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Chapter 9: Q.63 (page 721)

An epicycloid is another variation of a cycloid in which the point tracing the path is on the circumference of a wheel, but the wheel is rolling without slipping on the outside of another wheel, instead of along a horizontal track. If the radius of the rolling wheel is k and the radius of the fixed wheel is r, find parametric equations for the epicycloid.

Short Answer

Expert verified

As a response, the parametric equations

x=(r+k)cos胃-kcosrk+1胃y=(r+k)sin胃-rsinrk+1胃

Step by step solution

01

Step: 1 Given information

Consider an epicycloid in which the path is traced around the circumference of a wheel, but the wheel is not slipping on the outside of another wheel.

02

Step: 2: Calculation

The angles 胃,蠒are related to each other.

Let AA'd denote the arc length.

The arc length is equal to r胃.

AA'=r胃while A'p=k(胃+蠒)

We can conclude that蠒=rk+1胃

03

Step: 3: Further Calculation

The coordinates of pwith respect to origin are.

x=(r+k)cos胃-kcosrk+1胃y=(r+k)sin胃-rsinrk+1胃

The minus sign in the second component of the x-coordinate equation comes from the fact that the rolling circle's rotation and its center's motion are in the same direction.

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Most popular questions from this chapter

Use Cartesian coordinates to express the equations for the parabolas determined by the conditions specified in Exercises 22鈥�31.

directrixy=0,focus(0,1)

In Exercises 32鈥�47 convert the equations given in polar coordinates to rectangular coordinates.

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In Exercises 24鈥�31 find all polar coordinate representations for the point given in rectangular coordinates.

0,-3

Use polar coordinates to graph the conics in Exercises 44鈥�51.

r=24+cos胃

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