Center of an ellipse
= Center of an ellipse
{title2=$\mathbf c$}
The <center of an ellipse> is its unique <center of symmetry>. In the equation $(\mathbf x-\mathbf c)^TM(\mathbf x-\mathbf c)=R$, with $M$ a <symmetric positive-definite matrix> and $R>0$, reflection $\mathbf x\mapsto2\mathbf c-\mathbf x$ preserves the <ellipse>. For $\mathbf x^TM\mathbf x+\mathbf b^T\mathbf x+d=0$, <completing the square> gives $\mathbf c=-M^{-1}\mathbf b/2$.