Oval
An oval (from Latin ovum, "egg") is a closed curve in a plane which "loosely" resembles the outline of an egg. The term is not very specific, but in some areas (projective geometry, technical drawing, etc.) it is given a more precise definition, which may include either one or two axes of symmetry. In common English, the term is used in a broader sense: any shape which reminds one of an egg.
The 3-dimensional version of an oval is called an ovoid.
Oval in geometry
The term oval when used to describe curves in geometry is not well-defined, except in the context of projective geometry. Many distinct curves are commonly called ovals or are said to have an "oval shape". Generally, to be called an oval, a plane curve should resemble the outline of an egg or an ellipse. In particular, the common traits that these curves have are:
Ovoid (polar space)
In mathematics, an ovoid O of a (finite) polar space of rank r is a set of points, such that every subspace of rank 
intersects O in exactly one point.
Cases
Symplectic polar space
An ovoid of 
(a symplectic polar space of rank n) would contain 
points.
However it only has an ovoid if and only 
and q is even. In that case, when the polar space is embedded into 
the classical way, it is also an ovoid in the projective geometry sense.
Hermitian polar space
Ovoids of 
and 
would contain 
points.
Hyperbolic quadrics
An ovoid of a hyperbolic quadric
would contain 
points.
Parabolic quadrics
An ovoid of a parabolic quadric 
would contain points. For , it is easy to see to obtain an ovoid by cutting the parabolic quadric with a hyperplane, such that the intersection is an elliptic quadric. The intersection is an ovoid.
If q is even, 
is isomorphic (as polar space) with , and thus due to the above, it has no ovoid for 
.
Elliptic quadrics
An ovoid of an elliptic quadric 
would contain 
points.
Ovoid (projective geometry)
In the projective space PG(3,q), with q a prime power greater than 2, an ovoid is a set of 
points, no three of which are collinear (the maximum size of such a set). When 
the largest set of non-collinear points has size eight and is the complement of a plane.
An important example of an ovoid in any finite projective three-dimensional space are the points of an elliptic quadric (all of which are projectively equivalent).
When q is odd or 
, no ovoids exist other than the elliptic quadrics.
When 
another type of ovoid can be constructed : the Tits ovoid, also known as the Suzuki ovoid. It is conjectured that no other ovoids exist in PG(3,q). In fact "One of the most challenging open problems in finite geometry is the determination of ovoids in all finite three dimensional projective spaces".
Through every point P on the ovoid, there are exactly 
tangents, and it can be proven that these lines are exactly the lines through P in one specific plane through P. This means that through every point P in the ovoid, there is a unique plane intersecting the ovoid in exactly one point. Also, if q is odd or every plane which is not a tangent plane meets the ovoid in a conic.
Podcasts:
Ovoid
ALBUMS
- Trance - Psychedelic Flashbacks 4 released: 2000
- Goa Trance: Psychedelic Flashbacks 2 released: 1999
- Goa Trance 3 released: 1999
- Trance - Psychedelic Flashbacks 3 released:
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Evita
by: EvitaHigh flying, adored
So young, the instant queen
A rich, beautiful thing
Of all the talents
Across between
A fantasy of the bedroom
And a saint
You were just a backstreet girl
Hustling and fighting
Scratching and biting
High flying, adored
Did you believe
In your wildest moments
All this would be yours
That you'd become
The lady of them all?
Were there stars in your eyes
When you crawled in at night
From the bars, from the sidewalks
From the gutter-the-atrical?
Don't look down
It's a long, long way to fall
High flying, adored
What happens now?
Where do you go from here?
For someone on top of the world
The view is not exactly clear
A shame you did it all
At twentysix
There are no mysteries now
Nothing can thrill you
No one fulfill you
High flying, adored
I hope you come to terms with burden
So famous, so easily
So soon is not the wisest thing to be
You won't care if they love you
It's been done before
You'll despair if they hate you
You'll be drained of all energy
All the young who've made it
Would agree
High flying, adored
That's good to hear
But unimportant
My story's quite usual:
Local girl makes good
Weds famous man
I was slap in the right place
At the perfect time
Filled a gap - I was lucky
But one thing I say for me
No one else can fill it