Intermediate Level


The application illustrates the doubly ruled structure of a hyperbolic paraboloid by generating it through straight line segments that connect two points traveling on two non-complanar lines.

The hyperbolic paraboloid can be defined as the reunion of straight lines passing through two points that move with a constant velocity on two non-complanar straight lines.

Its canonical equation is \(z = \frac{x^2}{a^2} – \frac{y^2}{b^2}\) and its name is related with having hyperbolas and parabolas as sections.

In fact, by cutting this surface by the planes \(z = 1\) or \(z = -1\) we obtain the hyperbolas of equation \(\frac{x^2}{a^2} – \frac{y^2}{b^2} = 1\) or \(\frac{y^2}{b^2} – \frac{x^2}{a^2} = 1\) and, cutting by the coordinate planes \(y = 0\) or \(x = 0\), we obtain the parabolas \(z = \frac{x^2}{a^2}\) or \(z = -\frac{y^2}{b^2}\).

More generally, the intersections with planes parallel to the plane xOy are hyperboles and the intersections with planes parallel to the plane xOz or yOz are parabolas.

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    It was the construction of a roof with the form of a hyperbolic paraboloid that gave international recognition to the Spanish architect Felix Candela.

    In 1951 he used this form as a solution for a roof in the Cosmic Rays pavilion in Mexico City.This pavilion was designed to accomodate a laboratory, which required a roof no thicker than 1.5 cm. Candela proposed the use of hyperbolic paraboloids arguing that its geometry conferred the strength necessary for the construction of a roof with minimal thickness.

    Towards the end of his life, he left his mark in the Oceanarium of Valencia’s City of Arts and Sciences, using hyperbolic paraboloids to design the roof of the restaurant.