Quadratic Equations

1. Three squares with the same signs.

Ellipsoid: \(\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\)

2. Three squares with different signs.

Hyperboloid of 1 sheet: \(\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1\)

Cone: \(\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=0\)

Hyperboloid of 2 sheets: \(\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=-1\)

3. Two squares with the same signs

Paraboloid: \(\frac{z}{c}=\frac{x^2}{a^2}+\frac{y^2}{b^2}\)

4. Two squares with the different signs

Saddle: \(\frac{z}{c}=\frac{x^2}{a^2}-\frac{y^2}{b^2}\)

Source of images: Wikipedia