The curve is a circle centered at the origin. It is traced once, counterclockwise, and contains all points of the unit circle except for $(0,-1)$with $t\in \mathrm{ℝ}$.

The points of the unit circle except for $(0,-1)$ with $t\in \mathrm{ℝ}$

Consider a curve centered at the origin with $t\in \mathrm{ℝ}$.

The objective is to find the parametric equations which represent the given condition.

Given that the curve is centered at the origin, traced once in counter clockwise direction.

The curve is a unit circle. So the radius of the curve is 1 .

The curve not passes through the point$(0,-1)$.

The parametric equations for the curve which moves counter clockwise direction with center at origin is given by,

Thus, $x=\frac{2t}{1+t^2},y=\frac{1-t^2}{1+t^2}$ forms a circle with center $(0,0)$ and radius is 1 and the curve moves in counter clockwise direction, and not passes through the point $(0,-1)$.

Therefore, the required parametric equations are $x=\frac{2t}{1+t^2},y=\frac{1-t^2}{1+t^2}$.