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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 |
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The hyperbolism of a curve Analytically, with the line (D) x = a, the
transformation of |
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: |
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:
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The Newton transform of a couple of curves ( We get the hyperbolism with |
Cartesian parametrization of the Newton transform of
the curves |
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 |
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