I have the parametric equation of a circle: $$f(u) = \langle a \cos(u) + b, a \sin(u) + c\rangle,$$ and because the equation has $3$ unknowns $a,b$ and $c$, I have been given $3$ points $p_0, p_1$, and $p_2$. How to solve this system of equations?
EDIT
The circle equation was just an example of a parametric equation. I need a general approach of solving this kind of parametric equations which involves parameters.
On
Mechanical method: given ${\bf p}_i = (x_i,y_i)$, $i =1,2,3$, the equation of the circle passing through the points is given by: $$\begin{vmatrix} x^2+y^2 & x & y & 1 \\ x_1^2+y_1^2 & x_1 & y_1 & 1 \\ x_2^2+y_2^2 & x_2 & y_2 & 1 \\ x_3^2+y_3^2 & x_3 & y_3 & 1\end{vmatrix} = 0.$$
Terrible to remember this, though (I even took a look at a book here, to double check).
One can approach this geometrically: The center $(b, c)$ of the circle is the intersection of the respective perpendicular bisectors of the three line segments $P_0 P_1$, $P_1 P_2$, $P_2 P_0$ (of course, it suffices to find the intersection of any two bisectors). Then, one can the radius $a$ of the circle is just the distance from $(b, c)$ to any of the given points $P_k$.