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definition - Real_projective_plane

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Real projective plane

 The fundamental polygon of the projective plane. The Möbius strip with a single edge, can be closed into a projective plane by gluing opposite open edges together. In comparison the Klein bottle is a Möbius strip closed into a cylinder.

In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold, that is, a one-sided surface. It cannot be embedded in our usual three-dimensional space without intersecting itself. It has basic applications to geometry, since the common construction of the real projective plane is as the space of lines in R3 passing through the origin.

The plane is also often described topologically, in terms of a construction based on the Möbius strip: if one could glue the (single) edge of the Möbius strip to itself in the correct direction, one would obtain the projective plane. (This cannot be done in our three-dimensional space.) Equivalently, gluing a disk along the boundary of the Möbius strip gives the projective plane. Topologically, it has Euler characteristic 1, hence a demigenus (non-orientable genus, Euler genus) of 1.

Since the Möbius strip, in turn, can be constructed from a square by gluing two of its sides together, the real projective plane can thus be represented as a unit square (that is, [0,1] × [0,1] ) with its sides identified by the following equivalence relations:

(0, y) ~ (1, 1 − y)   for 0 ≤ y ≤ 1

and

(x, 0) ~ (1 − x, 1)   for 0 ≤ x ≤ 1,

as in the diagram on the right.

Examples

Topology is not concerned with flatness, and the real projective plane may be twisted up and placed in the Euclidean plane or 3-space in many different ways.[1] Some of the more important examples are described below.

The projective plane cannot be embedded (that is without intersection) in three-dimensional Euclidean space. The proof that the projective plane does not embed in three-dimensional Euclidean space goes like this: Assume that it does embed, it would bound a compact region in three-dimensional Euclidean space by the generalized Jordan curve theorem. The outward-pointing unit normal vector field would then give an orientation of the boundary manifold, but the boundary manifold would be projective space, which is not orientable. This is a contradiction, and so our assumption that it does embed must have been false.

The projective sphere

Consider a sphere, and let the great circles of the sphere be "lines", and let pairs of antipodal points be "points". It is easy to check that this system obeys the axioms required of a projective plane:

• any pair of distinct great circles meet at a pair of antipodal points; and
• any two distinct pairs of antipodal points lie on a single great circle.

If we identify each point on the sphere with its antipodal point, then we get a representation of the real projective plane in which the "points" of the projective plane really are points. This means that the projective plane is the quotient space of the sphere obtained by partitioning the sphere into equivalence classes under the equivalence relation ~, where x ~ y if y = −x. This quotient space of the sphere is homeomorphic with the collection of all lines passing through the origin in R3.

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. One can take the loop AB from the figure above to be the generator.

The projective hemisphere

Because the real projective plane covers the sphere twice, it may be represented as a hemisphere around whose rim opposite points are similarly identified.[2]

Boy's surface – an immersion

The projective plane can be immersed (local neighbourhoods do not have self-intersections) in 3-space. Boy's surface is an example of an immersion.

Polyhedral examples must have at least nine faces.[3]

Roman surface

An animation of the Roman Surface

Steiner's Roman surface is a more degenerate map of the projective plane into 3-space, containing a cross-cap.

The tetrahemihexahedron is a polyhedral representation of the real projective plane.

A polyhedral representation is the tetrahemihexahedron,[4] which has the same general form as Steiner's Roman Surface, shown to the right.

Hemi polyhedra

Looking in the opposite direction, certain abstract regular polytopeshemi-cube, hemi-dodecahedron, and hemi-icosahedron — can be constructed as regular figures in the projective plane; see also projective polyhedra.

Planar projections

Various planar (flat) projections or mappings of the projective plane have been described. In 1874 Klein described the mapping $k (x, y) = (1 + x^2 + y^2)^{1/2} . (x, y)$ [1]

Central projection of the projective hemisphere onto a plane yields the usual infinite projective plane, described below.

Homogeneous coordinates

The points in the plane can be represented by homogeneous coordinates. A point has homogeneous coordinates [x : y : z], where the coordinates [x : y : z] and [tx : ty : tz] are considered to represent the same point, for all nonzero values of t. The points with coordinates [x : y : 1] are the usual real plane, called the finite part of the projective plane, and points with coordinates [x : y : 0], called points at infinity or ideal points, constitute a line called the line at infinity. (The homogeneous coordinates [0 : 0 : 0] do not represent any point.)

The lines in the plane can also be represented by homogeneous coordinates. A projective line corresponding to the plane ax + by + c = 0 in R3 has the homogeneous coordinates (a : b : c). Thus, these coordinates have the equivalence relation (a : b : c) = (da : db : dc) for all nonzero values of d. Hence a different equation of the same line dax + dby + dc = 0 gives the same homogeneous coordinates. A point [x : y : z] lies on a line (a : b : c) if ax + by + c = 0. Therefore, lines with coordinates (a : b : c) where ab are not both 0 correspond to the lines in the usual real plane, because they contain points that are not at infinity. The line with coordinates (0 : 0 : 1) is the line at infinity, since the only points on it are those with z = 0.

The flat projective plane

In the projective plane P2, a point x is represented by the homogeneous coordinate (x1, x2, x3). If we think of (x1, x2, x3) as a point in real space R3 with the third value of the homogeneous coordinate as a value in the z direction, then P2 can be visualized as R3.

Points, rays, lines, and planes

A line in P2 can be represented by the equation ax + by + c = 0. If we treat a, b, and c as the column vector and x, y, 1 as the column vector x then the equation above can be written in matrix form as:

xT = 0 or Tx = 0.

Using vector notation we may instead write

x = 0 or x = 0.

The equation k(xT) = 0 (which k is a non-zero scalar) sweeps out a plane that goes through zero in R3 and k(x) sweeps out a ray, again going through zero. The plane and ray are subspaces in R3, which always go through zero.

Ideal points

In P2 the equation of a line is ax + by + c = 0 and this equation can represent a line on any plane parallel to the x, y plane by multiplying the equation by k.

If z = 1 we have a normalized homogeneous coordinate. All points that have z = 1 create a plane. Let's pretend we are looking at that plane (from a position further out along the z axis and looking back towards the origin) and there are two parallel lines drawn on the plane. From where we are standing (given our visual capabilities) we can see only so much of the plane, which we represent as the area outlined in red in the diagram. If we walk away from the plane along the z axis, (still looking backwards towards the origin), we can see more of the plane. In our field of view original points have moved. We can reflect this movement by dividing the homogeneous coordinate by a constant. In the image to the right we have divided by 2 so the z value now becomes 0.5. If we walk far enough away what we are looking at becomes a point in the distance. As we walk away we see more and more of the parallel lines. The lines will meet at a line at infinity (a line that goes through zero on the plane at z = 0). Lines on the plane when z = 0 are ideal points. The plane at z = 0 is the line at infinity.

The homogeneous point (0, 0, 0) is where all the real points go when you're looking at the plane from an infinite distance, a line on the z = 0 plane is where parallel lines intersect.

Duality

In the equation xT = 0 there are two column vectors. You can keep either constant and vary the other. If we keep the point constant x and vary the coefficients we create new lines that go through the point. If we keep the coefficients constant and vary the points that satisfy the equation we create a line. We look upon x as a point because the axes we are using are x, y, and z. If we instead plotted the coefficients using axis marked a, b, c points would become lines and lines would become points. If you prove something with the data plotted on axis marked x, y, and z the same argument can be used for the data plotted on axis marked a, b, and c. That is duality.

Lines joining points and intersection of lines (using duality)

The equation xT = 0 calculates the inner product of two column vectors. The inner product of two vectors is zero if the vectors are orthogonal. To find the line between the points x1 and x2 you must find the column vector that satisfies the equations x1T = 0 and x2T = 0, that is we must find a column vector that is orthogonal to x1 and x2. In the case of P2, the cross product will find such a vector. The line joining two points is given by the equation x1 × x2. To find the intersection of two lines you look to duality. If you plot in the coefficient space you get rays. To find the point x that is orthogonal to the two rays you find the cross product. That is 1 × 2.

While the cross product works in P2, it is not well-defined in arbitrary dimensions. However, this pair of equations is satisfied by

$\mathbf{x}_1^\mathrm{T} \ell - \lambda \mathbf{x}_2^\mathrm{T} \ell = 0$[citation needed]

Embedding into 4-dimensional space

The projective plane embeds into 4-dimensional Euclidean space. Consider $\mathbb RP^2$ to be the quotient of the two-sphere $S^2 = \{(x,y,z) \in \mathbb R^3 : x^2+y^2+z^2 = 1\}$ by the antipodal relation $(x,y,z)\sim (-x,-y,-z)\,$. Consider the function $\mathbb R^3 \to \mathbb R^4$ given by $(x,y,z)\longmapsto (xy,xz,y^2-z^2,2yz)$. This map restricts to a map whose domain is $S^2$ and, since it is a purely quadratic polynomial, it can be factorised to give a map $\mathbb RP^2 \to \mathbb R^4$. Moreover, this map is an embedding. Notice that this embedding admits a projection into $R^3$ which is the Roman surface.

Higher non-orientable surfaces

By glueing together projective planes successively we get non-orientable surfaces of higher demigenus. The glueing process consists of cutting out a little disk from each surface and identifying (glueing) their boundary circles. Glueing two projective planes creates the Klein bottle.

The article on the fundamental polygon describes the higher non-orientable surfaces.

References

1. ^ a b Apéry, F.; Models of the real projective plane, Vieweg (1987)
2. ^ Weeks, J.; The shape of space, CRC (2002), p 59
3. ^ Brehm, U.; "How to build minimal polyhedral models of the Boy surface", The mathematical intelligencer 12, No. 4 (1990), pp 51-56.
4. ^ (Richter)
• Coxeter, H.S.M. (1955), The Real Projective Plane, 2nd ed. Cambridge: At the University Press.
• Reinhold Baer, Linear Algebra and Projective Geometry,, Dover, 2005 (ISBN : 0-486-44565-8 )

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