12.6 Cylinder and Quadratic Surfaces

To sketch the graph on a surface, determine the curves of intersection of the surface with planes parallel to the coordinate planes. We have to hold one variable (either $x$, $y$, or $z$) constant. These curves are called "traces" (or "cross sections") of the surface. Put 2D traces together to make it 3D.

Cylinders

A cylinder is a surface that consists of all lines ("rulings") that are parallel to a given line and passes through a given plane. Parabolic cylinder is made of infinitely many shifted copies of the same parabola. If $x$, $y$, or $z$ is missing from teh equation of a surface, than it is a cylinder.

• $x^2+y^2=1$ represents a cylinder, not a circle, that runs downt the z-axis.

Quadratic Surfaces

A quadratic surface is a graph of a second degree equation with $x$, $y$, and $z$.

• For example: $Ax^2+By^2+Cz^2+Dxy+Eyz+Fez+Gx+Hy+Iz+J=0$ Quadratic surfaces are the counterparts in three dimensions of the conic sections (circle, ellipse, parabola, hyperbola) in the plane.