This article is about the geometrical figure or shape. For the species of mushrooms, see Psilocybe ovoideocystidiata.
In projective geometry an ovoid is a sphere like pointset (surface) in a projective space of dimension d ≥ 3. Simple examples in a real projective space are hyperspheres (quadrics). The essential geometric properties of an ovoid are:
Any line intersects in at most 2 points,
The tangents at a point cover a hyperplane (and nothing more), and
contains no lines.
To the definition of an ovoid: t tangent, s secant line
Property 2) excludes degenerated cases (cones,...). Property 3) excludes ruled surfaces (hyperboloids of one sheet, ...).
An ovoid is the spatial analog of an oval in a projective plane.
In a projective space of dimension d ≥ 3 a set of points is called an ovoid, if
(1) Any line g meets in at most 2 points.
In the case of , the line is called a passing (or exterior) line, if the line is a tangent line, and if the line is a secant line.
(2) At any point the tangent lines through P cover a hyperplane, the tangent hyperplane, (i.e., a projective subspace of dimension d − 1).
(3) contains no lines.
From the viewpoint of the hyperplane sections, an ovoid is a rather homogeneous object, because
For an ovoid and a hyperplane , which contains at least two points of , the subset is an ovoid (or an oval, if d = 3) within the hyperplane .
For finite projective spaces of dimension d ≥ 3 (i.e., the point set is finite, the space is pappian[1]), the following result is true:
If is an ovoid in a finite projective space of dimension d ≥ 3, then d = 3.
(In the finite case, ovoids exist only in 3-dimensional spaces.)[2]
In a finite projective space of order n >2 (i.e. any line contains exactly n + 1 points) and dimension d = 3 any pointset is an ovoid if and only if and no three points are collinear (on a common line).[3]
Replacing the word projective in the definition of an ovoid by affine, gives the definition of an affine ovoid.
If for an (projective) ovoid there is a suitable hyperplane not intersecting it, one can call this hyperplane the hyperplane at infinity and the ovoid becomes an affine ovoid in the affine space corresponding to . Also, any affine ovoid can be considered a projective ovoid in the projective closure (adding a hyperplane at infinity) of the affine space.
Examples
In real projective space (inhomogeneous representation)
(hypersphere)
These two examples are quadrics and are projectively equivalent.
Simple examples, which are not quadrics can be obtained by the following constructions:
(a) Glue one half of a hypersphere to a suitable hyperellipsoid in a smooth way.
(b) In the first two examples replace the expression x12 by x14.
Remark: The real examples can not be converted into the complex case (projective space over ). In a complex projective space of dimension d ≥ 3 there are no ovoidal quadrics, because in that case any non degenerated quadric contains lines.
But the following method guarantees many non quadric ovoids:
For any non-finite projective space the existence of ovoids can be proven using transfinite induction.[4][5]
Finite examples
Any ovoid in a finite projective space of dimension d = 3 over a field K of characteristic≠ 2 is a quadric.[6]
The last result can not be extended to even characteristic, because of the following non-quadric examples:
For odd and the automorphism
the pointset
is an ovoid in the 3-dimensional projective space over K (represented in inhomogeneous coordinates).
An ovoidal quadric has many symmetries. In particular:
Let be an ovoid in a projective space of dimension d ≥ 3 and a hyperplane. If the ovoid is symmetric to any point (i.e. there is an involutory perspectivity with center which leaves invariant), then is pappian and a quadric.[8]
An ovoid in a projective space is a quadric, if the group of projectivities, which leave invariant operates 3-transitively on , i.e. for two triples there exists a projectivity with .[9]
Let be an ovoid in a finite 3-dimensional desarguesian projective space of odd order, then is pappian and is a quadric.
Generalization: semi ovoid
Removing condition (1) from the definition of an ovoid results in the definition of a semi-ovoid:
A point set of a projective space is called a semi-ovoid if
the following conditions hold:
(SO1) For any point the tangents through point exactly cover a hyperplane.
(SO2) contains no lines.
A semi ovoid is a special semi-quadratic set[10] which is a generalization of a quadratic set. The essential difference between a semi-quadratic set and a quadratic set is the fact, that there can be lines which have 3 points in common with the set and the lines are not contained in the set.
Examples of semi-ovoids are the sets of isotropic points of an hermitian form. They are called hermitian quadrics.
As for ovoids in literature there are criteria, which make a semi-ovoid to a hermitian quadric. See, for example.[11]
Semi-ovoids are used in the construction of examples of Möbius geometries.
Panella, G. (1955), "Caratterizzazione delle quadriche di uno spazio (tridimensionale) lineare sopra un corpo finito", Boll. Un. Mat. Ital., 10: 507–513
External links
E. Hartmann: Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes. Skript, TH Darmstadt (PDF; 891 kB), S. 121-123.
January 07, 2023
ovoid, projective, geometry, this, article, about, geometrical, figure, shape, species, mushrooms, psilocybe, ovoideocystidiata, projective, geometry, ovoid, sphere, like, pointset, surface, projective, space, dimension, simple, examples, real, projective, spa. This article is about the geometrical figure or shape For the species of mushrooms see Psilocybe ovoideocystidiata In projective geometry an ovoid is a sphere like pointset surface in a projective space of dimension d 3 Simple examples in a real projective space are hyperspheres quadrics The essential geometric properties of an ovoid O displaystyle mathcal O are Any line intersects O displaystyle mathcal O in at most 2 points The tangents at a point cover a hyperplane and nothing more and O displaystyle mathcal O contains no lines To the definition of an ovoid t tangent s secant line Property 2 excludes degenerated cases cones Property 3 excludes ruled surfaces hyperboloids of one sheet An ovoid is the spatial analog of an oval in a projective plane An ovoid is a special type of a quadratic set Ovoids play an essential role in constructing examples of Mobius planes and higher dimensional Mobius geometries Contents 1 Definition of an ovoid 2 Examples 2 1 In real projective space inhomogeneous representation 2 2 Finite examples 3 Criteria for an ovoid to be a quadric 4 Generalization semi ovoid 5 See also 6 Notes 7 References 8 Further reading 9 External linksDefinition of an ovoid EditIn a projective space of dimension d 3 a set O displaystyle mathcal O of points is called an ovoid if 1 Any line g meets O displaystyle mathcal O in at most 2 points In the case of g O 0 displaystyle g cap mathcal O 0 the line is called a passing or exterior line if g O 1 displaystyle g cap mathcal O 1 the line is a tangent line and if g O 2 displaystyle g cap mathcal O 2 the line is a secant line 2 At any point P O displaystyle P in mathcal O the tangent lines through P cover a hyperplane the tangent hyperplane i e a projective subspace of dimension d 1 3 O displaystyle mathcal O contains no lines From the viewpoint of the hyperplane sections an ovoid is a rather homogeneous object because For an ovoid O displaystyle mathcal O and a hyperplane e displaystyle varepsilon which contains at least two points of O displaystyle mathcal O the subset e O displaystyle varepsilon cap mathcal O is an ovoid or an oval if d 3 within the hyperplane e displaystyle varepsilon For finite projective spaces of dimension d 3 i e the point set is finite the space is pappian 1 the following result is true If O displaystyle mathcal O is an ovoid in a finite projective space of dimension d 3 then d 3 In the finite case ovoids exist only in 3 dimensional spaces 2 In a finite projective space of order n gt 2 i e any line contains exactly n 1 points and dimension d 3 any pointset O displaystyle mathcal O is an ovoid if and only if O n 2 1 displaystyle mathcal O n 2 1 and no three points are collinear on a common line 3 Replacing the word projective in the definition of an ovoid by affine gives the definition of an affine ovoid If for an projective ovoid there is a suitable hyperplane e displaystyle varepsilon not intersecting it one can call this hyperplane the hyperplane e displaystyle varepsilon infty at infinity and the ovoid becomes an affine ovoid in the affine space corresponding to e displaystyle varepsilon infty Also any affine ovoid can be considered a projective ovoid in the projective closure adding a hyperplane at infinity of the affine space Examples EditIn real projective space inhomogeneous representation Edit O x 1 x d R d x 1 2 x d 2 1 displaystyle mathcal O x 1 x d in mathbb R d x 1 2 cdots x d 2 1 hypersphere O x 1 x d R d x d x 1 2 x d 1 2 point at infinity of x d axis displaystyle mathcal O x 1 x d in mathbb R d x d x 1 2 cdots x d 1 2 cup text point at infinity of x d text axis These two examples are quadrics and are projectively equivalent Simple examples which are not quadrics can be obtained by the following constructions a Glue one half of a hypersphere to a suitable hyperellipsoid in a smooth way b In the first two examples replace the expression x12 by x14 Remark The real examples can not be converted into the complex case projective space over C displaystyle mathbb C In a complex projective space of dimension d 3 there are no ovoidal quadrics because in that case any non degenerated quadric contains lines But the following method guarantees many non quadric ovoids For any non finite projective space the existence of ovoids can be proven using transfinite induction 4 5 Finite examples Edit Any ovoid O displaystyle mathcal O in a finite projective space of dimension d 3 over a field K of characteristic 2 is a quadric 6 The last result can not be extended to even characteristic because of the following non quadric examples For K G F 2 m m displaystyle K GF 2 m m odd and s displaystyle sigma the automorphism x x 2 m 1 2 displaystyle x mapsto x 2 frac m 1 2 the pointset O x y z K 3 z x y x 2 x s y s point of infinity of the z axis displaystyle mathcal O x y z in K 3 z xy x 2 x sigma y sigma cup text point of infinity of the z text axis is an ovoid in the 3 dimensional projective space over K represented in inhomogeneous coordinates Only when m 1 is the ovoid O displaystyle mathcal O a quadric 7 O displaystyle mathcal O is called the Tits Suzuki ovoid Criteria for an ovoid to be a quadric EditAn ovoidal quadric has many symmetries In particular Let be O displaystyle mathcal O an ovoid in a projective space P displaystyle mathfrak P of dimension d 3 and e displaystyle varepsilon a hyperplane If the ovoid is symmetric to any point P e O displaystyle P in varepsilon setminus mathcal O i e there is an involutory perspectivity with center P displaystyle P which leaves O displaystyle mathcal O invariant then P displaystyle mathfrak P is pappian and O displaystyle mathcal O a quadric 8 An ovoid O displaystyle mathcal O in a projective space P displaystyle mathfrak P is a quadric if the group of projectivities which leave O displaystyle mathcal O invariant operates 3 transitively on O displaystyle mathcal O i e for two triples A 1 A 2 A 3 B 1 B 2 B 3 displaystyle A 1 A 2 A 3 B 1 B 2 B 3 there exists a projectivity p displaystyle pi with p A i B i i 1 2 3 displaystyle pi A i B i i 1 2 3 9 In the finite case one gets from Segre s theorem Let be O displaystyle mathcal O an ovoid in a finite 3 dimensional desarguesian projective space P displaystyle mathfrak P of odd order then P displaystyle mathfrak P is pappian and O displaystyle mathcal O is a quadric Generalization semi ovoid EditRemoving condition 1 from the definition of an ovoid results in the definition of a semi ovoid A point set O displaystyle mathcal O of a projective space is called a semi ovoid ifthe following conditions hold SO1 For any point P O displaystyle P in mathcal O the tangents through point P displaystyle P exactly cover a hyperplane SO2 O displaystyle mathcal O contains no lines A semi ovoid is a special semi quadratic set 10 which is a generalization of a quadratic set The essential difference between a semi quadratic set and a quadratic set is the fact that there can be lines which have 3 points in common with the set and the lines are not contained in the set Examples of semi ovoids are the sets of isotropic points of an hermitian form They are called hermitian quadrics As for ovoids in literature there are criteria which make a semi ovoid to a hermitian quadric See for example 11 Semi ovoids are used in the construction of examples of Mobius geometries See also EditOvoid polar space Mobius planeNotes Edit Dembowski 1968 p 28 Dembowski 1968 p 48 Dembowski 1968 p 48 W Heise Bericht uber k displaystyle kappa affine Geometrien Journ of Geometry 1 1971 S 197 224 Satz 3 4 F Buekenhout A Characterization of Semi Quadrics Atti dei Convegni Lincei 17 1976 S 393 421 chapter 3 5 Dembowski 1968 p 49 Dembowski 1968 p 52 H Maurer Ovoide mit Symmetrien an den Punkten einer Hyperebene Abh Math Sem Hamburg 45 1976 S 237 244 J Tits Ovoides a Translations Rend Mat 21 1962 S 37 59 F Buekenhout A Characterization of Semi Quadrics Atti dei Convegni Lincei 17 1976 S 393 421 K J Dienst Kennzeichnung hermitescher Quadriken durch Spiegelungen Beitrage zur geometrischen Algebra 1977 Birkhauser Verlag S 83 85 References EditDembowski Peter 1968 Finite geometries Ergebnisse der Mathematik und ihrer Grenzgebiete Band 44 Berlin New York Springer Verlag ISBN 3 540 61786 8 MR 0233275Further reading EditBarlotti A 1955 Un estensione del teorema di Segre Kustaanheimo Boll Un Mat Ital 10 96 98 Hirschfeld J W P 1985 Finite Projective Spaces of Three Dimensions New York Oxford University Press ISBN 0 19 853536 8 Panella G 1955 Caratterizzazione delle quadriche di uno spazio tridimensionale lineare sopra un corpo finito Boll Un Mat Ital 10 507 513External links EditE Hartmann Planar Circle Geometries an Introduction to Moebius Laguerre and Minkowski Planes Skript TH Darmstadt PDF 891 kB S 121 123 Retrieved from https en wikipedia org w index php title Ovoid projective geometry amp oldid 998341228, wikipedia, wiki, book, books, library,