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Given canonical cone equation, how to find equation of cone's generatrix?

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Is the differential-geometry tag correct? –  Aryabhata Jan 17 '11 at 18:46

1 Answer 1

Consider a cone with caconical equation $x^2/a^2+y^2/b^2-z^2/c^2=0$. Generatrix satisfy the cone equation. Point $x=y=z=0$ satisfy the cone equation and thus satisfy the generatrix equation. Point $x=a y=b z=c\sqrt 2 $ satisfy both cone and generatrix equations. Thus, the direction vector for a certain generatrix will be $(a,b,c\sqrt2)$. Similarly, for any point $x=p, y=q, z=c/ab\sqrt{p^2 b^2+q^2 a^2}$ on a cone parametrized by $(p,q)$ there will be a generatrix with equation $px+qy+zc/ab\sqrt{p^2 b^2+q^2 a^2}=0$, or better, with equation $abpx+abqy+cz\sqrt{p^2 b^2+q^2 a^2}=0$. This is the canonical equation of a desired generatrix.

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