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Given is the parametric curve $K$ that satisfies $$\begin{cases}x=3\sin t \\ y=2\cos\left(t-\frac{1}{4}\pi\right)\end{cases}$$

How can you change the parametric equations if you want to turn the direction of movement? The answers say $t\mapsto t+\frac{1}{4}\pi$, but I don't understand why.. Could someone explain?

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  • $\begingroup$ What do you mean by "to turn the direction of movement"? $\endgroup$ – Aretino Oct 9 '15 at 21:19
  • $\begingroup$ @Aretino I mean that the curve starts at the same point for the same $t$, but moves the other way around. $\endgroup$ – Heinz Doofenschmirtz Oct 10 '15 at 6:56
  • $\begingroup$ To do that you have to change $t\to -t$. $\endgroup$ – Aretino Oct 10 '15 at 11:45
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The curve is an ellipse shown below. As $t$ increases the position of the general point $(x,y)$ rotates in an anticlockwise direction.

enter image description here

The transformation $t \rightarrow t+\frac \pi 4$ keeps the curve the same. It also keeps the direction of movement the same (anticlockwise). What does change is that each point associated with a particular time moves around the curve in the direction of movement.

enter image description here

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  • $\begingroup$ Ah, so my prof was incorrect.. What happens to $t$ if I want to start at the same point for $t=0$, but move the other way around (clockwise)? $\endgroup$ – Heinz Doofenschmirtz Oct 10 '15 at 6:57
  • $\begingroup$ Set $t \rightarrow -t$ $\endgroup$ – tomi Oct 11 '15 at 0:23

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