Ch 12.6: Cylinders and Quadric Surfaces

DefinitionA cylinder is a surface that consists of all lines (called rulings)that are parallel to a given line and pass through a given planecurve.

Example) Sketch the graph of surface z = x2.

Parabolic cylinderGiven z = x2,

ExamplesSketch the surfaces.

1. x2 + y2 = 12. y2 + z2 = 1

Quadric SurfacesA quadric surfaces is the graph of a second-degree equation inthree (3) variables x , y , and z . The standard form of the equationis

Ax2 + By2 + Cz2 + Dxy + Eyz + Fxz + Gx + Hy + Iz + J = 0

where A,B,C , · · · , J are constants.

Examples:

I x2 + y2

9 + z2

4 = 1I z = 4x2 + y2

I z = y2 − x2

ExampleUse traces to sketch the quadric equations x2 + y2

9 + z2

4 = 1.

Ellipsoid x2 + y2

9 + z2

4 = 1

ExampleUse traces to sketch the quadric equations z = 4x2 + y2.

Elliptic paraboloid: z = 4x2 + y 2

ExampleSketch the quadric equations z = y2 − x2.

Hyperbolic Paraboloid: z = y 2 − x2

Graphs of Quadric surfaces

Class ExerciseSketch the following surfaces:

1. 2x2 + y2 = 22. 2x2 − y2 + z2 = 0