Spiral array of Constant Breath Solid Rollers

thingiverse

Adds the ability to print arrays of rollers where the number of need not be composite, Any number of rollers may be used and keeping the packing tight should reduce print travel time. Additionally the Archimedean spiral is that the rollers print onto is interesting itself. About Constant Breath Solids I came across ‘Curves of Constant Breadth’ in an old book – ‘Riddles In Mathematics: a book of paradoxes’ by Eugene P. Northrop. Northrop introduces the idea of Curves of Constant Breadth by questioning whether a roller used under a sledge for moving heavy objects must be round. Surprisingly, the answer is “No.” Northrop produces the example of the Reuleaux triangle as a Curve of Constant Breadth, though he notes that the Reuleaux triangle makes a terrible roller due to its relatively sharp points. However, Northrop notes the points can be smoothed by prolonging the sides of the triangle some distance (s) and drawing a curve of radius (s) between the two extended sides. Where the breadth of the Reuleaux triangle is (L) (i.e. the length of each side of the underlying equilateral triangle), the breadth of the smoothed Curve of Constant Breadth is (2s+L). https://archive.org/details/riddles-in-mathematics-a-book-of-paradoxes-eugene-p-northrop/page/58/mode/2up A bit of math obscurity, Curves of Constant Breadth are the counter example to the oft asked question “Why are manhole covers round?” The traditional answer is “because only a round cover won’t fall in the hole.” The Reuleaux triangle is a counter example that shows that any Curve of Constant Breadth will make a serviceable manhole cover . . . although circular covers are probably easier to make and are certainly much more common. In contrast to stamped manhole covers, the beauty of 3-D printing is that objects with “complex” shapes are just as easy to slice and print as simpler ones, so with a little math and OpenSCAD we have an entire family of rollers made of Curves of Constant Breadth based off of an equilateral triangle. The OpenSCAD code/module included here takes the concept of a smoothed Reuleaux triangle a little further by introducing the concept of roundedness to quantify how much the Reuleaux triangle has been smoothed. An unsmoothed Curve of Constant Breadth would have a roundedness of 0 and in the OpenSCAD code/module would be a Reuleaux triangle, a fully smoothed Curve of Constant Breadth has a roundedness of 1 and would be a circle. Breadth (Diameter) =2s+L where L is the length of the side of the base equilateral triangle and the roundedness (Round) = 2s/D. Inputs for the module / customizer are: Diameter - or the Breadth of the solid Round - the roundedness of the roller, a number from 0 to 1. (Note: rollers with less roundness than .4 or .5 tend not to roll to well) About Archimedean spiral Code Archimedean spiral is adapted from the work of Justin Lin's as found on his helpful website "openhome.cc" specifically, the page "https://openhome.cc/eGossip/OpenSCAD/ArchimedeanSpiral.html" And used in https://www.thingiverse.com/thing:1763139 Spiral moving fish The original code was a great starting point, but it had two specific issues: 1) The accuracy of the predicted theta's was low below 4*PI radians 2) when the next theta was more that PI radians from the starting point the acos() function failed making staring values less than ~PI unusable for my purposes I wanted to pack as many objects into a tight spiral array as possible, to allow for any arbitrary number of objects to be printed in a tight space. Consequently, I needed reasonable accuracy at low theta values. The code solves issue 1 above by adding a second layer of recursion, making iteratively better and better guesses of theta2 until the number of tries is exhausted (default set to 50) or the requested error is achieved (default set to 1/100 of a percent) The code soles issue 2 by iterating based on length(L) vs Theta2 replacing the acos() with a cos(function) which is defined for all theta