Calculus III 12.6 Cylinders and Quadric Surfaces

If given an equation, how do you know that it is a cylinder in 3-D?

There are only 2 variables; 1 variable is missing

Which variable is the length/axis of the cylinder?

The missing variable

What are the 4 possibilities for what 2-D shape the cylinder will have?

Parabola, circle, ellipse, or hyperbola

If given an equation, how do you know that is it a quadric surface?

It has x, y, & z variables (2 or 3 of the variables will be to the 2nd power)

How do you find the x-trace of a quadric surface given the equation? (Similar procedure for y-trace & z-trace)

- Plug in x = k

- If both variables are to the 2nd power, put all the variable terms on the left side and the number/constant & k on the right side

- Determine which 2-D shape it is

(circle, ellipse, hyperbola)

- If one variable is to the 1st power and the other is to the 2nd power, then the 2-D shape is a parabola

Which variable is the length/axis of stacked 2-D shapes?

The missing variable (the variable you set equal to k)

Given the equation in the form:

x²/a² + y²/b² + z²/c² = 1

Name of this quadric surface and describe what it looks like.

Ellipsoid

A ball that is oval (football)

Given the equation in the form:

z/c = x²/a² + y²/b²

Name of this quadric surface and describe what it looks like.

Elliptic Paraboloid

A cup/bowl

Given the equation in the form:

z/c = x²/a² - y²/b²

Name of this quadric surface and describe what it looks like.

Hyperbolic Paraboloid

A saddle

Given the equation in the form:

z²/c² = x²/a² + y²/b²²

Name of this quadric surface and describe what it looks like.

Cone

Two cones that are mirror images of each other

Given the equation in the form:

x²/a² + y²/b² - z²/c² = 1

Name of this quadric surface and describe what it looks like.

Hyperboloid of One Sheet

Like the cone, except it is wider in the middle, like an hourglass

Given the equation in the form:

- x²/a² - y²/b² + z²/c² = 1

Name of this quadric surface and describe what it looks like.

Hyperboloid of Two Sheets

Two separate cups/bowls that are mirror images of each other

If you have an elliptic paraboloid, cone, or hyperboloid of one/two sheet(s), what is the only thing you need to know to sketch the surface?

The axis of the ellipse/circle in the surface. The rest comes from the chart.