Chapter 6.3, Problem 54E

Solution

a)

To plot: The graph simultaneously the hypocycloid and the circle of the radius 3.

The graph is shown in figure (1) to (4).

Given information:

The circle is $x^2+y^2=9$ and the hypocycloid $x=2\cos t+\cos 2t$ , $y=2\sin t-\sin 2t$ .

Calculation:

Represent the circle equation in the parametric form: $x=3\cos t$ , $y=3\sin t$ .

For $0\le t\le 1$, the graph of both circle and hypocycloid is given below:

  

Figure (1)

For $0\le t\le 2$, the graph of both circle and hypocycloid is given below:

  

Figure (2)

For $0\le t\le 4$, the graph of both circle and hypocycloid is given below:

  

Figure (3)

For $0\le t\le 6.5$, the graph of both circle and hypocycloid is given below:

  

Figure (4)

b)

To find:

Variant of Hypocycloid with parametric form: $x=3\cos t+\cos 3t$ , $y=3\sin t-\sin 3t$ .

Given information:

The circle of radius 4.

Calculation:

If replace all 2’s by 3 in the hypocycloid equation, the obtained equation is variant of the hypocycloid whose parametric form is given below:

  $x=3\cos t+\cos 3t$ , $y=3\sin t-\sin 3t$ .

  

Figure (5)

Therefore, the variant of Hypocycloid with parametric form: $x=3\cos t+\cos 3t$ , $y=3\sin t-\sin 3t$ .