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Concept
Steiner conic
Summary
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 uses a quadratic form (see Quadric (projective geometry)). 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 (i.e., ).
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 (figure 1)
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 (figure 2).
A projective mapping is a finite product of perspective mappings.
Simple example: If one shifts in the first diagram point and its pencil of lines onto and rotates the shifted pencil around by a fixed angle then the shift (translation) and the rotation generate a projective mapping of the pencil at point onto the pencil at . 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 , the rational numbers or the complex numbers . 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 only the images of 3 lines have to be given. These 5 items (2 points, 3 lines) uniquely determine the conic section.
Official source
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