How to find the center of a spiral given only the physical object?

Question

I have a spiral cam from a belt sander tension assembly that needs to have a hole bored in the exact center and I am unsure of how to find it mathematically. I have the object and can measure the diameter at various points but I am stuck at how to get it into a mathematical equation

Answer 1

I have assumed that the shape of the cam is an Archimedean spiral, as it is commonly used for converting uniform radial motion into uniform speed translation.

The polar equation of an Archimedean spiral is:

$$r=k\theta \ \ \ \text{with} \ \ \ k:=2a/\pi$$

meaning that each time the cam undergoes a $\pi/2$ ($ = 90°$) rotation , we have a new added length $a$, then $2a$, $3a$, $4a$, $a+4a=5a$, etc., this regularly added length explaining the regular translation.

See the theoretical curve in red with a rather good agreement with the shape you have given

Having just seen the proposal posted by Intelligenci Pauca, I see that our solutions should not differ that much.

Remark: the respective width ($14 a$ with an estimated extension of the cam) and height $12a$ of the "bounding box" of the cam should help to obtain an accurate positioning of the looked for center.

Edit: If the figure is at the right scale, as I have measured a 19.5 units in the horizontal direction and 17 in the vertical direction, ratio $19.5/17$ is in a very good agreement with ratio $14/12$.

Answer 2

Since it's not a circle, the center is not well-defined; there may be different points you could reasonably call the 'center' depending on what properties you like.

One reasonable choice is the centroid, i.e. center of mass. You could find the centroid by moving a line/straightedge until the shape balances perfectly on its end, and then do this again with another line at an angle to it. The intersection of the two lines is the centroid. You can check that the shape should balance perfectly on this point. See plumb-line method at wikipedia.

If you want a purely mathematical way to find the center of mass, you will need to provide precise equations for the shape. Then you can use calculus methods.

Answer 3

Playing around with GeoGebra (which I'd suggest you to try) I found this one as my best logarithmic spiral:

and this other as my best archimedean spiral:

The log spiral looks better fitting with the sketch you gave.

Hope this helps.