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 xy
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 = x^{2} .
Then we choose x = t and y = t^{2} . since we know z = 1  x^{2}  y^{2}
we then define z(t) = 1  t^{2}  t^{4 } 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  x^{2} +y^{2} .
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  cos^{2}(t) + sin^{2}(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 merrygoround.
Animation MerrygoRound
Notes Solution to MerrygoRound 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 3D
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 BunnyA very short story by Kurt Vonnegut Jr.
©20082010
Kelly Liakos
