Intermediate Level

Architecture

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.