projektivienti

Projektivienti, also known as projective geometry, is a branch of geometry that studies properties of figures that are invariant under projective transformations. These transformations include projections, which are mappings that preserve the straightness of lines but not necessarily their lengths or angles. Projektivienti is a generalization of Euclidean geometry, which studies properties invariant under rigid motions such as translations and rotations.

In projektivienti, the concept of parallel lines does not exist in the same way as in Euclidean geometry. Instead, any two lines in a projective plane intersect at a point, which may be at infinity. This point at infinity allows for the extension of Euclidean geometry to include parallel lines as intersecting at a point at infinity.

One of the fundamental concepts in projektivienti is the projective plane. A projective plane can be constructed from a Euclidean plane by adding a line at infinity, where all parallel lines in the Euclidean plane intersect. This line at infinity contains points that represent the directions of the parallel lines.

Projektivienti has applications in various fields, including computer graphics, where projective transformations are used to map three-dimensional objects onto a two-dimensional screen. It is also used in photography and cinematography to correct for lens distortion and to create perspective effects. Additionally, projektivienti plays a role in the study of algebraic geometry and the classification of surfaces.