Parameterizing a non-centered circle

Question

So basically I am stuck with parameterizing a curve. Half of a unit circle is centered at $(1,0)$ in the first quadrant and traced clockwise from $(0,0)$ to $(2,0)$. I am not so sure how to parameterize this.

Answer 1

centre is (1,0) and radius is 1

$X = X_0 + r* \cos z = 1 + \cos z$
$Y = Y_0 + r* \sin z = 0 + \sin z$

You Just want Y to be non negative.

So, just add a constraint on z, since $y = \sin z$

and $sin z < 0$ for $z < 2*\pi$ and $z> \pi$

Finally,$ (1+ \cos z, \sin z), 0<= z <= \pi$

Answer 2

Start with the a parametrization of a unit circle centered at the origin (mouseover to reveal):

$r_1(t) = \langle \cos t, \sin t \rangle$, $t \in \mathbb{R}$

Then fix the orientation so we travel clockwise:

$r_2(t) = \langle \sin t, \cos t \rangle$, $t \in \mathbb{R}$

Then restrict the values of $t$ to get only the top half:

$r_3(t) = \langle \sin t, \cos t \rangle$, $t \in [\pi, 2 \pi]$

And then shift everything right by one unit:

$r_4(t) = \langle \sin t, \cos t \rangle + \langle 1, 0 \rangle = \langle 1 + \sin t, \cos t \rangle$ , $t \in [\pi, 2 \pi]$.