Double egg

DOUBLE EGG CURVE

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.

next curve previous curve 2D curves 3D curves surfaces fractals polyhedra

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)