Summary of Quadric Surfaces

Essential Concepts

  • 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 [latex]xy[/latex]-, [latex]yz[/latex]-, or [latex]xz[/latex]-planes, set [latex]z=0[/latex], [latex]x=0[/latex], or [latex]y=0[/latex], 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 [latex]Ax^2+By^2+Cz^2+Dxy+Exz+Fyz+Gx+Hy+Jz+K=0[/latex].
  • 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 Figure 8 and Figure 9.

Glossary

cylinder
a set of lines parallel to a given line passing through a given curve
ellipsoid
a three-dimensional surface described by an equation of the form [latex]\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1[/latex] all traces of this surface are ellipses
elliptic cone
a three-dimensional surface described by an equation of the form [latex]\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=0[/latex] traces of this surface include ellipses and intersecting lines
elliptic paraboloid
a three-dimensional surface described by an equation of the form [latex]z=\frac{x^2}{a^2}+\frac{y^2}{b^2}[/latex] traces of this surface include ellipses and parabolas
hyperboloid of one sheet
a three-dimensional surface described by an equation of the form [latex]\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1[/latex] traces of this surface include ellipses and parabolas
hyperboloid of two sheets
a three-dimensional surface described by an equation of the form [latex]\frac{z^2}{c^2}-\frac{x^2}{a^2}-\frac{y^2}{b^2}=1[/latex] traces of this surface include ellipses and parabolas
quadric surfaces
surfaces in three dimensions having the property that the traces of the surface are conic sections (ellipses, hyperbolas, and parabolas)
rulings
parallel lines that make up a cylindrical surface
trace
the intersection of a three-dimensional surface with a coordinate plane