Cylinders and Quadric Surfaces Section 13.6 Cylinders and Quadric Surfaces

SURFACES The graph of an equation in three variables (x, y, and z) is normally a surface. Two examples we have already seen are planes and spheres. Graphing surfaces can be complicated. The best way is by finding the intersections of the surface with well-chosen planes (e.g., the coordinate planes). The intersections are called cross sections; those intersections with the coordinate planes are called traces.

CYLINDERS In Calculus the term cylinder denotes a much wider class of surfaces than the familiar right circular cylinder. A cylinder is a surface that consists of all lines (called rulings) that are parallel to a given line and pass through a given plane curve (called the generating curve) An equation of a cylinder is easy to recognize since it will contain only two of the three variables x, y, and z.

QUADRIC SURFACES If a surface is the graph in three-space of an equation of second degree, it is called a quadric surface. Cross sections of quadric surfaces are conics. Through rotations and translations, any general second degree equation can be reduced to either Ax2 + By2 + Cz2 + J = 0 or Ax2 + By2 + Iz = 0.

SIX QUADRIC SURFACES For graphs of these surfaces, see page 872 in the text.