Geometrie analitica si diferentiala

Geometrie analitica si diferentiala

paraboloid hiperbolic

Se numeste paraboloid hiperbolic 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}=2z,@d unde $$a>0,b>0$$.

Tot paraboloizi hiperbolici sunt si cuadricele de ecuatii @d\frac{x^2}{a^2}-\frac{z^2}{c^2}=2y\text{ sau }\frac{y^2}{b^2}-\frac{z^2}{c^2}=2x.@d

Paraboloidul hiperbolic admite doua familii de generatoare rectilinii:

@dd_{\alpha,\beta}:\begin{cases}\displaystyle\alpha\left(\frac{x}{a}+\frac{y}{b}\right)=2\beta z\\
\displaystyle\beta\left(\frac{x}{a}-\frac{y}{b}\right)=\alpha\end{cases}\text{ si }d_{\lambda,\mu}:\begin{cases}\displaystyle\lambda\left(\frac{x}{a}+\frac{y}{b}\right)=\mu\\
\displaystyle\mu\left(\frac{x}{a}-\frac{y}{b}\right)=2\lambda z\end{cases}.@d

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