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; in other words, a one-sided surface. It cannot be embedded in standard 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 **R**^{3} passing through the origin.

- Examples
- The projective sphere
- The projective hemisphere
- Boy's surface – an immersion
- Roman surface
- Hemi polyhedra
- Planar projections
- Cross-capped disk
- Homogeneous coordinates
- Points, lines, and planes
- Ideal points
- Duality
- Embedding into 4-dimensional space
- Higher non-orientable surfaces
- See also
- References
- External links

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 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 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 leftmost diagram shown here.

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.^{ [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: Assuming 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 the projective plane, which is not orientable. This is a contradiction, and so our assumption that it does embed must have been false.

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 or y = −x. This quotient space of the sphere is homeomorphic with the collection of all lines passing through the origin in **R**^{3}.

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.

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.^{ [2] }

The projective plane can be immersed (local neighbourhoods of the source space 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] }

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

A polyhedral representation is the tetrahemihexahedron,^{ [4] } which has the same general form as Steiner's Roman Surface, shown here.

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

Various planar (flat) projections or mappings of the projective plane have been described. In 1874 Klein described the mapping:^{ [1] }

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

A closed surface is obtained by gluing a disk to a cross-cap. This surface can be represented parametrically by the following equations:

where both *u* and *v* range from 0 to 2*π*.

These equations are similar to those of a torus. Figure 1 shows a closed cross-capped disk.

Figure 1. Two views of a cross-capped disk. |

A cross-capped disk has a plane of symmetry 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.

Figure 2. Two views of a cross-capped disk which has been sliced open. |

Once this exception is made, it will be seen that the sliced cross-capped disk is homeomorphic to a self-intersecting disk, as shown in Figure 3.

Figure 3. Two alternative views of a self-intersecting disk. |

The self-intersecting disk is homeomorphic to an ordinary disk. The parametric equations of the self-intersecting disk are:

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 reflections of each other. The disks have centers at the 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 is identified with the point (*u* + π, 1) whose coordinates are . 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}.

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* + *cz* = 0 in **R**^{3} 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.

A line in **P**^{2} 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:

**x**^{T}**ℓ**= 0 or**ℓ**^{T}**x**= 0.

Using vector notation we may instead write **x**⋅**ℓ** = 0 or **ℓ**⋅**x** = 0.

The equation *k*(**x**^{T}**ℓ**) = 0 (which k is a non-zero scalar) sweeps out a plane that goes through zero in **R**^{3} and *k*(*x*) sweeps out a line, again going through zero. The plane and line are linear subspaces in **R**^{3}, which always go through zero.

In **P**^{2} the equation of a line is *ax* + *by* + *cz* = 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 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 *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.

In the equation **x**^{T}**ℓ** = 0 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.

The equation **x**^{T}**ℓ** = 0 calculates the inner product of two column vectors. The inner product of two vectors is zero if the vectors are orthogonal. In **P**^{2}, the line between the points **x**_{1} and **x**_{2} may be represented as a column vector **ℓ** that satisfies the equations **x**_{1}^{T}**ℓ** = 0 and **x**_{2}^{T}**ℓ** = 0, or in other words a column vector **ℓ** that is orthogonal to **x**_{1} and **x**_{2}. The cross product will find such a vector: the line joining two points has homogeneous coordinates given by the equation **x**_{1} × **x**_{2}. The intersection of two lines may be found in the same way, using duality, as the cross product of the vectors representing the lines, **ℓ**_{1} × **ℓ**_{2}.

The projective plane embeds into 4-dimensional Euclidean space. The real projective plane **P**^{2}(**R**) is the quotient of the two-sphere

**S**^{2}= {(*x*,*y*,*z*) ∈**R**^{3}:*x*^{2}+*y*^{2}+*z*^{2}= 1}

by the antipodal relation (*x*, *y*, *z*) ~ (−*x*, −*y*, −*z*). Consider the function **R**^{3} → **R**^{4} given by (*x*, *y*, *z*) ↦ (*xy*, *xz*, *y*^{2} − *z*^{2}, 2*yz*). This map restricts to a map whose domain is **S**^{2} and, since each component is a homogeneous polynomial of even degree, it takes the same values in **R**^{4} on each of any two antipodal points on **S**^{2}. This yields a map **P**^{2}(**R**) → **R**^{4}. Moreover, this map is an embedding. Notice that this embedding admits a projection into **R**^{3} which is the Roman surface.

By gluing together projective planes successively we get non-orientable surfaces of higher 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.

In classical mathematics, **analytic geometry**, also known as **coordinate geometry** or **Cartesian geometry**, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.

In topology, a branch of mathematics, the **Klein bottle** is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down. Other related non-orientable objects include the Möbius strip and the real projective plane. While a Möbius strip is a surface with boundary, a Klein bottle has no boundary. For comparison, a sphere is an orientable surface with no boundary.

A **sphere** is a geometrical object in three-dimensional space that is the surface of a ball.

In mathematics, a **Möbius strip**, **band**, or **loop**, also spelled **Mobius** or **Moebius**, is a surface with only one side and only one boundary curve. The Möbius strip is the simplest non-orientable surface. It can be realized as a ruled surface. Its discovery is attributed independently to the German mathematicians Johann Benedict Listing and August Ferdinand Möbius in 1858, though similar structures can be seen in Roman mosaics c. 200–250 AD. Möbius published his results in his articles "Theorie der elementaren Verwandtschaft" (1863) and "Ueber die Bestimmung des Inhaltes eines Polyëders" (1865).

In mathematics, a **3-sphere**, or **glome**, is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensions is an ordinary sphere, the boundary of a ball in four dimensions is a 3-sphere. A 3-sphere is an example of a 3-manifold and an n-sphere.

A **great circle**, also known as an **orthodrome**, of a sphere is the intersection of the sphere and a plane that passes through the center point of the sphere. A great circle is the largest circle that can be drawn on any given sphere. Any diameter of any great circle coincides with a diameter of the sphere, and therefore all great circles have the same center and circumference as each other. This special case of a circle of a sphere is in opposition to a *small circle*, that is, the intersection of the sphere and a plane that does not pass through the center. Every circle in Euclidean 3-space is a great circle of exactly one sphere.

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 of the projective plane; however, the figure resulting from removing six singular points is one. Its name arises because it was discovered by Jakob Steiner when he was in Rome in 1844.

An **ellipsoid** is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.

In geometry, a **hyperboloid of revolution**, sometimes called a **circular hyperboloid**, is the surface generated by rotating a hyperbola around one of its principal axes. A **hyperboloid** is the surface obtained from a hyperboloid of revolution by deforming it by means of directional scalings, or more generally, of an affine transformation.

In mathematics, the **complex plane** or ** z-plane** is the plane associated with complex coordinate system, formed or established by the

**Elliptic geometry** is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines are usually assumed to intersect at a single point. Because of this, the elliptic geometry described in this article is sometimes referred to as *single elliptic geometry* whereas spherical geometry is sometimes referred to as *double elliptic geometry*.

In mathematics, the concept of a **projective space** originated from the visual effect of perspective, where parallel lines seem to meet *at infinity*. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally, an affine space with points at infinity, in such a way that there is one point at infinity of each direction of parallel lines.

In mathematics, **homogeneous coordinates** or **projective coordinates**, introduced by August Ferdinand Möbius in his 1827 work *Der barycentrische Calcul*, are a system of coordinates used in projective geometry, as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points, including points at infinity, can be represented using finite coordinates. Formulas involving homogeneous coordinates are often simpler and more symmetric than their Cartesian counterparts. Homogeneous coordinates have a range of applications, including computer graphics and 3D computer vision, where they allow affine transformations and, in general, projective transformations to be easily represented by a matrix.

In linear algebra, linear transformations can be represented by matrices. If is a linear transformation mapping to and is a column vector with entries, then

In geometry, a surface *S* is **ruled** if through every point of *S* there is a straight line that lies on *S*. Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, the right conoid, the helicoid, and the tangent developable of a smooth curve in space.

In geometry, a (**general**) **conical surface** is the unbounded surface formed by the union of all the straight lines that pass through a fixed point — the *apex* or *vertex* — and any point of some fixed space curve — the *directrix* — that does not contain the apex. Each of those lines is called a *generatrix* of the surface.

A **cone** is a three-dimensional geometric shape that tapers smoothly from a flat base to a point called the apex or vertex.

In geometry, the notion of **line** or **straight line** was introduced by ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects, which are often described in terms of two points or referred to using a single letter.

The **Lambert azimuthal equal-area projection** is a particular mapping from a sphere to a disk. It accurately represents area in all regions of the sphere, but it does not accurately represent angles. It is named for the Swiss mathematician Johann Heinrich Lambert, who announced it in 1772. "Zenithal" being synonymous with "azimuthal", the projection is also known as the **Lambert zenithal equal-area projection**.

In mathematics, a **surface** is a generalization of a plane, which is not necessarily flat – that is, the curvature is not necessarily zero. This is analogous to a curve generalizing a straight line. There are many more precise definitions, depending on the context and the mathematical tools that are used to analyze the surface.

- 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 )
- Richter, David A.,
*Two Models of the Real Projective Plane*, retrieved 2010-04-15

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