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HYPERBOLISM AND ANTIHYPERBOLISM OF A CURVE

NEWTON TRANSFORMATION

Notion studied by Newton. |

Cartesian equation | Cartesian parametrization | |

initial curve | ||

hyperbolism with respect to O and the line x
= a |
||

antihyperbolism |

The hyperbolism of a curve
with respect to a point O and a line (D) is the curve ,
locus of the point M defined as follows: given a point (M_{0})
on , the
line (OM_{0})
cuts (D) at P; M is the projection of P on
the line parallel to (D) passing by M_{0}.
Analytically, with the line ( |

The inverse transformation ,
is referred to as *antihyperbolism*.

Examples:

antihyperbolism | O and (D)
(for the initial curve) |
O et (D) (for
the final curve) |
hyperbolism |

circle | O on the circle, (D)
tangent to the circle opposite to O |
O on the "middle" of the
asymptote and (D) tangent at the summit |
witch of Agnesi |

circle | O centre of the circle,
(D) tangent to the circle |
O at the centre and (D)
tangent at the summit |
Külp quartic |

circle | (D) perpendicular to the line joining O
to the centre of the circle
(this case includes the previous ones) |
Granville egg | |

circle | O on the circle, (D) parallel to the diameter
passing by O |
O at the centre and (D) passing by the
intersection points with the circle |
anguinea |

lemniscate of Gerono: (up to scaling in one direction) | O at the centre, (D) tangent at a vertex |
O at the centre, (D) tangent to the circle |
circle |

piriform quartic | O at the cusp, (D)
perpendicular to the symmetry axis |
O on the circle, (D)
parallel to the tangent to the circle at this point |
circle |

rational divergent parabola: | O at the centre, (D):
x
= a |
O at the centre and (D):
x
= a |
parabola |

cubical parabola: | O and line x
= a |
O and line x =
a |
trident: |

visiera: | O and line x = a |
O and line x = a |
visiera: |

If the straight line (*D*) is replaced by any given
curve, we get the more general transformation of Newton:

The Newton transform of a couple of curves (,)
with respect to a frame Oxy is the curve ,
locus of the point M defined as follows: a line (D) passing
by O cuts
at P and
at Q; M is the intersection point between the line parallel
to Ox passing by P and the line parallel to Oy passing
by Q.
We get the hyperbolism with a line parallel to Oy. |

Cartesian parametrization of the Newton transform of
the curves
and with
respect to Oxy: . |

Examples:

first curve (G_{1}) |
second curve (G_{2}) |
transform |

circle with centre O |
circle with centre O |
ellipse (obtained by "reduction of the ordinates") |

circle centred on Ox passing through O |
circle with centre O |
eight-like curve ,
dilatation of a lemniscate of Gerono
(not dilated when a = b, i.e. when the circles are tangent). |

swap the previous circles | arc of a parabola | |

circle centred on Ox |
circle with centre O |
Hügelschäffer egg |

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© Robert FERRÉOL 2017