|
In mathematics, the real projective plane is the space of lines in R3 passing through the origin. It is a non-orientable two-dimensional manifold, that is, a surface, that has basic applications to geometry, but which cannot be embedded in our usual three-dimensional space without intersecting itself. It has Euler characteristic of 1 giving a genus of 1. Image File history File links ProjectivePlaneAsSquare. ...
Image File history File links ProjectivePlaneAsSquare. ...
In mathematics, each closed surface in the sense of geometric topology can be constructed from an even-sided oriented polygon, called a fundamental polygon, by pairwise identification of its edges. ...
For other meanings of mathematics or math, see mathematics (disambiguation). ...
Projective plane - Wikipedia, the free encyclopedia /**/ @import /skins-1. ...
The torus is an orientable surface. ...
On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ...
An open surface with X-, Y-, and Z-contours shown. ...
Table of Geometry, from the 1728 Cyclopaedia. ...
In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ...
It has been suggested that Vertex/Face/Edge relation in a convex polyhedron be merged into this article or section. ...
In mathematics, the genus has few different meanings Topology The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. ...
It is often described intuitively, in relation with a Möbius strip: it would result if one could glue the single edge of the strip to itself in the correct direction. Or in other words, a square [0,1] × [0,1] with sides identified by the relations: A Möbius strip made with a piece of paper and tape. ...
In plane (Euclidean) geometry, a square is a polygon with four equal sides, four right angles, and parallel opposite sides. ...
- (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.
Formal construction
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 it obeys the axioms required of a projective plane: A sphere (< Greek σφαίÏα) is a perfectly symmetrical geometrical object. ...
For the Brisbane bus routes known collectively as the Great Circle Line (598 & 599), see the following list of Brisbane Transport routes A great circle on a sphere A great circle is a circle on the surface of a sphere that has the same diameter as the sphere, dividing the...
In mathematics, the antipodal point of a point on the surface of a sphere is the point which is diametrically opposite it — so situated that a line drawn from the one to the other passes through the centre of the sphere and forms a true diameter. ...
Projective plane - Wikipedia, the free encyclopedia /**/ @import /skins-1. ...
- 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.
This is the real projective plane.
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.
The resulting surface, a 2-dimensional compact non-orientable manifold, is a little hard to visualize, because it cannot be embedded in 3-dimensional Euclidean space without intersecting itself. In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ...
This article or section should be merged with Orientability. ...
Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...
Trying to embed the real projective plane in three-space The projective plane cannot strictly be embedded (that is without intersection) in three-dimensional space. However, it can be immersed (local neighbourhoods do not have self-intersections). In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ...
In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ...
Boy's surface is an example of an immersion. The Roman surface is another interesting example, but this contains cross-caps so it is not technically an immersion. The same goes for a sphere with a cross-cap. In geometry, Boys surface is an immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901. ...
The Roman surface (so called because Jakob Steiner was in Rome when he thought of it) is a self-intersecting immersion of the real projective plane into three-dimensional space, with an unusually high degree of symmetry. ...
In mathematics, a cross-cap is a two-dimensional surface that is topologically equivalent to a Möbius strip. ...
A polyhedral representation is the Tetrahemihexahedron. A polyhedron is a geometric shape which in mathematics is defined by three related meanings. ...
In geometry, the tetrahemihexahedron is a concave uniform polyhedron, indexed as U4. ...
Homogeneous coordinates The set of lines in the plane can be represented using homogeneous coordinates. A line ax+by+c=0 can be represented as (a:b:c). These coordinates have the equivalence relation (a:b:c) = (da:db:dc) for all non zero values of d. Hence a different representation of the same line dax+dby+dc=0 has the same coordinates. The set of coordinates (a:b:1) gives the usual real plane, and the set of coordinates (a:b:0) defines a line at infinity. In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius, allow affine transformations to be easily represented by a matrix. ...
...
In geometry and topology, the line at infinity is a line which is added to the real (affine) plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane. ...
Higher genus The article on the fundamental polygon provides a description of the real projective planes of higher genus. In mathematics, each closed surface in the sense of geometric topology can be constructed from an even-sided oriented polygon, called a fundamental polygon, by pairwise identification of its edges. ...
In mathematics, the genus has few different meanings Topology The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. ...
See also |
Feeds |
Contact