Lemniscate of Bernoulli

LEMNISCATE OF BERNOULLI

Curve studied by Jacques Bernoulli in 1694 and Fagnano in 1750.
Jacques Bernoulli (1654 -1705): Swiss mathematician.
Other name : Bernoullian lemniscate.

Bipolar equation: (where d is the half-distance between the poles F and F', foci of the lemniscate).
Tripolar equation: (O middle of F and F').
Polar equation: (with , F(d, 0), F'(-d,0)).

Cartesian equation: .
Rational bicircular quartic.
Cartesian parametrization: ().
Rational Cartesian parametrization: (, ),
hence the complex parametrization: .
Another Cartesian parametrization: ().
Cartesian parametrization of the complexified curve: .
Complex equation: .
Pedal equation: .
Polar tangential angle: .
Curvilinear abscissa: .
Radius of curvature: .
Intrinsic equation:.
Length: , where , elliptic integral of the first kind, is the constant of the lemniscate (variation of the letter "pi"), to relate to .
We also have: . Total area: a².

The lemniscate of Bernoulli is a challenger, along with the cardioid, for the record number of memberships to various families of remarkable curves.

It is indeed:

- a special case of Cassinian oval (see the bipolar equation)

- a special case of Booth curve.

- a special case of sinusoidal spiral (see the polar equation)

- as all rational bicircular quartic, at the same time, the pedal with respect to O and the inverse (reference circle with diameter [A(a,0) ; A'(-a,0)]) of the rectangular hyperbola with centre O and vertices A and A'; F and F' are the inverses of the foci of this hyperbola and the tangents at the origin are the inverses of the asymptotes.

- it is also, as a pedal curve, the envelope of the circles with diameters the ends of which are the centre and a point of this hyperbola.

- as well as the locus of the centre of a hyperbola rolling without slipping on an equal hyperbola, with coinciding vertices.

- the cissoid of the circle with centre F passing by O and the circle with centre and radius a.

- the cissoid with pole O of the circles (C) and (C') with centres F and F' and radii a/2.
In dotted lines, the circles (C) and (C'), in blue their homothetic image, the median of which is the lemniscate.

- the locus of the middles of segment lines of length 2d the ends of which describe two circles with radius a centred on F and F'.

Therefore, the lemniscate is a curve of the three-bar, in the special case of the Watt curve; according to the principle of the slider-crank exchange, there exists a second construction with an articulated quadrilateral:

- The section of a torus, with revolution radius d and meridian radius d/2, by a plane located at distance d/2 from the axis (the lemniscate is therefore a spiric of Perseus)

- the curve passing by O the curvature of which is proportional to the distance to O (compare to the elastic curve, the curvature of which is proportional to the distance to a fixed line)

- the locus of the points M such that

- the projection on the plane xOy of the biquadratic: , intersection of a cone of revolution by a paraboloid of revolution:

Furthermore:

- the asymptotic curves of the Plücker conoid are projected on lemniscates of Bernoulli.

- the lemniscate of Bernoulli is a synodal curve of all the intersecting lines passing by the double point:

The evolute of the lemniscate of Bernoulli is parametrized by . Notice that the two vertices correspond to maxima of the curvature...
...as opposed to the lemniscate of Gerono, where they correspond to minima...
Generalisation: the pedal of the rectangular hyperbola with respect to a point on the symmetry axis is a distorted lemniscate, parametrized by .

Watt mechanism to construct the lemniscate

Families of orthogonal lemniscates

See here how to "thicken" a lemniscate:.

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- a special case of Cassinian oval (see the bipolar equation)
- a special case of Booth curve.
- a special case of sinusoidal spiral (see the polar equation)
- as all rational bicircular quartic, at the same time, the pedal with respect to O and the inverse (reference circle with diameter [A(a,0) ; A'(-a,0)]) of the rectangular hyperbola with centre O and vertices A and A'; F and F' are the inverses of the foci of this hyperbola and the tangents at the origin are the inverses of the asymptotes.
- it is also, as a pedal curve, the envelope of the circles with diameters the ends of which are the centre and a point of this hyperbola.
- as well as the locus of the centre of a hyperbola rolling without slipping on an equal hyperbola, with coinciding vertices.
- the cissoid of the circle with centre F passing by O and the circle with centre and radius a.
- the cissoid with pole O of the circles (C) and (C') with centres F and F' and radii a/2. In dotted lines, the circles (C) and (C'), in blue their homothetic image, the median of which is the lemniscate.
- the locus of the middles of segment lines of length 2d the ends of which describe two circles with radius a centred on F and F'.
Therefore, the lemniscate is a curve of the three-bar, in the special case of the Watt curve; according to the principle of the slider-crank exchange, there exists a second construction with an articulated quadrilateral:
- The section of a torus, with revolution radius d and meridian radius d/2, by a plane located at distance d/2 from the axis (the lemniscate is therefore a spiric of Perseus)
- the curve passing by O the curvature of which is proportional to the distance to O (compare to the elastic curve, the curvature of which is proportional to the distance to a fixed line)
- the locus of the points M such that
- the projection on the plane xOy of the biquadratic: , intersection of a cone of revolution by a paraboloid of revolution: