TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the real
projective plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
is an example of a compact non-
orientable In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
two-dimensional
manifold In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

; in other words, a one-sided
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight Visual perce ...
. It cannot be embedded in standard three-dimensional space without intersecting itself. It has basic applications to
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

, 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

: if one could glue the (single) edge of the Möbius strip to itself in the correct direction, one would obtain the
projective plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. (This cannot be done in three-dimensional space without the surface intersecting itself.) Equivalently, gluing a disk along the boundary of the Möbius strip gives the projective plane. Topologically, it has
Euler characteristic #REDIRECT Euler characteristic#REDIRECT Euler characteristic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry) ...
1, hence a demigenus (non-orientable genus, Euler genus) of 1. Since the Möbius strip, in turn, can be constructed from a
square In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method ...

by gluing two of its sides together, the ''real'' projective plane can thus be represented as a unit square (that is,
, 1 The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
× ,1) with its sides identified by the following
equivalence relation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
s: :(0, ''y'') ~ (1, 1 − ''y'')   for 0 ≤ ''y'' ≤ 1 and :(''x'', 0) ~ (1 − ''x'', 1)   for 0 ≤ ''x'' ≤ 1, as in the leftmost diagram shown here.

# Examples

Projective geometry is not necessarily concerned with curvature and the real projective plane may be twisted up and placed in the Euclidean plane or 3-space in many different ways.Apéry, F.; ''Models of the real projective plane'', Vieweg (1987) Some of the more important examples are described below. The projective plane cannot be embedded (that is without intersection) in three-dimensional
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
. The proof that the projective plane does not embed in three-dimensional Euclidean space goes like this: Assuming that it does embed, it would bound a compact region in three-dimensional Euclidean space by the . The outward-pointing unit normal vector field would then give an
orientation Orientation may refer to: Positioning in physical space * Map orientation, the relationship between directions on a map and compass directions * Orientation (housing), the position of a building with respect to the sun, a concept in building desi ...
of the boundary manifold, but the boundary manifold would be the
projective plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, 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 A sphere (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appr ...

, and let the
great circle A great circle, also known as an orthodrome, of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a circle in two-dimensional sp ...

s of the sphere be "lines", and let pairs of
antipodal point In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s be "points". It is easy to check that this system obeys the axioms required of a
projective plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
: *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 or y = −x. This quotient space of the sphere is
homeomorphic In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its populat ...
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 In mathematics, specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space C to a topological space X such that each point in X has an Neighborhood_(mathematics)#Definitions, ...

. It follows that the
fundamental group In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

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 sphere covers the real projective plane twice, the plane may be represented as a closed hemisphere around whose rim opposite points are similarly identified.

## Boy's surface – an immersion

The projective plane can be immersed (local neighbourhoods of the source space do not have self-intersections) in 3-space.
Boy's surface In geometry, Boy's surface is an immersion (mathematics), immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901. He discovered it on assignment from David Hilbert to prove that the projective plane ''could not'' ...
is an example of an immersion. Polyhedral examples must have at least nine faces.Brehm, U.; "How to build minimal polyhedral models of the Boy surface", ''The mathematical intelligencer'' 12, No. 4 (1990), pp 51-56.

## Roman surface

Steiner's
Roman surface The Roman surface or Steiner surface is a self-intersecting mapping of the real projective plane into three-dimensional space, with an unusually high degree of symmetry. This mapping is not an immersion (mathematics), immersion of the projective ...
is a more degenerate map of the projective plane into 3-space, containing a
cross-cap In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. A polyhedral representation is the , which has the same general form as Steiner's Roman Surface, shown here.

## Hemi polyhedra

Looking in the opposite direction, certain abstract regular polytopes – hemi-cube,
hemi-dodecahedron A hemi-dodecahedron is an abstract polytope, abstract regular polyhedron, containing half the faces of a regular dodecahedron. It can be realized as a projective polyhedron (a tessellation of the real projective plane by 6 pentagons), which can be ...

, and
hemi-icosahedron A hemi-icosahedron is an abstract regular polyhedron, containing half the faces of a regular icosahedron In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") i ...

– can be constructed as regular figures in the ''projective plane;'' see also
projective polyhedra Projective may refer to Mathematics * Projective geometry *Projective space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra) ...
.

## Planar projections

Various planar (flat) projections or mappings of the projective plane have been described. In 1874 Klein described the mapping: : $k \left(x, y\right) = \left\left(1 + x^2 + y^2\right\right)^\frac \beginx \\ y\end$ Central projection of the projective hemisphere onto a plane yields the usual infinite projective plane, described below.

## Cross-capped disk

A closed surface is obtained by gluing a
disk Disk or disc may refer to: * Disk (mathematics) * Disk storage Music * Disc (band), an American experimental music band * Disk (album), ''Disk'' (album), a 1995 EP by Moby Other uses * Disc (galaxy), a disc-shaped group of stars * Disc (magazin ...
to a
cross-cap In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. This surface can be represented parametrically by the following equations: :$\begin X\left(u,v\right) &= r \, \left(1 + \cos v\right) \, \cos u, \\ Y\left(u,v\right) &= r \, \left(1 + \cos v\right) \, \sin u, \\ Z\left(u,v\right) &= -\operatorname\left\left(u - \pi \right\right) \, r \, \sin v, \end$ where both ''u'' and ''v'' range from 0 to 2''π''. These equations are similar to those of a
torus In geometry, a torus (plural tori, colloquially donut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanarity, coplanar with the circle. If the axis of revolution does not to ...

. Figure 1 shows a closed cross-capped disk. A cross-capped disk has a
plane of symmetry In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
which passes through its line segment of double points. In Figure 1 the cross-capped disk is seen from above its plane of symmetry ''z'' = 0, but it would look the same if seen from below. A cross-capped disk can be sliced open along its plane of symmetry, while making sure not to cut along any of its double points. The result is shown in Figure 2. Once this exception is made, it will be seen that the sliced cross-capped disk is homeomorphism, homeomorphic to a self-intersecting disk, as shown in Figure 3. The self-intersecting disk is homeomorphic to an ordinary disk. The parametric equations of the self-intersecting disk are: :$\begin X\left(u, v\right) &= r \, v \, \cos 2u, \\ Y\left(u, v\right) &= r \, v \, \sin 2u, \\ Z\left(u, v\right) &= r \, v \, \cos u, \end$ where ''u'' ranges from 0 to 2''π'' and ''v'' ranges from 0 to 1. Projecting the self-intersecting disk onto the plane of symmetry (''z'' = 0 in the parametrization given earlier) which passes only through the double points, the result is an ordinary disk which repeats itself (doubles up on itself). The plane ''z'' = 0 cuts the self-intersecting disk into a pair of disks which are mirror Reflection (mathematics), reflections of each other. The disks have centers at the Origin (mathematics), origin. Now consider the rims of the disks (with ''v'' = 1). The points on the rim of the self-intersecting disk come in pairs which are reflections of each other with respect to the plane ''z'' = 0. A cross-capped disk is formed by identifying these pairs of points, making them equivalent to each other. This means that a point with parameters (''u'', 1) and coordinates $\left(r \, \cos 2u, r \, \sin 2u, r \, \cos u\right)$ is identified with the point (''u'' + π, 1) whose coordinates are $\left(r \, \cos 2 u, r \, \sin 2 u, - r \, \cos u\right)$. But this means that pairs of opposite points on the rim of the (equivalent) ordinary disk are identified with each other; this is how a real projective plane is formed out of a disk. Therefore, the surface shown in Figure 1 (cross-cap with disk) is topologically equivalent to the real projective plane ''RP''2.

# 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 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'' + ''dcz'' = 0 gives the same homogeneous coordinates. A point [''x'' : ''y'' : ''z''] lies on a line (''a'' : ''b'' : ''c'') if ''ax'' + ''by'' + ''cz'' = 0. Therefore, lines with coordinates (''a'' : ''b'' : ''c'') where ''a'', ''b'' 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.

## Points, lines, and planes

A line in P2 can be represented by the equation ''ax'' + ''by'' + ''cz'' = 0. If we treat ''a'', ''b'', and ''c'' as the column vector ℓ and ''x'', ''y'', ''z'' as the column vector x then the equation above can be written in matrix form as: :xTℓ = 0 or ℓTx = 0. Using vector notation we may instead write x ⋅ ℓ = 0 or ℓ ⋅ x = 0. The equation ''k''(xTℓ) = 0 (which k is a non-zero scalar) sweeps out a plane that goes through zero in R3 and ''k''(''x'') sweeps out a line, again going through zero. The plane and line are linear subspaces in real coordinate space, R3, which always go through zero.

## Ideal points

In P2 the equation of a line is and this equation can represent a line on any plane parallel to the ''x'', ''y'' plane by multiplying the equation by ''k''. If 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 adjacent image 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 ). Lines on the plane when are ideal points. The plane at is the line at infinity. The homogeneous point is where all the real points go when you're looking at the plane from an infinite distance, a line on the plane is where parallel lines intersect.

## Duality

In the equation there are two column vectors. You can keep either constant and vary the other. If we keep the point x constant and vary the coefficients ℓ 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 calculates the dot product, inner product of two column vectors. The inner product of two vectors is zero if the vectors are orthogonal. In P2, the line between the points x1 and x2 may be represented as a column vector ℓ that satisfies the equations and , or in other words a column vector ℓ that is orthogonal to x1 and x2. The cross product will find such a vector: the line joining two points has homogeneous coordinates given by the equation . The intersection of two lines may be found in the same way, using duality, as the cross product of the vectors representing the lines, .

# Embedding into 4-dimensional space

The projective plane embeds into 4-dimensional Euclidean space. The real projective plane P2(R) is the Quotient space (topology), quotient of the two-sphere :S2 = by the antipodal relation . Consider the function given by . This map restricts to a map whose domain is S2 and, since each component is a homogeneous polynomial of even degree, it takes the same values in R4 on each of any two antipodal points on S2. This yields a map . Moreover, this map is an embedding. Notice that this embedding admits a projection into R3 which is the
Roman surface The Roman surface or Steiner surface is a self-intersecting mapping of the real projective plane into three-dimensional space, with an unusually high degree of symmetry. This mapping is not an immersion (mathematics), immersion of the projective ...
.

# Higher non-orientable surfaces

By gluing together projective planes successively we get non-orientable surfaces of higher genus (mathematics), demigenus. The gluing process consists of cutting out a little disk from each surface and identifying (''gluing'') their boundary circles. Gluing two projective planes creates the Klein bottle. The article on the fundamental polygon describes the higher non-orientable surfaces.

*Real projective space *Projective space *Pu's inequality, Pu's inequality for real projective plane *Smooth projective plane

# References

*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 ( ) *