An Exploration of the Cycloid
By
Kenneth E. Montgomery
Description
For a wheel
rolling along a line (Figure 1), we let the point of tangency for the circle
and line be point T. The circle has center, C and radius . is
the line segment congruent to the radius, k that is perpendicular to the
line on which the circle is rolling, which we have taken to be the x-axis. is
the line segment, parallel to the x-axis, formed by endpoints R
and S, which is the segment’s intersection with .
We then have the right triangle,,
with .
The locus of points formed by R, as the circle rolls along the x-axis
is known as the cycloid and is given by the pink trace in Figure 1.
Open this GSP file to explore the cycloid of a circle,
rolling along the x-axis.
Figure 1: Cycloid of a wheel
rolling along the x-axis
Derivation of Parametric
Equations for the Cycloid
Let be the parameter. Then , since the circle is rolling along the x-axis.
We have the center, C given by:
From , we have
Thus, we have
The parametric equations for the
cycloid are therefore given by
We then plot these
parametric equations in Graphing Calculator 3.2 to obtain the graph in Figure
2, for four revolutions of the wheel.
Figure 2: ,
Open
this GCF file to explore the cycloid in Graphing
Calculator.
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