The following figures summarizes the most important ones.
Characteristics of Common Quadratic Surfaces: Ellipsoid,
Hyperboloid of One Sheet ,
Hyperboloid of Two Sheets .Characteristics of Common Quadratic Surfaces:
Elliptic Cone ,
Elliptic Paraboloid , Hyperbolic Paraboloid.
Identifying equations of quadric surfaces
Identify the surfaces represented by the given equations.
The
and
terms are all squared, and are all positive, so this is probably an ellipsoid. However, let’s put the equation into the standard form for an ellipsoid just to be sure. We have
Dividing through by 144 gives
So, this is, in fact, an ellipsoid, centered at the origin.
We first notice that the
term is raised only to the first power, so this is either an elliptic paraboloid or a hyperbolic paraboloid. We also note there are
terms and
terms that are not squared, so this quadric surface is not centered at the origin. We need to complete the square to put this equation in one of the standard forms. We have
A set of lines parallel to a given line passing through a given curve is called a
cylinder , or a
cylindrical surface . The parallel lines are called
rulings .
The intersection of a three-dimensional surface and a plane is called a
trace . To find the trace in the
xy -,
yz -, or
xz -planes, set
respectively.
Quadric surfaces are three-dimensional surfaces with traces composed of conic sections. Every quadric surface can be expressed with an equation of the form
To sketch the graph of a quadric surface, start by sketching the traces to understand the framework of the surface.
Important quadric surfaces are summarized in
[link] and
[link] .
For the following exercises, sketch and describe the cylindrical surface of the given equation.
[T]
The surface is a cylinder with the rulings parallel to the
y -axis.
