If you have a parametric equation grapher, graph the...

If you have a parametric equation grapher, graph the equations over the given intervals. a. Epicycloid $$ x=9 \cos t-\cos 9 t, \quad y=9 \sin t-\sin 9 t ; \quad 0 \leq t \leq 2 \pi $$ b. Hypocycloid $$ x=8 \cos t+2 \cos 4 t, \quad y=8 \sin t-2 \sin 4 t ; \quad 0 \leq t \leq 2 \pi $$ c. Hypotrochoid $$ x=\cos t+5 \cos 3 t, \quad y=6 \cos t-5 \sin 3 t ; \quad 0 \leq t \leq 2 \pi $$

If you have a parametric equation grapher, graph the equations over the given intervals.
a. Epicycloid
$$
x=9 \cos t-\cos 9 t, \quad y=9 \sin t-\sin 9 t ; \quad 0 \leq t \leq 2 \pi
$$
b. Hypocycloid
$$
x=8 \cos t+2 \cos 4 t, \quad y=8 \sin t-2 \sin 4 t ; \quad 0 \leq t \leq 2 \pi
$$
c. Hypotrochoid
$$
x=\cos t+5 \cos 3 t, \quad y=6 \cos t-5 \sin 3 t ; \quad 0 \leq t \leq 2 \pi
$$

Key Concepts

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of an independent parameter, usually denoted by t. This method allows for a flexible representation of complex curves that might be difficult to describe using a single equation in Cartesian coordinates. By varying the parameter over a specified interval, one can generate the entire curve, which is particularly useful for plotting and analyzing motion and closed curves in the plane.

Epicycloid

An epicycloid is a type of curve generated by the path of a point on the circumference of a circle as it rolls without slipping around the exterior of another circle. This curve is defined using parametric equations where the motion is a combination of the circular movement of the rolling circle and the rolling action itself. Epicycloids often exhibit cusps and beautiful symmetries, reflecting the underlying periodic nature of the trigonometric functions involved.

Hypocycloid

A hypocycloid is a curve traced by a point on the circumference of a circle that rolls without slipping along the inside of a larger fixed circle. The parametric equations for hypocycloids combine the movement of the rolling circle with the rotation of the point on its edge. These curves are known for their elegant, star-like shapes and are characterized by a specific number of cusps that depend on the ratio of the radii of the two circles.

Hypotrochoid

A hypotrochoid is formed by the trajectory of a point attached to a circle as it rolls along the inside of another circle, but in this case the point does not lie on the edge of the rolling circle. The resulting curve is described by parametric equations incorporating different coefficients for the cosine and sine terms, representing the varying distances from the center of the circle to the point. Hypotrochoids can create a wide variety of intricate and looping patterns, underpinning their importance in both mathematical theory and artistic designs.

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