Geometric difference between $x^TAx$ and $x^TAx + b^Tx + c$

Question

What is the difference between $x^TAx$ and $x^TAx + b^Tx + c$ geometrically? Some analogous examples from quadriatic equations would be great.

Answer 1

A special case: $A=I, b=[1,1]$ and assume the vectors lie in 2-d. Thus, we have $x^Tx=1$ and $x^Tx+b^Tx=1$. Former is a circle with center at origin and latter another circle with a different center.. In general, for positive definite $A$ former one will be a ellipsoid with origin as center and the latter will also be a ellipsoid with a different center. Other aspects like foci will also be different.

Answer 2

Simply, the first case is an homogeneous polynomial of degree 2; the second is a non homogeneous polynomial of degree 2.

EDIT: Maybe I can't really understand your question. I hope this comment can be helpful. Consider that if $x \in \mathbb{R}^{n}$ you can rewrite ${x}^{T} A x + {b}^{T} x +c$ in this way: $(x^{T},1) B (x^{T},1)^{T}$ where B is $\left(\begin{array}{c|c} A & b^{T} \\ \hline 0 & c \end{array}\right)$