Roulette

Compute roulette curves (derived by rolling one curve against another), in matlab.

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Description

This package contains code for drawing "roulettes" which are mathematical curves created when one shape is rolled against another and we trace a location relative to the curve as it rolls (think spirographs).

Exact algorithms and sources are documented in the code.

Dependencies

Currently it can either compute elliptic integrals by numeric integration (which is a bit crude), or using the package at http://code.google.com/p/elliptic/. If you want to use the latter, then you need to set the path to find it in common.m. Some of the tests assume you have the second more accurate method for doing calculations of arc lengths.

Install

Change into the main directory, and run "common.m" to put the subdirectories on the path.

Edit elliptic_int_test.m to show the path to the elliptical integral code if this is installed, or leave it blank if not (results will be potentially faster and more accurate with that code).

Then run the tests to see if everything works, and use them as an example to get things going.

Contents

The code contains:

• roulette1.m: calculates roulettes of specified curve when rolled along the x-axis
• roulette2.m: calculates roulettes of one curve rolled against another

In each of these cases curves are specified by a two functions:

• one takes a parameters $t$ (or vector of parameters) and returns the appropriate $(x,y)$ Cartesian coordinates along the curve.
• the other returns the derivatives $(dx/dt, dy/dt)$ at the same points

There are several examples of such functions in the directory Curves/

• ellipse.m and ellipsed.m (which can do a circle as a special case)
• hyperbola.m and hyperbolad.m
• parabola.m and parabolad.m

There are also several precalculated roulettes to use as test functions in the directory Roulettes/

• `cycloid.m``: roulette of a circle on a line (includes prolate and curtate cases)
• epitrochoid.m: roulette of a circle on the outside of another circle (epicycloid is a special case)
• hypotrochoid.m: roulette of a circle on the inside of another circle (hypocycloid is a special case)
• undulary.m: gives the roulette of an ellipse along a straight line
• nodary.m: gives the roulette of a hyperbola, NB: there are two curves created by this, one for each focus of the hyperbola
• catenary.m: gives the roulette of a parabola
• cissoid.m: of Dioclese: the roulette of two parabolas

There is one function defined to help in calculations

• arclength.m numerically calculates arclengths along curves in whatever parameterisation is being used

There are a couple of test scripts, which also provide some examples of how to use the code in the directory 'Tests_and_Examples/'

• arclength_test.m: provides tests of the arclength code. It does some diagnostic plots, and outputs differences between numerical and theoretical arclengths. Some deviations at about the 1.0e-6 level occur.
• roulette1_test.m: shows a selection of standard roulettes both calculated using roulette1, and using the theoretical curve
• roulette2_test.m: tests a selection of standard roulettes formed from two curves

The two directories:

• Plots/
• Animations/

contain output from examples.

Examples

For examples of how to use the code see

• roulette1_test.m
• roulette2_test.m
• roulette_animate.m

The test code does a few plots, which go in the Plots/ directory.

There is also an script that will create some animations in the Animations/ directory.

Matt Roughan, 2015