Given:

The given diagram is shown in Figure 1

Figure 1

Calculation:

For the circle of the radius r , the parametric equation is given by,

  x=rcosθy=rsinθ

Consider that |OT| increases to the point R and construct the line segment from the T parallel to the x axis and the line segment from the P parallel to the y axis.

Since, both the lines intersect each other at the point C. Then, from the figure is,

  ∠PCT=π2

Then,

  ∠RTC=∠TOX=θ

Then, the angle ∠OTP=π2 . Then, ∠RTP=π2

From the above given diagram,

  ∠CTP=∠RTP−∠RTC=π2−θ

Consider the coordinates of the point T are (rcosθ, rsinθ) and the initial position is (r,0) .

Then, |TP| is the radius angle is at the centre of the circle.

Then, the value of |TP| is,

  |TP|=rθ

Then for the triangle CTP is,

  |CT|=rθcos(π2−θ)=rθsinθ

For CP,

  |CP|=rθsin(π2−θ)=rθcosθ

Then, the coordinates of the point P is,

  x=rcosθ+|CT|=rcosθ+rθsinθ=r(cosθ+θsinθ)

Also,

  y=rcosθ−|CP|=rsinθ−rθcosθ=r(sinθ−θcosθ)

Thus, the required parametric equation is,

  x=r(cosθ+θsinθ)y=r(sinθ−θcosθ)