| TWO-SHEETED HYPERBOLOIDS |
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two-sheeted elliptical hyperboloid \(-\frac{x^2}{a^2}-\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\) Real plane intersections are: the ellipses, the circles, the parabolas and the hyperbolas. |
hyperboloid of revolution, two-sheeted circular hyperboloid \(-\frac{x^2}{a^2}-\frac{y^2}{a^2}+\frac{z^2}{c^2}=1\) It is obtained by rotating a hyperbola about its transverse axis. Real plane intersections are: the ellipses,the circles, the parabolas and the hyperbolas. |
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| ONE-SHEETED HYPERBOLOID |
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one-sheeted elliptical hyperboloid \(\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1\) A ruled surface with two systems of rulings - there are two lines of the surface through each point of this surface. Real plane intersections are: the ellipses, the circles, the parabolas, the hyperbolas and degenerated conics - pairs of two real lines (intersections with the tangent planes). |
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one-sheeted hyperboloid of revolution \(\frac{x^2}{a^2}+\frac{y^2}{a^2}-\frac{z^2}{c^2}=1\) A special case of a one-sheeted elliptic hyperboloid (for \( a=b\)). It is obtained by rotating a hyperbola about its conjugate axis or by rotating one line about another line when these lines are skew. The rotated line is a ruling belonging to both systems of rulings of this ruled surface. |
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Created by Sonja Gorjanc, translated by Helena Halas and Iva Kodrnja - 3DGeomTeh - Developing project of the University of Zagreb