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A projective plane of order N is a Steiner S(2, N + 1, N 2 + N + 1) system (see Steiner system). Conversely, one can prove that all Steiner systems of this form (λ = 2) are projective planes. The number of mutually orthogonal Latin squares of order N is at most N − 1. N − 1 exist if and only if there is a projective plane of order N.
A muzzle brake or recoil compensator is a device connected to, or a feature integral (ported barrel) to the construction of, the muzzle or barrel of a firearm or cannon that is intended to redirect a portion of propellant gases to counter recoil and unwanted muzzle rise. [1] Barrels with an integral muzzle brake are often said to be ported.
The standard examples are the nondegenerate projective conic sections. In pappian projective planes of even order greater than four there are ovals which are not conics. In an infinite plane there exist ovals, which are not conics. In the real plane one just glues a half of a circle and a suitable ellipse smoothly.
The quotient map from the sphere onto the real projective plane is in fact a two sheeted (i.e. two-to-one) covering map. It follows that the fundamental group of the real projective plane is the cyclic group of order 2; i.e., integers modulo 2.
If P is a finite set, the projective plane is referred to as a finite projective plane. The order of a finite projective plane is n = k – 1, that is, one less than the number of points on a line. All known projective planes have orders that are prime powers. A projective plane of order n is an ((n 2 + n + 1) n + 1) configuration. The smallest ...
Hanfried Lenz gave a classification scheme for projective planes in 1954, [6] which was refined by Adriano Barlotti in 1957. [7] This classification scheme is based on the types of point–line transitivity permitted by the collineation group of the plane and is known as the Lenz–Barlotti classification of projective planes.
Qvist's theorem [3] [4]. Let Ω be an oval in a finite projective plane of order n. (a) If n is odd, every point P ∉ Ω is incident with 0 or 2 tangents. (b) If n is even, there exists a point N, the nucleus or knot, such that, the set of tangents to oval Ω is the pencil of all lines through N.
If any of the lines is removed from the plane, along with the points on that line, the resulting geometry is the affine plane of order 2. The Fano plane is called the projective plane of order 2 because it is unique (up to isomorphism). In general, the projective plane of order n has n 2 + n + 1 points and the same number of lines; each line ...