Parametric Curves

Above we showed how Mathematica plots the parametric circle

[Graphics:../Images/index_gr_69.gif]
[Graphics:../Images/index_gr_70.gif]

Many interesting curves can be built from vector combinations of circular and linear motion.  Seeing the vectors move helps understand the parametrizations.  The following links open a number of animated parameterizations.

Circle

Gears

Cycloid

Epicycloid

Helix

Hypocycloid

Skewed Circle

Unwinding thread from a Spool

Curve around the torus
 

Position, Velocity, and Acceleration

Parametric curves can represent motion in space.  We might have a position measured by a vector [Graphics:../Images/index_gr_72.gif]  at time [Graphics:../Images/index_gr_73.gif].  As time varies the position changes.  In this case, the derivative of the position is the velocity of the moving particle and the derivative of  velocity (or second derivative of position) is acceleration.  A curve with position, velocity, and acceleration at one time are shown in the next figure.  Mathematica can animate this idea and show moving position, velocity, and acceleration changing with time as in the link below.  There are several important links between geometry, motion, and the product and chain rules of calculus related to this motion.

[Graphics:../Images/index_gr_74.gif],[Graphics:../Images/index_gr_75.gif], [Graphics:../Images/index_gr_76.gif]

[Graphics:../Images/index_gr_77.gif]

Load the Animation 1  Position, Velocity, & Acceleration
Load the Animation 2  A moving Frenet Frame with the Derivative Vectors tugging on the Frame
Converted by Mathematica      May 7, 2001