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DOUBLE EGG
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 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