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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 (G0) with respect to a point O and a line (D) is the curve (G), locus of the point M defined as follows: given a point (M0) on (G0), 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 (G0) into (G) 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 O at the centre 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 ((G1),(G2)) with respect to a frame Oxy is the curve (G), locus of the point M defined as follows: a line (D) passing by O cuts (G1) at P and (G2) 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 (G1) 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

 
 
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© Robert FERRÉOL 2017