next surface  previous surface  2D curves  3D curves  surfaces  fractals  polyhedra 
MILK CARTON
Could also be named "humbug"...
Surface studied by Cundy and Rollett in 1951 [Cundy Rollett p 185 à 188] See also a model from the National Museum of American History. 
Cartesian parametrization: .
Cartesian equation: (proving that the contour lines are ellipses), i.e. . Quartic surface. Volume of the milk carton: . Area of the milk carton when k = 1/2: » 7,29a^{2}. 
Here, (D_{1}) is , (D_{2}) is and the radius of (C) is ka.
Here is the (more) complete surface:
The milk carton is also the ruled surface generated by
the lines (M_{1}M_{2}),
and with
two orthogonal sinusoidal motions in quadrature; the part shaped like a
milk carton is the reunion of the segment lines [M_{1}M_{2}].
The length of the segment line [M_{1}M_{2}] then remains constant equal to ; the milk carton can therefore also be defined as the ruled surface generated by a line two fixed points of which slide on two fixed orthogonal nonintersecting lines. All the points on the line describe ellipses (which constitutes a generalization of the Proclus ellipsograph). 

The projection of the segment line [M_{1}M_{2}] on xOy also maintains a constant length: the view from above of a milk carton is therefore a full astroid. 

We also get a generalization of the milk carton by considering
the conoidal surface generated by the lines (M_{1}M_{2}), ,
having two orthogonal sinusoidal motions with any phase difference.
phase difference equal to  zero phase difference: we get a hyperbolic paraboloid  phase opposition: another hyperbolic paraboloid 



Be careful, a milk carton like the one opposite made of paper is a developable surface, made from a tetrahedron template by bending the edges... 
Compare to the conocuneus,
as well as the Cayley cubic surface.
next surface  previous surface  2D curves  3D curves  surfaces  fractals  polyhedra 
© Robert FERRÉOL, Alain ESCULIER 2017