Projective in a sentence

Some disciplines in mathematics restrict the meaning of projective plane to only this type of projective plane since otherwise general statements about projective spaces would always have to mention the exceptions when the geometric dimension is two.

The real (or complex) projective plane and the projective plane of order 3 given above are examples of Desarguesian projective planes.

Another longstanding open problem is whether there exist finite projective planes of prime order which are not finite field planes (equivalently, whether there exists a non-Desarguesian projective plane of prime order).

By the end of the 19th century, projective geometers were studying more general kinds of transformations on figures in projective space.

Dualizing this theorem and the first two axioms in the definition of a projective plane shows that the plane dual structure C* is also a projective plane, called the dual plane of C. If C and C* are isomorphic, then C is called self-dual.

Every projective algebraic set may be uniquely decomposed into a finite union of projective varieties.

Finite projective planes It can be shown that a projective plane has the same number of lines as it has points (infinite or finite).

If is a matrix, : is equivalent to the following : The above-mentioned augmented matrix is called affine transformation matrix, or projective transformation matrix (as it can also be used to perform projective transformations ).

In this construction, each "point" of the real projective plane is the one-dimensional subspace through the origin in a 3-dimensional vector space, and a "line" in the projective plane arises from a plane through the origin in the 3-space.

Projective profinite groups A profinite group is projective if it has the lifting property for every extension.

Since there are projective planes in which Desargues' theorem fails ( non-Desarguesian planes ), these planes can not be embedded in a higher-dimensional projective space.

Swan's theorem states that, via Γ, the category of vector bundles is equivalent to the category of finitely generated projective R-modules ("projective" corresponds to local trivialization.) In application, one often cooks up a ring by summing up modules.

The affine planes formed by the removal of different lines of the projective plane will be isomorphic if and only if the removed lines are in the same orbit of the collineation group of the projective plane.

Then each is an abelian category and we have an inclusion functor identifying the simple projective, simple injective and indecomposable projective-injective modules.

There is a projective plane of order N if and only if there is an affine plane of order N. When there is only one affine plane of order N there is only one projective plane of order N, but the converse is not true.

Thus, the connected sum of three real projective planes is homeomorphic to the connected sum of the real projective plane with the torus.

Affine planes Projectivization of the Euclidean plane produced the real projective plane.

A finite projective plane will produce a finite affine plane when one of its lines and the points on it are removed.

Another construction of the same projective plane shows that no line can be distinguished (on geometrical grounds) from any other.

Any connected sum involving a real projective plane is nonorientable.