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

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

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

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

For ${\mathcal {V}}(K)=K^{n}$ and ${\vec {x}}=\sum _{i=1}^{n}x_{i}{\vec {e}}_{i}$ ($\{{\vec {e}}_{1},\ldots ,{\vec {e}}_{n}\}$ is a base of ${\mathcal {V}}(K)$) $\rho$ has the form

$\rho ({\vec {x}})=\sum _{1=i\leq k}^{n}a_{ik}x_{i}x_{k}{\text{ with }}a_{ik}:=f({\vec {e}}_{i},{\vec {e}}_{k}){\text{ for }}i\neq k{\text{ and }}a_{ik}:=\rho ({\vec {e}}_{i}){\text{ for }}i=k$ and

$f({\vec {x}},{\vec {y}})=\sum _{1=i\leq k}^{n}a_{ik}(x_{i}y_{k}+x_{k}y_{i})$.

For example:

$n=3,\ \rho ({\vec {x}})=x_{1}x_{2}-x_{3}^{2},\ f({\vec {x}},{\vec {y}})=x_{1}y_{2}+x_{2}y_{1}-2x_{3}y_{3}.$

Definition and properties of a quadric

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

${\mathcal {P}}=\{\langle {\vec {x}}\rangle \mid {\vec {0}}\neq {\vec {x}}\in V_{n+1}(K)\}$

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

${\mathcal {G}}=\{\{\langle {\vec {x}}\rangle \in {\mathcal {P}}\mid {\vec {x}}\in U\}\mid U{\text{ 2-dimensional subspace of }}V_{n+1}(K)\}$

the set of lines.

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

${\mathcal {Q}}=\{\langle {\vec {x}}\rangle \in {\mathcal {P}}\mid \rho ({\vec {x}})=0\}$

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

$P^{\perp }:=\{\langle {\vec {x}}\rangle \in {\mathcal {P}}\mid f({\vec {p}},{\vec {x}})=0\}$

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: ${\mathcal {Q}}\neq \emptyset$.

Example: For $\rho ({\vec {x}})=x_{1}x_{2}-x_{3}^{2}$ 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) $g\cap {\mathcal {Q}}=\emptyset$ 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 $\operatorname {char} K\neq 2$ we have ${\mathcal {R}}={\mathcal {S}}$.

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

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

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

$\varphi :{\vec {x}}\rightarrow {\vec {x}}-{\frac {f({\vec {p}},{\vec {x}})}{\rho ({\vec {p}})}}{\vec {p}}$

induces an involutorial central collineation with axis $P^{\perp }$ and centre $P$ which leaves ${\mathcal {Q}}$ invariant.
In case of $\operatorname {char} K\neq 2$ mapping $\varphi$ gets the familiar shape $\varphi :{\vec {x}}\rightarrow {\vec {x}}-2{\frac {f({\vec {p}},{\vec {x}})}{f({\vec {p}},{\vec {p}})}}{\vec {p}}$ with $\varphi ({\vec {p}})=-{\vec {p}}$ and $\varphi ({\vec {x}})={\vec {x}}$ for any $\langle {\vec {x}}\rangle \in P^{\perp }$.

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 $\Pi ({\mathcal {Q}})$ the group of projective collineations of ${\mathfrak {P}}_{n}(K)$ which leaves ${\mathcal {Q}}$ invariant. We get

Lemma: $\Pi ({\mathcal {Q}})$ 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: $i\leq {\frac {n+1}{2}}$.

Let be ${\mathcal {Q}}$ a non-degenerate quadric in ${\mathfrak {P}}_{n}(K),n\geq 2$, 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 $\rho ({\vec {x}})=x_{1}x_{2}-x_{3}^{2}$ 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 $\rho ({\vec {x}})=x_{1}^{2}+a_{0}x_{1}x_{2}+b_{0}x_{2}^{2}-x_{3}x_{4}$ gives rise of a non-degenerate quadric ${\mathcal {Q}}$ in ${\mathfrak {P}}_{3}(K)$.

c) In ${\mathfrak {P}}_{3}(K)$ the quadratic form $\rho ({\vec {x}})=x_{1}x_{2}+x_{3}x_{4}$ 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