From the invariant definition of this parametrization one can construct coordinate systems and parametrizations of the conic, by using the idea of stereographic projection. In fact, take an arbitrary line (e) and map every point X of the conic to the point Y = f

I call these also

Define the analogous map f

The argument can be reversed: Consider two points C, D and a homographic relation y=g(x) on the points of a line (e). The intersection points X of the lines Cx, Dy, joining corresponding points of the line build a conic. This again is obvious, since eliminating the line coordinate from the equation defining X leads to a quadratic equation for the coordinates X(x

Some of the applications of the idea of good parametrization are the definitions of

The inverse of a good parametrization is a rational representation of the curve in terms of the (projective) coordinates of a line. Indeed, setting (u,v) for a system of projective coordinates of the line (e) and working out the inverse of f

Good parametrizations generalize the

(++) Good parametrizations establish a bijection (homeomorphism) between projective conics and projective lines, showing that conic sections are bendings of lines. A further common characteristic with the lines is that all (projective) conics are equivalent under projectivities. There are many ways to see that. A simple one is through a particular coordinate system, adapted to an arbitrary conic, in which that conic is represented through the equation x

Chasles_Steiner_Envelope.html

Complex_Cross_Ratio.html

Complex_Cross_Ratio2.html

CrossRatio.html

GoodParametrizationInverse.html

Harmonic_Bundle.html

HomographicRelation.html

HomographicRelationExample.html

ParabolaProperty.html

ProjectiveBase.html

ProjectiveCoordinates.html

ProjectivePlane.html

RectHypeRelation.html

Stereographic.html

Berger Marcel, Pansu Pierre, Berry Jean-Pic, Saint-Raymond Xavier

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