Calc 3 Lecture Notes Section 11.6 Page 6 of 6

Section 11.6: Parametric Surfaces

Big idea: A curve is one-dimensional, and thus always can be parameterized in terms of one independent parameter. A surface is two-dimensional, and thus requires two independent parameters to describe it.

Big skill: You should be able to recognize a surface given its parametric representation, and derive the parametric representations of simple surfaces.

Quadric Surfaces: review from section 10.6

A quadric surface is any 3D surface described by a quadratic equation in three variables:

.

General Procedure for Converting Quadric Surface Equations to Parametric Equations:

Recall these identities:

cos2(u) + sin2(u) = 1

cosh2(u) – sinh2(u) = 1

If the equation describing a surface involves each of the variables squared, then the parametric representation for each variable will be a product of circular and/or hyperbolic functions.

If the equation describing a surface involves two of the variables squared and a linear occurrence of the third variable, then the parametric representation for the squared variables will be a product of a circular or hyperbolic functions of one of the parameters and a linear factor of the other parameter.


Cylindrical Surfaces

An equation that is supposed to represent a 3D graph, but that only has two of the three variables in it (like x2 + y2 = 1) creates a graph called a cylindrical surface.

Practice:

  1. Sketch the 3D surface described by the equation x2 + y2 = 9, then find a parametric representation of the surface. Then find a parametric representation of a cylinder of radius 3 extending from -5 to 5 in the z direction with end caps.
  1. Sketch the 3D surface described by the equation z = y2, then find a parametric representation of the surface.


Ellipsoid centered at origin (includes spheres):

Quadric Form of the Surface:

Parameterized form of the Surface: x = bxcos(q)sin(j), y = bysin(q)sin(j), z = bzcos(j)

Practice:

  1. Find a parametric representation of the surface described by the equation .
  1. Find a parametric representation of the surface described by the equation .


Elliptic Paraboloid with vertex at origin:

Quadric Form of the Surface:

Parameterized form of the Surface: x = bxvcos(q), y = byvsin(q), z = v2

Practice:

  1. Find a parametric representation of the surface described by the equation .

Elliptic Cone centered at origin:

Quadric Form of the Surface:

Parameterized form of the Surface: x = bxvcos(u), y = byvsin(u), z = v

Practice:

  1. Find a parametric representation of the surface described by the equation .


Hyperboloid of One Sheet centered at origin:

Quadric Form of the Surface:

Parameterized form of the Surface: x = bxcos(q)cosh(v), y = bysin(q)cosh(v), z = bzsinh(v)

Practice:

  1. Find a parametric representation of the surface described by the equation .

Hyperboloid of Two Sheets centered at origin:
Quadric Form of the Surface:

Parameterized form of the Surface: x = bxcos(q)sinh(v), y = bysin(q)sinh(v), z = ±bzcosh(v)

Practice:

  1. Find a parametric representation of the surface described by the equation .


Hyperbolic Paraboloid centered at origin:

Quadric Form of the Surface:

Parameterized form of the Surface:

x = bxcosh(u)v, y = bysinh(u)v, z = v2

AND

x = bxsinh(u)v, y = bycosh(u)v, z = –v2

Practice:

  1. Find a parametric representation of the surface described by the equation .