Calculus 3

Parametric Equations

The only differences in parametric equations in 3 space are:

1. There are 3 equations x =x(t) y =y(t) and z = z(t)

2. We have to determine the surface or surfaces the trajectory is restricted to.
We'll see several egs to see how this is done

Example 1 --- A Helix

Here we let x = cos(t) y -sin(t) and z = t

1 We see that x² + y² =1 so the trajectory is on the surface of a vertical cylinder

2. The inititial point is (1,0,0)

3. x'(t) =-sin(t) y' (t) = cos(t) and z'(t) = 1

therefore the trajectory is smooth and initially the particle moves back, to the right and up.

In other words the particle travels counterclockwise with a constant rate of increase in the +z direction

Helix z increasing

Example 2 -- Another Helix

Let x(t) =cos(t) and y(t) =sin(t) as before but Let z(t) = e ^-t .

1. The initial pt is now (1,0,1)

2. z '(t) = - e ^-t

Therefore the particle winds down the cylinder approaching circular motion in the x-y plane

Helix z decreasing

Example 3 Motion on a Parabaloid

Let x(t) = t y(t)= t z(t) = 1-2t² .

1 This is a little tricky but notice z = 1 - x² - y² the eqn of a parabaloid with vertex (0,0,1) opening down. Futher y = x which is a vertical plane; therefore the trajectory is on the curve of
intersection of the parabaloid and and the vertical plane

2. Initial pt (0.0.1)

3 x' = 1 y' =1 z' = -4t
Therefore the motion is forward, right, and down

Motion on a Parabaloid

Inducing a Parameterization

Of course we want to be able to create parametric equations, not just determine a trajectory from given eqns.

Suppose we think of z = f(x,y) as the temperature at any pt in the x-y plane.

In our previous example we can think of the trajectory as the temperature as we move along the line y =x in the xyplane.

Suppose we choose any path to travel in the plane say along the curve y = x² .

Then we choose x = t and y = t² . since we know z = 1 - x² - y² we then define
z(t) = 1 - t² - t⁴ as our 3d eqn.

This is called inducing a parameterization.
See the animation below

Motion on a Parabaloid 2

Motion on a saddle

Suppose we have the saddle z = 1 - x² +y² .

For our path in the plane le's take the circle x = cos(t) y = sin(t)

Then the induced parameterization for z is z = 1 - cos²(t) + sin²(t) = 1 - cos(2t)

View below

Motion on a Saddle

The following Animation shows if you move along a level curve f does not change

Motion on a Level Curve

In the following Animation and Notes we solve the problem of dropping a rock off

of a spinning merry-go-round.

Animation- Merry-go-Round

Notes- Solution to Merry-go-Round Problem

The following notes and animation deals with projectile motion in 3 dimensions.

The example used is a rocket fired from ground level.

Notes- Projectile motion in 3-D

Animation - 3D Projectile Motion

There is an intimate relationship between parametric equations and vector valued functions. You may want to see the Vevtor value page at this point.

For a discussion of the parameterization of lines in 2 and 3spce go to the vector valued function page

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The Smart Bunny-A very short story by Kurt Vonnegut Jr.

Send an email regarding this site to Connie Liakos Evers, Kelly's sister (no math questions, please).