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Curve studied by Villalpando in 1606 and by Münger in 1894.
Other names: bioval, proportionatrix curve (name given by Villalpando).

Polar equation:.
Cartesian equation: .
Curvilinear abscissa: .
Radius of curvature:.
Length of an egg: .
Area of an egg: .
Rational sextic.

Double egg and its evolute; the curvature is infinite at the centre.

The double egg curve is the Clairaut curve with the above polar equation, which shows that it is a conchoid of the quatrefoil.
It is also the inverse of the Kampyle of Eudoxus with respect to its centre:
And it is obtained by the following construction: given a segment line of constant length constrained to have its ends moving on two perpendicular lines intersecting at O, the projection of O on a line perpendicular to the segment at one of its ends describes the double egg.
The double egg is the glissette of the tip of a cardioid constrained to stay tangent to a fixed line at a fixed point.
It can also be obtained by an ellipse rolling on a quadrifolium.
The magnetic field lines created by a magnetic dipole are double eggs; the orthogonal lines are the curves with polar equations .
The road associated to the wheel shaped like a double egg so that the centre has a linear motion is the image of a cycloid by a 1/2 scaling in one direction (animation by Alain Esculier)

Compare to the simple folium ( instead of ) and look at other eggs at ovoid.

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© Robert FERRÉOL  2017