# Turn the direction of movement of a parametric curve

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?

• What do you mean by "to turn the direction of movement"? – Aretino Oct 9 '15 at 21:19
• @Aretino I mean that the curve starts at the same point for the same $t$, but moves the other way around. – Heinz Doofenschmirtz Oct 10 '15 at 6:56
• To do that you have to change $t\to -t$. – Aretino Oct 10 '15 at 11:45

The curve is an ellipse shown below. As $t$ increases the position of the general point $(x,y)$ rotates in an anticlockwise direction.
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.
• 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)? – Heinz Doofenschmirtz Oct 10 '15 at 6:57
• Set $t \rightarrow -t$ – tomi Oct 11 '15 at 0:23