Section 12.4

Quadric Surfaces

The 3D equivalents of conic sections: Ellipsoids, Paraboloids, and Hyperboloids.

Introduction

A Quadric Surface is the graph of a second-degree equation in three variables. They are the 3D counterparts to parabolas, ellipses, and hyperbolas. The shape is determined by the signs of the squared terms and the powers of the variables.

Interactive: The Hyperbolic Paraboloid (Saddle)

Graph of . Notice it goes up in and down in .

Standard Forms

Ellipsoid

All terms positive. (Sphere if a=b=c)

Cone

Variables on one side equal square of other.

Elliptic Paraboloid

One linear variable, two squared (same sign).

Hyperbolic Paraboloid

Saddle shape. Squared terms have opposite signs.

Hyperboloid (1 Sheet)

One negative sign. Opens along negative axis.

Hyperboloid (2 Sheets)

Two negative signs. Two separate pieces.

Worked Examples

Example 1: Identifying Surfaces (Level 1)

Identify the surface given by .

Step 1: Move Constant

Step 2: Make RHS = 1

Divide by -12:

Step 3: Analyze Signs

Signs are . Two negatives.
Conclusion: Hyperboloid of Two Sheets (opening along y-axis).

Example 2: Analyzing Traces (Level 2)

Sketch traces for (Elliptic Paraboloid).

z = k (Horizontal Traces):
. For , these are Ellipses.
x = 0 (Vertical Trace):
. This is a Parabola.
y = 0 (Vertical Trace):
. This is a Parabola.

This confirms the shape is a bowl opening up along the z-axis.

Example 3: Cylinder vs Surface (Level 3)

Classify .

Key Detail: The variable is missing!

In the -plane, this is an ellipse.
Since can be anything, we extend this ellipse along the y-axis.
Result: Elliptic Cylinder centered on the y-axis.