Parametric Equations

Can you tell what the parameterization of the spiral on the left is?   Click the graph to see an animation.

Here is a graph with animated (moving) points showing the four position functions demonstrated in class.  The functions are given below with t going from 0 to 1.  Each point has a different color tangent vector (not velocity vector) attached to it.  See if you can match each color to the functions given below.

xt1 = -2 + 4t,     yt1 = (-2 + 4t)²

xt2 = -2 + 4(sin(pi*t/2))     yt2 = (-2 + 4(sin(pi*t/2)))²

xt3 = -2 + 4(tan(pi*t/2))     yt3 = (-2 + 4(tan(pi*t/2)))²

xt4 = (-2 + 4t)³     yt4 = (-2 + 4t)⁶

Click on each parametric representation of a curve below to see its graph and an animation showing the changing velocity vectors (green) and acceleration vectors (red).  The numbers inside the brackets indicate the t interval for the graph.  The figure below shows the velocity vectors (green) and acceleration vectors (red) at t = -1, t = 0, and t = 1 along the path described parametrically by x = t,  y = 4-t².  Click on the picture to see an animation that will also include the principal unit normal vector N (magenta).

x = t ,   y = t2     [-2,2] x =  t ,   y = t3/3 - t     [-2,2] x = 3cos(t) ,   y = 2sin(t)    [0,2pi] x =  t3 ,   y = 2sin(t)    [-2pi,2pi] x = tsin(t) ,   y = tcos(t)    [0,6pi] x =  t2 ,   y =  2sin(t) + cos(t)     [-2pi,2pi] x = t3 ,   y =  2sin(t) + cos(t)     [-2pi,2pi] x = t3 ,   y =  2sin(t) + cos(t)     [-4pi,4pi]

The examples below are from vector valued position functions in Calculus III.  They will be modified to parametric representations of curves soon.

Below on the left is a simple variation of the butterfly curve developed by Temple H. Fay.  Click on the picture to see a picture with an animated point tracing around the curve.  Quicktime Version  Click here for three animated point butterflies with a black background.  Below on the right is a butterfly constructed from four variations of the Fay butterfly curve.  The vector valued function graphed below on the left is

At the right is a picture of the epicycloid described in example 3 on page 842.  Click on the picture to animate the point.

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        Lane Vosbury, Math Chair, Seminole Community College   email:  vosburyl@scc-fl.edu

        This page was last updated on 10/08/08          Copyright 2002          webstats