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LAMÉ CURVE
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Curves studied by Lamé
in 1818.
Gabriel Lamé (1795-1870): French mathematician and engineer. Other names for |
Cartesian equation of Cartesian parametrization of Area delimited by |
The Lamé curves
and
are
defined by their Cartesian equation above.
For rational values of ,
the curve
,
the part of
located in the
quadrant, is a portion of an algebraic curve
of degree pq ?, and equation
? (when p is even,
and
coincide);
the same holds for the curves
.
Examples of curves with a = b:
Lamé curve |
associated algebraic curve |
figure: the Lamé curve in red, the associated algebraic curve in green. | |
square: |
line: |
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circle: |
ditto |
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Lamé cubic: |
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union of 4 arcs of parabolas: |
parabola: |
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astroid: |
ditto |
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union of 4 branches of rectangular hyperbolas: |
rectangular hyperbola: |
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crosscurve: |
ditto |
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Lamé curve |
associated algebraic curve |
figure: Lamé curve in red, the associated algebraic curve in green. | |
eight half-lines: |
line: |
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rectangular hyperbola: |
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union of 8 arcs of parabolas: |
parabola: |
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ditto; it is the union of two evolutes of hyperbola. |
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union of 8 branches of rectangular hyperbolas: |
rectangular hyperbola: |
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bullet nose
curve: |
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When a = b =1 and a
= n is an integer,
is the Fermat curve.
See also the Lamé surfaces.
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© Robert FERRÉOL 2017