The projective plane over any alternative division ring is a Moufang plane.

Structures analogous to those found in continuous geometries ( Euclidean plane, real projective space, etc.

Since traditional " Euclidean " space never reaches infinity, the projective equivalent, called extended Euclidean space, must be formed by adding the required ' plane at infinity '.

Ellipses also arise as images of a circle under parallel projection and the bounded cases of perspective projection, which are simply intersections of the projective cone with the plane of projection.

Other related non-orientable objects include the Möbius strip and the real projective plane.

* Real projective plane

The complex projective plane P < sup > 2 </ sup >( C ) is related to M-theory in 11 dimensions.

In mathematics, a projective plane is a geometric structure that extends the concept of a plane.

A projective plane can be thought of as an ordinary plane equipped with additional " points at infinity " where parallel lines intersect.

Thus any two lines in a projective plane intersect in one and only one point.

The archetypical example is the real projective plane, also known as the Extended Euclidean Plane.

There are many other projective planes, both infinite, such as the complex projective plane, and finite, such as the Fano plane.

A projective plane is a 2-dimensional projective space, but not all projective planes can be embedded in 3-dimensional projective spaces.

To turn the ordinary Euclidean plane into a projective plane proceed as follows:

The center of the special unitary group has order gcd ( n, q + 1 ) and consists of those unitary scalars which also have order dividing n. The quotient of the unitary group by its center is called the projective unitary group, PU ( n, q² ), and the quotient of the special unitary group by its center is the projective special unitary group PSU ( n, q² ).

A projective geometry of dimension 1 consists of a single line containing at least 3 points.

Its structure has been analysed since the nineteenth century, but it is ' large ' ( while the corresponding group for the projective line consists only of Möbius transformations determined by three parameters ).

* In a projective plane, a projective frame consists of four points, no three of which lie on a projective line.

Dirac's conjectured lower bound is asymptotically the best possible, since there is a proven matching upper bound for even n greater than four. The construction, due to Károly Böröczky, that achieves this bound consists of the vertices of a regular m-gon in the real projective plane and another m points ( thus, ) on the line at infinity corresponding to each of the directions determined by pairs of vertices ; although there are pairs, they determine only m distinct directions.

The other counterexample, due to McKee, consists of two regular pentagons joined edge-to-edge together with the midpoint of the shared edge and four points on the line at infinity in the projective plane ; these 13 points have among them 6 ordinary lines.

In projective geometry, an ellipse can be defined as the set of all points of intersection between corresponding lines of two pencils of lines which are related by a projective map.

Some topics represented here by a significant number of papers are: set theory ( including measurable cardinals and abstract measures ), topology, transformation theory, ergodic theory, group theory, projective algebra, number theory, combinatorics, and graph theory.

In projective geometry, Plücker coordinates refer to a set of homogeneous co-ordinates introduced initially to embed the set of lines in three dimensions as a quadric in five dimensions.

* Arc ( projective geometry ), a particular type of set of points of a projective plane

Another technique is to use projective velocity blending, which is the blending of two projections ( last known and current ) where the current projection uses a blending between the last known and current velocity over a set time.

There are five known ways to build a set of point on an Edwards curve: the set of affine points, the set of projective points, the set of inverted points, the set of extended points and the set of completed points.

The set is the real projective line, which is a one-point compactification of the real line.

This set is analogous to the real projective line, except that it is based on the field of complex numbers.

For every positive integer n, the zero set of a non-singular homogeneous degree n + 2 polynomial in the homogeneous coordinates of the complex projective space CP < sup > n + 1 </ sup > is a compact Calabi – Yau n-fold.

However, it is an elementary result, which can be seen concretely as follows: the set of normal subgroups of a given index p form a projective space, namely the projective space

Recall that n-dimensional projective space is defined to be the set of equivalence classes of non-zero points in by identifying two points that differ by a scalar multiple in k. The elements of the polynomial ring are not functions on because any point has many representatives that yield different values in a polynomial ; however, for homogeneous polynomials the condition of having zero or nonzero value on any given projective point is well-defined since the scalar multiple factors out of the polynomial.

In mathematics a projective space is the set of lines through the origin of a vector space V. The cases when V = R < sup > 2 </ sup > or V = R < sup > 3 </ sup > are the projective line and the projective plane, respectively, where R denotes the field of real numbers, R < sup > 2 </ sup > denotes pairs of real numbers, and R < sup > 3 </ sup > denotes triplets of real numbers.

* Projective cone, the union of all lines that intersect a projective subspace and an arbitrary subset of some other disjoint subspace

The concept of duality here is closely related to the duality in projective geometry, where lines and edges are interchanged ; in fact it is often mistakenly taken to be a particular version of the same.

By projective duality, an ellipse can be defined also as the envelope of all lines that connect corresponding points of two lines which are related by a projective map.

* Complete quadrangle ( projective geometry ), a configuration with four points and six lines.

A finite projective plane of order q, with the lines as blocks, is an S ( 2, q + 1, q < sup > 2 </ sup >+ q + 1 ), since it has q < sup > 2 </ sup >+ q + 1 points, each line passes through q + 1 points, and each pair of distinct points lies on exactly one line.

To see how this works algebraically, in projective space, the lines x + 2y = 3 and x + 2y = 5 are represented by the homogeneous equations x + 2y-3z = 0 and x + 2y-5z = 0.

In the projective plane, points, denoted by, are ' the same ' as lines in a threedimensional space that go through the origin.

This lines are in the projective plane the ' points of infinity ' that are used in the affine-plane above.

Geometric objects, such as points, lines, or planes, can be given a representation as elements in projective spaces based on homogeneous coordinates.

In a projective representation of lines and points, however, such an intersection point exists even for parallel lines, and it can be computed in the same way as other intersection points.