Quadratic surfaces

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Definition: Cylinder

A set of parallel lines passing trough a fixed curve is cylindrical surface or cylinder.

Definition: Traces

The trace of a surface is the cross section of the surface with a plane that is parallel to one of the coordinate planes.

Definition: Quadratic surface

Quadratic surface is the set of points satisfying

$$Ax^2 + By^2 + Cz^2+Dxy + Exz + Ryz + Gx + Hy + Jz +K = 0$$

Definition: Ellipsoid

An ellipsoid is a quadratic surface described by an equation of the form

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

Definition: Elliptic paraboloid

An elliptic paraboloid is a quadratic surface described by an equation of the form

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

Definition: Hyperboloid of one sheet

A hyperboloid of one sheet is a quadratic surface described by an equation of the form

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

Definition: Hyperboloid of two sheet

A hyperboloid of two sheets is a quadratic surface described by an equation of the form

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

Definition: Elliptic cone

An elliptic cone is a quadratic surface described by an equation of the form

$$\frac{x^2}{a^2}+ \frac{y^2}{b^2} - \frac{z^2}{c^2} =0$$

Definition: Hyperbolic paraboloid

A hyperbolic paraboloid is a quadratic surface described by an equation of the form

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