Thiele's interpolation formula

In projective geometry a quadric is the set of points of a projective space where a certain quadratic form on the homogeneous coordinates becomes zero. We shall restrict ourself to the case of finite dimensional projective spaces.

Quadratic forms

Let be $K$ a field and $\mathcal{𝒱}(K)$ a vector space over $K$. A mapping $\rho$ from $\mathcal{𝒱}(K)$ to $K$ such that

(Q1) $\rho (x\overrightarrow{x})=x^2\rho (\overrightarrow{x})$ for any $x\in K$ and $\overrightarrow{x}\in \mathcal{𝒱}(K)$.

(Q2) $f(\overrightarrow{x},\overrightarrow{y}):=\rho (\overrightarrow{x}+\overrightarrow{y})-\rho (\overrightarrow{x})-\rho (\overrightarrow{y})$ is a bilinear form.

is called quadratic form. (the bilinear form $f$ is even symmetric!)

In case of $charK\ne 2$ we have $f(\overrightarrow{x},\overrightarrow{x})=2\rho (\overrightarrow{x})$, i.e. $f$ and $\rho$ are mutually determined in a unique way.
In case of $charK=2$ we have always $f(\overrightarrow{x},\overrightarrow{x})=0$, i.e. $f$ is symplectic.

For $\mathcal{𝒱}(K)=K^n$ and $\overrightarrow{x}=\sum _{i=1}^n{x}_i{\overrightarrow{e}}_i$ ($\{{\overrightarrow{e}}_1,\dots ,{\overrightarrow{e}}_n\}$ is a base of $\mathcal{𝒱}(K)$) $\rho$ has the form

$\rho (\overrightarrow{x})=\sum _{1=i\le k}^n{a}_{ik}{x}_i{x}_k\text{ with }{a}_{ik}:=f({\overrightarrow{e}}_i,{\overrightarrow{e}}_k)\text{ for }i\ne k\text{ and }{a}_{ik}:=\rho ({\overrightarrow{e}}_i)\text{ for }i=k$ and

$f(\overrightarrow{x},\overrightarrow{y})=\sum _{1=i\le k}^n{a}_{ik}(x_i{y}_k+x_k{y}_i)$.

For example:

$n=3,\rho (\overrightarrow{x})=x_1{x}_2-x_3^2,f(\overrightarrow{x},\overrightarrow{y})=x_1{y}_2+x_2{y}_1-2x_3{y}_3.$

Definition and properties of a quadric

Below let $K$ be a field, $2\le n\in \mathbb{\mathbb{N}}$, and ${\mathfrak{P}}_n(K)=(\mathcal{𝒫},\mathcal{𝒢},\in )$ the n-dimensional projective space over $K$, i.e.

$\mathcal{𝒫}=\{⟨\overrightarrow{x}⟩\mid \overrightarrow{0}\ne \overrightarrow{x}\in V_{n+1}(K)\}$

the set of points. ($V_{n+1}(K)$ is a (n + 1)-dimensional vectorspace over field $K$ and $⟨\overrightarrow{x}⟩$ is the 1-dimensional subspace generated by $\overrightarrow{x}$),

$\mathcal{𝒢}=\{\{⟨\overrightarrow{x}⟩\in \mathcal{𝒫}\mid \overrightarrow{x}\in U\}\mid U\text{ 2-dimensional subspace of }{V}_{n+1}(K)\}$

the set of lines.

Additionally let be $\rho$ a quadratic form on vector space $V_{n+1}(K)$. A point $⟨\overrightarrow{x}⟩\in \mathcal{𝒫}$ is called singular if $\rho (\overrightarrow{x})=0$. The set

$\mathcal{𝒬}=\{⟨\overrightarrow{x}⟩\in \mathcal{𝒫}\mid \rho (\overrightarrow{x})=0\}$

of singular points of $\rho$ is called quadric (with respect to the quadratic form $\rho$). For point $P=⟨\overrightarrow{p}⟩\in \mathcal{𝒫}$ the set

$P^{\perp }:=\{⟨\overrightarrow{x}⟩\in \mathcal{𝒫}\mid f(\overrightarrow{p},\overrightarrow{x})=0\}$

is called polar space of $P$ (with respect of $\rho$). Obviously $P^{\perp }$ is either a hyperplane or $\mathcal{𝒫}$.

For the considerations below we assume: $\mathcal{𝒬}\ne \mathrm{\varnothing }$.

Example: For $\rho (\overrightarrow{x})=x_1{x}_2-x_3^2$ we get a conic in ${\mathfrak{P}}_2(K)$.

For the intersection of a line with a quadric $\mathcal{𝒬}$ we get:

Lemma: For a line $g$ (of $P_n(K)$) the following cases occur:

a) $g\cap \mathcal{𝒬}=\mathrm{\varnothing }$ and $g$ is called exterior line or

b) $g\subset \mathcal{𝒬}$ and $g$ is called tangent line or

b') $|g\cap \mathcal{𝒬}|=1$ and $g$ is called tangent line or

c) $|g\cap \mathcal{𝒬}|=2$ and $g$ is called secant line.

Lemma: A line $g$ through point $P\in \mathcal{𝒬}$ is a tangent line if and only if $g\subset P^{\perp }$.

Lemma:

a) $\mathcal{ℛ}:=\{P\in \mathcal{𝒫}\mid P^{\perp }=\mathcal{𝒫}\}$ is a (projective) subspace. $\mathcal{ℛ}$ is called f-radical of quadric $\mathcal{𝒬}$.

b) $\mathcal{𝒮}:=\mathcal{ℛ}\cap \mathcal{𝒬}$ is a (projective) subspace. $\mathcal{𝒮}$ is called singular radical or $\rho$-radical of $\mathcal{𝒬}$.

c) In case of $\mathrm{char}K\ne 2$ we have $\mathcal{ℛ}=\mathcal{𝒮}$.

A quadric is called non-degenerate if $\mathcal{𝒮}=\mathrm{\varnothing }$.

Remark: An oval conic is a non-degenerate quadric. In case of $\mathrm{char}K=2$ its knot is the f-radical, i.e. $\mathrm{\varnothing }=\mathcal{𝒮}\ne \mathcal{ℛ}$.

A quadric is a rather homogeneous object:

Lemma: For any point $P\in \mathcal{𝒫}\setminus (\mathcal{𝒬}\cup \mathcal{ℛ})$ there exists an involutorial central collineation ${\sigma }_P$ with center $P$ and ${\sigma }_P(\mathcal{𝒬})=\mathcal{𝒬}$.

Proof: Due to $P\in \mathcal{𝒫}\setminus (\mathcal{𝒬}\cup \mathcal{ℛ})$ the polar space $P^{\perp }$ is a hyperplane.

The linear mapping

$\phi :\overrightarrow{x}\to \overrightarrow{x}-\frac{f(\overrightarrow{p},\overrightarrow{x})}{\rho (\overrightarrow{p})}\overrightarrow{p}$

induces an involutorial central collineation with axis $P^{\perp }$ and centre $P$ which leaves $\mathcal{𝒬}$ invariant.
In case of $\mathrm{char}K\ne 2$ mapping $\phi$ gets the familiar shape $\phi :\overrightarrow{x}\to \overrightarrow{x}-2\frac{f(\overrightarrow{p},\overrightarrow{x})}{f(\overrightarrow{p},\overrightarrow{p})}\overrightarrow{p}$ with $\phi (\overrightarrow{p})=-\overrightarrow{p}$ and $\phi (\overrightarrow{x})=\overrightarrow{x}$ for any $⟨\overrightarrow{x}⟩\in P^{\perp }$.

Remark:

a) The image of an exterior, tangent and secant line, respectively, by the involution ${\sigma }_P$ of the Lemma above is an exterior, tangent and secant line, respectively.

b) $\mathcal{ℛ}$ is pointwise fixed by ${\sigma }_P$.

Let be $\mathrm{\Pi }(\mathcal{𝒬})$ the group of projective collineations of ${\mathfrak{P}}_n(K)$ which leaves $\mathcal{𝒬}$ invariant. We get

Lemma: $\mathrm{\Pi }(\mathcal{𝒬})$ operates transitively on $\mathcal{𝒬}\setminus \mathcal{ℛ}$.

A subspace $\mathcal{𝒰}$ of ${\mathfrak{P}}_n(K)$ is called $\rho$-subspace if $\mathcal{𝒰}\subset \mathcal{𝒬}$ (for example: points on a sphere or lines on a hyperboloid (s. below)).

Lemma: Any two maximal $\rho$-subspaces have the same dimension $m$.

Let be $m$ the dimension of the maximal $\rho$—subspaces of $\mathcal{𝒬}$. The integer $i:=m+1$ is called index of $\mathcal{𝒬}$.

Theorem: (BUEKENHOUT) For the index $i$ of a non-degenerate quadric $\mathcal{𝒬}$ in ${\mathfrak{P}}_n(K)$ the following is true: $i\le \frac{n+1}{2}$.

Let be $\mathcal{𝒬}$ a non-degenerate quadric in ${\mathfrak{P}}_n(K),n\ge 2$, and $i$ its index.

In case of $i=1$ quadric $\mathcal{𝒬}$ is called sphere (or oval conic if $n=2$).

In case of $i=2$ quadric $\mathcal{𝒬}$ is called hyperboloid (of one sheet).

Example:

a) Quadric $\mathcal{𝒬}$ in ${\mathfrak{P}}_2(K)$ with form $\rho (\overrightarrow{x})=x_1{x}_2-x_3^2$ is non-degenerate with index 1.

b) If polynomial $q(\xi )={\xi }^2+a_0\xi +b_0$ is irreducible over $K$ the quadratic form $\rho (\overrightarrow{x})=x_1^2+a_0{x}_1{x}_2+b_0{x}_2^2-x_3{x}_4$ gives rise of a non-degenerate quadric $\mathcal{𝒬}$ in ${\mathfrak{P}}_3(K)$.

c) In ${\mathfrak{P}}_3(K)$ the quadratic form $\rho (\overrightarrow{x})=x_1{x}_2+x_3{x}_4$ gives rise of a hyperboloid.

Remark: It is not reasonable to define formally quadrics for vector spaces over genuine skew fields (division rings). Because one would get secants bearing more than 2 points of the quadric which is totally different to usual quadrics. The reason is the following statement.

Theorem: A division ring $K$ is commutative if and only if any equation $x^2+ax+b=0,a,b\in K$ has at most two solutions.

There are generalizations of quadrics: quadratic sets. A quadratic set is a set of points of a projective plane/space, which bears the same geometric properties as a quadric: any line intersects a quadratic set in no or 1 or two lines or is containt in the set.

External links

• Lecture Note Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes, p. 117