The Steiner conic or more precisely Steiner's 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. The usual definition of a conic in projective space uses a quadratic form. Another alternative definition of a conic uses a hyperbolic polarity. It is due to K. G. C. von Staudt and sometimes called a von Staudt conic. The disadvantage of von Staudt's definition is that it only works when the underlying field has odd characteristic.
Definition of a Steiner conic Given two pencils B ( U ) , B ( V ) {\displaystyle B(U),B(V)} of lines at two points U , V {\displaystyle U,V} (all lines containing U {\displaystyle U} and V {\displaystyle V} resp.) and a projective but not perspective mapping π {\displaystyle \pi } of B ( U ) {\displaystyle B(U)} onto B ( V ) {\displaystyle B(V)} . Then the intersection points of corresponding lines form a non-degenerate projective conic section (figure 1)
A perspective mapping π {\displaystyle \pi } of a pencil B ( U ) {\displaystyle B(U)} onto a pencil B ( V ) {\displaystyle B(V)} is a bijection (1-1 correspondence) such that corresponding lines intersect on a fixed line a {\displaystyle a} , which is called the axis of the perspectivity π {\displaystyle \pi } (figure 2). A projective mapping is a finite product of perspective mappings. Simple example: If one shifts in the first diagram point U {\displaystyle U} and its pencil of lines onto V {\displaystyle V} and rotates the shifted pencil around V {\displaystyle V} by a fixed angle φ {\displaystyle \varphi } then the shift (translation) and the rotation generate a projective mapping π {\displaystyle \pi } of the pencil at point U {\displaystyle U} onto the pencil at V {\displaystyle V} . From the inscribed angle theorem one gets: The intersection points of corresponding lines form a circle. Examples of commonly used fields are the real numbers R {\displaystyle \mathbb {R} } , the rational numbers Q {\displaystyle \mathbb {Q} } or the complex numbers C {\displaystyle \mathbb {C} } . The construction also works over finite fields, providing examples in finite projective planes. 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 that, for the Steiner generation of a conic section, besides two points U , V {\displaystyle U,V} only the images of 3 lines have to be given. These 5 items (2 points, 3 lines) uniquely determine the conic section. Remark: The notation "perspective" is due to the dual statement: The projection of the points on a line a {\displaystyle a} from a center Z {\displaystyle Z} onto a line b {\displaystyle b} is called a perspectivity (see below).
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