Cycloid

*In assignment 10, we are
exploring the parametric curves.*

*A parametric curve in the
xy-plane is determined by a pair of functions x(t) and y(t). In
other words, the curve is given parametrically by the pair of
functions x(t) and y(t).*

*The two equations are usually
called the parametric equations of a curve. Parametric equations
are a set of equations and each function is expressed by independent
variables, which are called parameters. For example, while the
equation of a circle in Cartesian coordinates can be given by*

*one set of parametric equations
for the circle are given by*

*Here I will investigete
the cycloid. The cycloid is the locus of a point on the rim of
a circle that rolls along a straight line. It was studied and
named by Galileo in 1599. First, let's construct the cycloid with
GSP and find the parametric equations for the cycloid.*

*If the cycloid has a cusp
at the origin and its humps are oriented upward, its parametric
equation is*

*Let's take a=1.25 and plot
the cycloid. Here I used the Graphing Calculator 3.2.*

*Next let's have more interesting
cycloids. How about moving the circle(let's call this blue circle
'wheel') on the circle insteady of the straight line.*

*Trisect the radius of the
big circle and move the wheel around the small circle, whose radius
is 2/3 of the radius of the big circle. Here the radius of the
wheel is 1/3 of the big circle, so the wheel is moving around
the big circle while the center of the wheel is moving around
the small circle.*

*What is going on if we bisect
the radius of the big circle?*

*Let's bisect the radius
of the big circle and put the wheel which the radius is 1/2 of
the radius of the big circle.*

*Now, let's play the animation
and find the trace.*

*Here the cycloid is a line,
which is a diameter of the big circle. Isn't it interesting?*