11.6 Surfaces in Space

Definition of a Cylinder Let C be a curve in a plane and let L be a line not in a parallel plane. The set of all lines parallel to L and intersecting C is called a cylinder. C is called the generating curve (or directrix), and the parallel lines are called rulings. Note: If one of the variables is missing from the equation of a cylinder, its rulings are parallel to the coordinate axis of the missing variable.

Examples:

Quadric Surfaces The equation of a quadric surface in space is a second-degree equation of the form There are six basic types of quadric surfaces:

1) Ellipsoid Standard Form

2) Hyperboloids of One Sheet Standard Equation:

3) Hyperboloid of Two Sheets Standard equation

4) Elliptic Cone Standard Equation: The axis of the cone corresponds to the variable whose coefficient is negative. The traces in the coordinate planes parallel to this axis are intersecting lines.

5) Elliptic Paraboloid Standard Equation: Two traces are parabolas and one is an ellipse. The axis of the paraboloid corresponds to the variable raised to the first power.

6) Hyperbolic Paraboloid Standard Equation: The axis of the paraboloid corresponds to the variable rasied to the first power.

To Sketch a Quadric Surface Write the surface in standard form. Determine the traces in the coordinate planes by setting each variable =0 For example: To get the trace in the xy-plane, set z=0. To get the trace in the xz-plane, set y=0, etc. If needed, find the traces in planes that are parallel to coordinate planes by holding a variable constant.

Examples: Identify and Sketch: 1) 2) #6, 19, 30

#42 Sketch the region bounded by the graphs of the equations. X=0, y=0, z=0