Quadric Surface

Conic Section but in 3 dimensions.

Ellipsoid

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$$

Paraboloid

Elliptic Paraboloid

$$\frac{z}{c}=\frac{x^2}{a^2}+\frac{y^2}{b^2}$$

Hyperbolic Paraboloid

$$\frac{z}{c}=\frac{x^2}{a^2}-\frac{y^2}{b^2}$$

Cone

$$\frac{z^2}{c^2}=\frac{x^2}{a^2}+\frac{y^2}{b^2}$$

Hyperboloid

Hyperboloid of One Sheet

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$$

Hyperboloid of Two Sheets

$$-\frac{x^2}{a^2}-\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$$