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More on Quadrics

Bruno Autin
Organization: MathSoft Overseas
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This notebook illustrates the concept of volume limited by various quadrics. The ellipsoid has a closed shape and its volume is finite. The other quadrics extend to infinity, and the surface of the quadric alone is no longer sufficient to define a finite volume. An intuitive approach consists, in general, of viewing a quadric as a surface of revolution of axis Δ. This would be exact if the section of the quadric by a plane perpendicular to Δ were a circle. In fact, for a general quadric, the section is an ellipse. There is, however, an exception to that approach because the hyperbolic paraboloid has no elliptical section but the axis Δ still exists. The volume is then limited by the quadric and two planes perpendicular to Δ. The direct access to the volume form of a quadric is provided by the canonical form, the simplest representation of a quadric obtained when the reference frame coincides with the axes of symmetry of the quadric. Since a volume cannot be represented, it is suggested by a sequence of surfaces that fill it, hence the term Russian dolls, which are objects boxed into one another. Important also in this notebook are the manipulation of the parameters and their range of variations within the contexts of surfaces or volumes.

*Mathematics > Geometry
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quadrics.nb (4.7 MB) - Mathematica Notebook