# Making cuts on a spiral so that all segments are of the same length

Imagine a roll of magnetic strip (about 2 cm in width) which is rolled in a roll of about 30 windings and about 10m length. The strip is about 2cm in width.

The roll is put on a Rod which the user unwinds by hand and with a knife he cuts segments of about 5 cm in length.

Another way of doing this would be to perform the cuts directly on the roll.

The issue is that the inner windings are smaller than the outside windings thus if we make the cuts directly on the roll we come up with different length segments all around.

Is their a geometrically correct way of making the cuts so that all segments will come out the same length?

This is a naive visualisation of the cuts; naturally the segments will come out in different lengths as we approach the centre of the spiral.

And this is how the user cuts them up until now:

There should be a mathematic formula which dictates where to make the cuts directly on the spiral(the roll) so all segments comes out at the same length.

If the rolls are round and the windings evenly spaced (radially), I do not believe this is possible. Most arbitrary lengths, relative to the diameter of the roll, produce scattered cuts (red marks below) that do not align:

Certain lengths result in cuts that more or less align, but they do not form straight lines:

If my simulation is not troubled by numerical inaccuracy it seems local optimums for alignment become worse as segment length increases. An example of the best I could find in the region of this length:

• well thanks for the simulation. I guess the solution is not practical enough in my environment. One could use a laser cutter to make the cuts instead of a knife thus allowing for uneven distribution of cuts. Jan 29, 2014 at 20:23

I have a slightly different approach. Basically Archimedes Spiral (as given in the answer by Mr. Wizard) is not accurately describing the curve of the windings... I did not calculate how much difference it makes, but that would depend on the specific parameters. My model does not take the flexibility of the strip into account though!

I modelled the situation by composing circular curve segments with straight lines so that the strip winds around the roll until the tangent line of the inner side of the strip points towards the outer side of the circular part of last winding. This makes the model curve a composition of

1. circular segments (magenta in my model) centered at the center og the roll
2. straight parallel lines (green in my model) defined by the first tangent of the core circle that intersects the beginning of the first circular segment mentioned above
3. circular segments (orange in my model) centered at the beginning of the first circular segment mentioned in 1.

This can be seen in my model document in the following link (made with GeoGebra):

GeoGebra-model of the situation

As Mr. Wizard alreay noted, the cuts only become practical for certain matching parameters. In the example given when opening my document it is actually possible to cut most of the segments succesfully from one half of the roll and then one could cut the remaining multiples of the segment length afterwards.

Otherwise you could calculate in some waste in order to have a distance between the cuts that meets the diameter of the roll and thickness of the strip well so that the cuts can be initially of the wrong length but with little waste and then afterwards stacking and cutting these segments. There are many possibilities to consider!

• That's a nice refinement, and the first time I've seen GeoGebra too. +2 if I could! Feb 8, 2014 at 16:51