Geometrie analitica si diferentiala

Geometrie analitica si diferentiala

Hiperboloid cu o panza

Se numeste hiperboloid cu o panza o cuadrica pentru care exista un reper ortogonal in spatiu in raport cu care suprafata are ecuatia canonica
@d\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}-1=0,@d unde $$a>0,b>0,c>0$$.

Tot hiperboloizi cu o panza sunt si cuadricele de ecuatii @d\frac{x^2}{a^2}-\frac{y^2}{b^2}+\frac{z^2}{c^2}-1=0\text{ sau }\frac{x^2}{a^2}-\frac{y^2}{b^2}-\frac{z^2}{c^2}+1=0.@d

Hiperboloidul cu o panza admite doua familii de generatoare rectilinii:

@dd_{\alpha,\beta}:\begin{cases}\displaystyle\alpha\left(\frac{x}{a}+\frac{z}{c}\right)=\beta\left(1+\frac{y}{b}\right)\\
\displaystyle\beta\left(\frac{x}{a}-\frac{z}{c}\right)=\alpha\left(1-\frac{y}{b}\right)\end{cases}\text{ si }d_{\lambda,\mu}:\begin{cases}\displaystyle\lambda\left(\frac{x}{a}+\frac{z}{c}\right)=\mu\left(1-\frac{y}{b}\right)\\
\displaystyle\mu\left(\frac{x}{a}-\frac{z}{c}\right)=\lambda\left(1+\frac{y}{b}\right)\end{cases}.@d

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