Question

Let $P$ be a point at a distance $d$ from the center of a circle of radius $r$. The curve traced out by $P$ as the circle rolls along a straight line is called a trochoid. (Think of the motion of a point on a spoke of a bicycle wheel.) The cycloid is the special case of a trochoid with $d=r$. Using the same parameter $\theta$ as for the cycloid, and assuming the line is the $x$ -axis and $\theta=0$ when $P$ is at one of its lowest points, show that parametric equations of the trochoid are $$ x=r \theta-d \sin \theta \quad y=r-d \cos \theta $$ Sketch the trochoid for the cases $d<r$ and $d>r$.

          Let $P$ be a point at a distance $d$ from the center of a circle of radius $r$. The curve traced out by $P$ as the circle rolls along a straight line is called a trochoid. (Think of the motion of a point on a spoke of a bicycle wheel.) The cycloid is the special case of a trochoid with $d=r$. Using the same parameter $\theta$ as for the cycloid, and assuming the line is the $x$ -axis and $\theta=0$ when $P$ is at one of its lowest points, show that parametric equations of the trochoid are
$$
x=r \theta-d \sin \theta \quad y=r-d \cos \theta
$$
Sketch the trochoid for the cases $d<r$ and $d>r$.

Added by Kristin M.

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