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Informal names: HP bowl, or horse saddle , or Pringles.

Cartesian equation: .
Doubly ruled quadric.
Cartesian parametrization: 
the coordinate lines of which constitute the double family of straight lines (which are also the asymptotic lines): 
the coordinate lines of which constitute the double family of generatrix parabolas as a translation surface:
the coordinate lines of which constitute the family of hyperbolas and one of the 2 families of parabolas: or
the coordinate lines of which are the curvature lines:
The projections of the curvature lines on xOy form a double lattice of hyperbolas one of which is given by .

Striction lines: the 2 parabolas sections by the planes  (linear and located in the plane z = 0 in the rectangular case).

The paraboloid is said to be rectangular if a = b (the generatrices of each family are then perpendicular two by two).
Cartesian equation in the rectangular case:  (frame turned by  around Oz).
Cylindrical equation: .
Cartesian parametrizations:
the coordinate lines of which give the family of hyperbolas and a family of parabolas: or 
the coordinate lines of which give a family of straight lines and a family of horopter curves:
First fundamental quadratic form: .
Surface element: .
Second fundamental quadratic form:.
Gaussian curvature: ; all the points are hyperbolic.
Mean curvature: .

The hyperbolic paraboloid can be defined as the ruled surface generated by the straight lines
    - meeting two lines that are non coplanar and remaining parallel to a fixed plane (secant to these two lines) called directrix plane of the paraboloid
    - meeting three lines that are two by two non coplanar, but parallel to a given plane (when it is not the case, we get the one-sheeted hyperboloid).
A hyperbolic paraboloid can also be defined as the union of the lines joining two points moving at constant speed on two non coplanar lines.
The 4 sides of any skew quadrilateral are therefore included in a unique HP (see this link for details); unfortunately, this HP is not the surface with minimal area supported on this contour (see a special case at Schwarz surface).

See also milk carton.

A portion of paraboloid can therefore be produced by tightening elastic bands between two linear rods (with a smooth connection between the elastic bands and the rods).

In the equations above, the paraboloid is the union of the lines  parallel to the directrix plane (which is also an asymptote) (P):  and also the union of the lines  parallel to the directrix plane (P'): .
The hyperbolic paraboloid is doubly a conoid; more precisely, it is a conoid with axis one of the lines , directrix plane (P') and directrix another line , and a conoid with axis one of the lines , directrix plane (P) and directrix another line .
The rectangular case (a = b) corresponds to the case of the right conoid, and it is a special case of generalized Zindler conoid.

It is also a translation surface (translation of a parabola along another one, oriented in opposite directions).

The sections by planes parallel to Oz are parabolas, and the other planar sections are hyperbolas or lines.

In the rectangular case, the sections by cylinders with axis Oz are pancake curves.
The projections on xOy of the geodesic lines of the equilateral HP  z = xy are the curves solutions of the differential equation  : .
You can see on this picture the geodesic lines coming from O and the corresponding "geodesic circles".

Confocal paraboloids and triple orthogonal family of paraboloids.

If a > b, then the paraboloids with equation  are such that the sections by the plane xOz are confocal parabolas (i.e. with the same focus ).
For  we get a first family, composed of elliptic paraboloids, for  we get a second family, composed of hyperbolic paraboloids, for , we get a third family, composed again of elliptic paraboloids.
Moreover these three families constitute a triple orthogonal family, which means that every surface of each family cuts perpendicularly every surface of the other two families (3D generalization of the 2D orthogonal trajectories).
The intersection lines are the curvature lines (Dupin theorem).
Opposite, an example of each family.

See also the flippable surfaces, the Bezier surfaces and the monkey saddle.

Playground for children, boulevard Richard Lenoir, Paris.
The ropes follow (more or less) the Cartesian parametrization by the straight lines...

Here, Place Bellecour, Lyon.
The ropes follow the polar parametrization, by parabolas and pancake curves.


Structure composed of 12 portions of hyperbolic paraboloids; the 8 oblique generatrices (joining 2 middles of a side of a square) are generatrices of a hyperboloid of revolution.
If the hyperbolic paraboloids are replaced by the minimal surface supported on the boundary, we get the fundamental patch of the Schwarz P minimal surface.

Sculpture by Angel DUARTE made of pieces of hyperbolic paraboloids (Lausanne, Switzerland) using this structure. It has 36 portions of hyperbolic paraboloids.

Explanation of this structure.
Compare to the Schwarz P minimal surface.


Restaurant Los Manantiales, Mexico
Architect: F. Candela 1958

Vaults inside the Sagrada Familia, Barcelona.
Architect: Antoni Gaudi, who used hyperbolic paraboloids a lot.

Church of Becerril de la Sierra, Spain
Architect: Friar Francisco Coello de Portugal

Palais du Cnit, Paris, the shape of which reminds the hyperbolic paraboloid,
is in fact, as a first approximation, the union of three parabolic cylinders.

See more beautiful pictures on the mathourist's page.
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© Robert FERRÉOL  2017