Course: Calculus III
Topic: Vectors and Geometry of Space
Subtopic: Quadric Surfaces


Beyond planes, more 3D shapes surface (pun intended)! Quadratic surfaces are also called quadric surfaces or quadrics for short. They have general equation ax2+by2+cz2+2dx+2ey+2fz+2gxy+2hxz+2iyz+j=0 (where a, b, ..., j are constants). A quadric surface intersects every axes plane in a (possibly degenerate) conic section. Examples of quadratic surfaces include the cone, cylinder, ellipsoid, elliptic cone, elliptic cylinder, elliptic hyperboloid, elliptic paraboloid, hyperbolic cylinder, hyperbolic paraboloid, paraboloid, sphere, and spheroid.


By the end of this topic you should know and be prepared to be tested on:


Terms you should be able to define: quadric surface, all 12 quadrics hyperlinked in the overview above, trace, xy-trace, xz-trace, yz-trace

Formulae to have in your notes: equations of quadric surfaces

Supplemental Resources (recommended)

Download/Print: Prof. Keely's Quadric Surfaces Formula Sheet

Need a 3D grapher? See recommendations at "Online 3D Graphers (free!)" at Where can I find a free online grapher, graphing software, or graphing app?.

Supplemental Resources (optional)

Paul's OL Notes - Calc III: Quadric Surfaces

More tutorial videos if you need them are listed at James Sousa's MathIsPower4U - Calc II. Scroll down right column to "Quadric, Surfaces, Cylindrical Coordinates and Spherical Coordinates". There are lots of related titles in the first half of that list.