Normal form for quadrics

Question

In the classification theorem of the wiki's page on quadric, the normal form is obtained by action of rigid body transformations (called also as change of Cartesian coordinates).

If homotheties are considered, then there wouldn't be so many orbits.

If we consider all affine transformations, then there would be only one orbit so the classification wouldn't be interesting.

Yet, in the proposed theorem it seems that Rigid body transformations are the same as change of Cartesian coordinates, but an homothety is a change of Cartesian coordinates (changing an orthonormal basis onto an other orthonormal basis), for instance $\varphi(x,y,z)=2(x,y,z)$, but is not a rigid body transformation.

1. When they speak about Euclidean transformations, do they exclude reflections ? But I wonder if it makes sense if we take into account reflections ?

2. Do you have some references containing a proof of the calculations of this normal form.

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Since all equations are invariant by reflections, we can take them into account.

Answer 1

… rigid body transformations (called also as change of Cartesian coordinates)

I'd read that as a rigid body transformation is (a specific case of) a change of coordinates. But not everything that changes coordinates is a rigid body transformation. In particular, a general homothety is not.

When they speak about Euclidean transformations, do they exclude reflections?

Probably yes, but it makes no difference since every quadric has at least one plane of symmetry, so it's reflections will end up in the same orbit.

Do you have some references containing a proof of the calculations of this normal form.

The Wikipedia article dies cite two references. But let's see. Start with $xQx^T+Px^T+R=0$ from the beginning of the article. Moving the center of symmetry of the quadric to the center of the coordinate system removes linear terms, so you end up with something of the form $xQx^T+R=0$. (Note that the $Q$ and $R$ in these two equations are not the same unless $P=(0,0,0)$.) The matrix $Q$ can be chosen to be symmetric, and is likely real if you are dealing with Euclidean space. So quoting the Wikipedia article on Real symmetric matrices:

The finite-dimensional spectral theorem says that any symmetric matrix whose entries are real can be diagonalized by an orthogonal matrix.

That orthogonal matrix is just a Euclidean rotation. So you can find a representation (i.e. an element from the orbit) where $P$ is null and $Q$ is diagonal. Expand this form and you get

$$Q_{11}x^2+Q_{22}y^2+Q_{33}z^2+R=0\;.$$

So now consider all possible signs for these numbers. At fisr there are $3^4=81$ possible sign combinations. But since you can exchange coordinates freely (just a rotation or reflection), a lot of them are essentially the same. And some are too degenerate to be considered quadric, although that decision might depend on context. Try to come up with a reduced list, then find a way to map each sign combination to one and only one of the 17 normal forms. Also check out which of the 17 normal forms are not among those you found, and figure out which step in my explanation was wrong for them. Come up with a better case distinction. Feel free to post your findings as a separate answer.