Hyperbolic paraboloid . It is a doubly ruled surface so it can be built from straight lines. It is one of the most used in works by Antoni Gaudí and Felix Candela .
Summary
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- 1 Extended definition
- 2 equation
- 3Propiedades
- 4 Applications
- 1 Examples
- 5 Its construction
- 6 Sources
Definition expanded
The Hyperbolic Paraboloid is also known under the names of saddle or mountain pass due to its geometric conformation, since it is a surface that in one direction has the parabola- shaped sections with the sides up and, in the perpendicular section, sections are parabola-shaped with sides down. The concept can be simplified by claiming that it is a warped plane.
The sections according to planes perpendicular to the previous two (according to the third dimension of space) are hyperbola- shaped . If they are below the point of the chair, in the center of the figure, the sides of the hyperbola are shaped like valleys. If they are above this point, the sections of the hyperbola shape the peaks that flank the pass.
Equation
Cartesian equation:
Hyperbolic paraboloid
- With the z = k: planes, hyperbolas that change axis with the sign of k.
- If k = 0 it reduces to a pair of lines
- With x = k: axis parabolas with descending branches and ascending vertices.
- With y = k: certain parabolas of z axis with ascending branches and descending vertices remain.
The surface is shaped like a saddle.
Thus the origin appears to be a local maximum from one direction, but a local minimum from a different direction. Such a point on a surface is called a saddle point.
Properties
Hyperbolic Paraboloid has the following properties:
- Even being a curved surface, it can be built with straight lines.
- Given four points in space that are not on the same plane, there is a single hyperbolic paraboloid that passes precisely through these four points.
Applications
The Hyperbolic Paraboloid has been one of the most widely applied surfaces in architecture . Gaudí was one of those who used it, but the one who has worked the most has been Félix Candela. Within the surface fauna, this curve is a specimen already known to the Greeks.
The really important property, which motivated the interest of both Gaudí and Candela, is the fact that the hyperbolic paraboloid, even being a curved surface, can be built with straight lines. All you have to do is vary the angle of inclination of a line that moves on top of another curve. Geometrists call these surfaces ruled surfaces and there are examples in sufficient quantity in other art , in sculpture .
Presumably, this property is what allowed Gaudí to give precise instructions to his workers and the foreman when they had to build a hyperbolic paraboloid on the roof of the Sagrada Familia (started in 1883 ).
Gaudí used the hyperbolic paraboloid and also other doubly ruled surfaces such as the Hyperboloid of revolution. The architect of Spanish origin, exiled to Mexico and later nationalized North American, Félix Candela was the one who showed a sublime mastery in its use.
Examples
The best example can be found at
Restaurant Los manatiales
Los Manantiales restaurant (1958) in the Choximilco park in Mexico City. The roof is made up of eight hyperbolic paraboloids. The same structure can now be found in the new Oceanogràfic (2002) of the City of Arts and Sciences in Valencia. Another example was El Parque_Güell designed by the architect Gaudí with codes of the Catalan modernism style , with Catalan vault covers in the form of a hyperbolic paraboloid.
His construction
Given four points in space that are not on the same plane, there is a single hyperbolic paraboloid that passes precisely through these four points. This is the same property that says that two points determine a single line.
What the workers had to do was connect one of the pairs of dots on one side and the opposite pair on the other with bars. Then you just have to let another bar slide on the previous two, maintaining a constant speed at the ends.
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