How can I understand this parametrization belongs to an ellipse?
$$\vec{r}(t) = (5 \cos t, 5 \sin t, 10t); \; 0 < t < 2\pi$$
My attempt:
$$x(t) = 5 \cos t$$
$$y(t) = 5 \sin t$$
$$z(t) = 10t$$
I know the O-centered ellipse parametrization is
$$\vec{r}(t) = (a \cos t, b \sin t); \; t \in [0,2\pi]$$
While I'm not familiar yet with curves in $\mathbb{R}^{3}$, what about the z-component? How can I plot this curve?
Thanks in advance
Best Answer
It is not an ellipse, it is a vertical helix.
More in detail the projection onto x-y plane is a circle centered at the origin with radius $R=5$ indeed
$$x^2+y^2=(5\cos t)^2+(5\sin t)^2=25$$