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- Steiner's_theorem_(geometry) abstract "The Steiner theorem, or Steiner generation of a conic , named after the Swiss mathematician Jakob Steiner, is an alternative method to define a non-degenerate projective conic section in a projective plane over a field:Given two pencils of lines at two points (all lines containing and resp.) and a projective but not perspective mapping of onto . Then the intersection points of corresponding lines form a non-degenerate projective conic section (1. picture)A perspective mapping of a pencil onto a pencil is a bijection (1-1 correspondence) such that corresponding lines intersect on a fixed line , which is called the axis of the perspectivity (2. picture).A projective mapping is a finite sequence of perspective mappings.Examples of commonly used fields are the real numbers , the rational numbers or the complex numbers . Even finite fields are allowed.Remark:The fundamental theorem for projective planes states, that a projective mapping in a projective plane over a field (pappian plane) is uniquely determined by prescribing the images of three lines. That means for the Steiner generation of a conic section: besides two points only the images of 3 lines have to be given. From these 5 items (2 points, 3 lines) the conic section is uniquely determined.Remark:The notation "perspective" is due to the dual statement: The projection of the points on a line from a center onto a line is called perspective.".
- Steiner's_theorem_(geometry) thumbnail Steiner-erz-def.png?width=300.
- Steiner's_theorem_(geometry) wikiPageID "41207528".
- Steiner's_theorem_(geometry) wikiPageRevisionID "591911907".
- Steiner's_theorem_(geometry) subject Category:Conic_sections.
- Steiner's_theorem_(geometry) subject Category:Theorems_in_projective_geometry.
- Steiner's_theorem_(geometry) comment "The Steiner theorem, or Steiner generation of a conic , named after the Swiss mathematician Jakob Steiner, is an alternative method to define a non-degenerate projective conic section in a projective plane over a field:Given two pencils of lines at two points (all lines containing and resp.) and a projective but not perspective mapping of onto . Then the intersection points of corresponding lines form a non-degenerate projective conic section (1.".
- Steiner's_theorem_(geometry) label "Satz von Steiner (Geometrie)".
- Steiner's_theorem_(geometry) label "Steiner's theorem (geometry)".
- Steiner's_theorem_(geometry) sameAs Satz_von_Steiner_(Geometrie).
- Steiner's_theorem_(geometry) sameAs m.0zdlr2q.
- Steiner's_theorem_(geometry) sameAs Q15844090.
- Steiner's_theorem_(geometry) sameAs Q15844090.
- Steiner's_theorem_(geometry) wasDerivedFrom Steiner's_theorem_(geometry)?oldid=591911907.
- Steiner's_theorem_(geometry) depiction Steiner-erz-def.png.
- Steiner's_theorem_(geometry) isPrimaryTopicOf Steiner's_theorem_(geometry).
