found.

Let's say you have a circular object, and you want to mark equal spacing along it's perimeter. Why would you want to do that? to make a clock, for example. Or to show off your pizza cutting capabilities. get creative!

So to do that is to you'll need a compass and a straight angle ruler. First, you'll need to find the middle of the circle. which you already know how. Now, mark a vertical and horizontal diameters (chords that intersect the center point), with the straight angle ruler. Then take a compass, and adjust the span of the compass to a radius (needle on the perimeter of the circle, pencil point on the circle center). Now, with the compass needle at one of the intersection points between the diameters and the perimeter, draw an imaginary arc with the compass, until the arc intersects the perimeter of the circle, and mark that intersection point. Notice that are 2 of those intersection points. repeat the process with the other intersection points between the diameter and the perimeter, and you should end up with 12 perfectly spaced marks along the perimeter of the circle.

The math behind this trick is as followed: every time you mark that imaginary arc, you're actually creating a virtual equilateral triangle, made of a chord between the points you've marked, and 2 radii. In such a triangle, all the angles are of 60 degrees. And since a circle is in fact a 360 angle, you divide the 'path' of the perimeter to 6 equal parts (360/6). So, If you were to use only one diameter instead of the 2, you would divide the circle into 6 equal parts, not 12. but because you have 2 'starts' (2 diameters to start from), you double the division.

One of the most widely used theorems in geometry is the Pythagoras theorem. It says that in a right triangle, the square length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squared lengths of the other two sides. Consequently, if a triangle satisfies this condition, then it must be right triangle, right?

We can use that rule in order to check for perpendicularity between 2 perpendicular elements. Let's say you've welded 2 steel tubes together, but can't use a right angle ruler because of the fillet weld (see photos). Or, more likely in my case, you just can't seem to remember where you've left the damn thing.

So how do we use Pythagoras to check that our angle is right? we can measure and mark 2 known distances from the connection point, and then measure the imaginary hypotenuse between them. If it satisfies Pythagoras theorem, i.e. it's equal to the square root of the sum of squares of the other distances, the we have indeed a right angle. You must be thinking right now "sure, that's cool, but I don't have a fancy calculator in my shop and I'm not going to waste time on this mumbo jumbo calculations!", and you're absolutely right.

Luckily, there's a cheat: one of the cool things about the Pythagoras theorem, is that there are a few famous series of numbers that are known to satisfy the theorem - (3,4,5), (5,12,13), (7,24,25) and more. The first series, (3,4,5), is by far the easiest to remember and use.

So back to our welded beams. If we measure and mark, say, 30 cm to one direction, and measure and mark 40 cm in the other direction, then if the "air distance" between the marks (which is the hypotenuse) is 50 cm, the it is a right angle!

This is a simple trick to check whether a rectangular frame you've just built is indeed rectangular or is it warped (like a parallelogram). You can also look at it as a way to check for squareness of all the corners at ones. The geometric principle is super simple - in a rectangle, the diagonals are equal. In a parallelogram, they're not.

So just measure the diagonals of your frame, and if they're equal, your built frame is rectangular (meaning that all the corner angles are right angles). If not, you need to adjust or rebuild the frame so that it does.

Thank you for checking out my instructable. As mentioned, it's my first, so I would appreciate any feedback. In addition, if you've liked it, please vote for it in the contest.

Goodbye for now!