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QUADRIC

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Originally appearing in Volume V11, Page 704 of the 1911 Encyclopedia Britannica.
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QUADRIC SURFACES § 91. The conics, the cones of the second See also:

order, and the ruled quadric surfaces See also:complete the figures which can be generated by projective rows or See also:flat and axial pencils, that is, by those aggregates of elements which are of one See also:dimension (§§ 5, 6). We shall now consider the simpler figures which are generated by aggregates of two dimensions. The space at our disposal will not, however, allow us to do more than indicate a few of the results. § 92. We establish a See also:correspondence between the lines and planes in pencils in space, or reciprocally between the points and lines in two or more planes, but consider principally pencils. In two pencils we may either make planes correspond to planes and lines to lines, or else planes to lines and. lines to planes. If hereby the See also:condition be satisfied that to a flat, or axial, See also:pencil corresponds in the first See also:case a projective flat, or axial, pencil, and in the second a projective axial, or flat, pencil, the pencils are said to be See also:pro ective in the first case and reciprocal in the second. For instance, two pencils which join two points Si and Si to the different points and lines in a given See also:plane ir are projective (and in See also:perspective position), if those lines and planes be taken as to the directrix than to the See also:focus. In a See also:parabola the vertex lies halfway between directrix and focus. It follows in an See also:ellipse the ratio between the distance of a point from the focus to that from the directrix is less than unity, in the parabola it equals unity, and in the See also:hyperbola it is greater than unity. It is here the same which focus we take, because the two foci See also:lie symmetrical to the See also:axis of the conic.

If now P is any point on the conic having the distances r, and See also:

r2 from the foci and the distances di and d2 from the corresponding directrices, then ri/di=r2/d2=e, where e is See also:constant. Hence alsodi d2 =e. In the ellipse, which lies between the directrices, dl+d2 is constant, therefore also ri+ri. In the hyperbola on the other See also:hand di–d2 is constant, equal to the distance between the directrices, therefore in this case ri–r2 is constant. If we See also:call the distances of a point on a conic from the focus its See also:focal distances we have the theorem: In an ellipse the sum of the focal distances is constant; and in an hyperbola the difference of the fecal distances is constant. This constant sum or difference equals in both cases the length of the See also:principal axis.

End of Article: QUADRIC

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