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

Overview

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.

Objectives

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

• 13.7.1 Find equation and graph (manually and electronically) a sphere and cylinder in R3
• 13.7.2 Identify a quadric surface from its equation or graph
• 13.7.3 Find the general equation of a given quadratic surface
• 13.7.4 Find the axis intercept points of a quadric surface
• 13.7.5 Find the trace of a quadric surface (ie, the xy-trace, xz-trace, and yz-trace)
• 13.7.6 Graph a quadric surface electronically

Terminology

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 free 3D grapher? [ CalcPlot3D Grapher | CalcPlot3D Info ]

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.