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