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 (M0) on , the line (OM0) cuts (D) at P; M is the projection of P on the line parallel to (D) passing by M0. Analytically, with the line (D) x = a, the transformation of  into  can be written ; it is quadratic, so an algebraic curve of degree n is transformed into an algebraic curve of degree £ 2n.

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 (G1) second curve (G2) 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

© Robert FERRÉOL 2017