DIFFERENTIAL GEOMETRY: EPITROCHOIDS

The epitrochoid is the curve generated by a point p on the flange of a wheel of radius b  rolling around a fixed circle of radius a.  If h is the distance of the point from the centre of the wheel,  then the epitrochoid has eqn:  gamma(t) =((a+b)cos t - h cos((a+b)t/b),  (a+b)sin t - h sin((a+b)t/b)).

For the example on the Example sheet, a=18, b=2, h=6.  We plot this using a single Maple command:

>    plot([20*cos(t)-6*cos(10*t),20*sin(t)-6*sin(10*t),t=0..2*Pi],-30..30,-30..30,scaling=constrained,legend=Epitrochoid1);

[Maple Plot]

Note how easy it is to use Maple, just edit the command below to display other parametrized curves in the plane.  The -30..30,-30..30 gives the range of x and y, `scaling=constrained' means that the axes have the same scale, omit this if you want Maple to decide on scales.

On the next page, we vary the parameters.  Firstly, we put  h=b, getting the curve generated by the point ON a circle which is rolling round another circle.  This is called an epicycloid.   Then we take h less than b, getting the curve generated by a point INSIDE a disk which is rolling round another circle.

>    plot([20*cos(t)-2*cos(10*t),20*sin(t)-2*sin(10*t),t=0..2*Pi],-30..30,-30..30,scaling=constrained, legend=epicycloid);

[Maple Plot]

>    plot([20*cos(t)-1*cos(10*t),20*sin(t)-1*sin(10*t),t=0..2*Pi],-30..30,-30..30,scaling=constrained, legend=Epitrochoid2);

[Maple Plot]