I'm doing a seminar of geometry. We're learning how to classify quadrics with Maple, and there's a steps we have to follow in order to find what kind of quadric we have.
Initially, they give me this:
$$Q(x,y,x)=x^2+2xy-2xz-3y^2+2yz+z^2-2x-2y+6z+9.$$
We complete the squares until we have this equivalent expression of $Q$:
$$Q(x,y,z)=(x+y-z-1)^2-4(y-z(\frac{1}{2}))^2+(z+2)^2+4.$$
We haven't done a single exercise of quadric classification, and from here, after finding this expression, they're asking me to find what kind of quadric it is, by looking at the expression above. How can we classify the quadric only by looking at it?
I thought that this could be the answer:
$$Q(x,y,z)=(x+y-z-1)^2-4(y-z(\frac{1}{2}))^2+(z+2)^2+4 \sim (X)^2-4(Z)^2+(Y)^2+4,$$
changind the variables properly. So, it's like the canonic form
$$R(x,y,z)=x^2+y^2-z^2+1,$$
identifying $X$ with $x$, $Z$ with $y$ and $Y$ with $z$. This canonic form corresponds to an hiperboloid of two sheets.
Is it correct?
If it's correct, how I can find a change of variables (or an affinity $f$) such that $R(f(x,y,z))=\mu Q(x,y,z)$?
Any hint will be really appreciated.
Thanks.
You have almost solved it by yourself. Instead of rushing to introduce $X,Y,Z$, do one more simplification on $Q$, rewriting it in the form
$$Q(x,y,z) = 4 \left[ \left( \frac {x+y-z-1} 2 \right)^2 - \left( y - \frac z 2\right)^2 + \left( \frac {z+2} 2 \right)^2 + 1 \right] .$$
Define
$$f(x,y,z) = \left( \begin{matrix} X \\ Y \\ Z \end{matrix} \right) = \left( \begin{matrix} \frac {x+y-z-1} 2 \\ \frac {z+2} 2 \\ y - \frac z 2 \end{matrix} \right) = \left( \begin{matrix} \frac 1 2 & \frac 1 2 & -\frac 1 2 \\ 0 & 0 & \frac 1 2 \\ 0 & 1 & -\frac 1 2 \end{matrix} \right) \left( \begin{matrix} x \\ y \\ z \end{matrix} \right) + \left( \begin{matrix} -\frac 1 2 \\ 1 \\ 0 \end{matrix} \right).$$
You can see that $f$ first acts by multiplication by a linear transformation (the $3 \times 3$ matrix), then translates this result by a constant vector, therefore $f$ is an affine transformation.
Note that $Q(x, y, z) = 4 R(f(x, y, z))$, so $\mu = \dfrac 1 4$.