Parametric Curves

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Parametric Equations

Video introduction

Example: come up with parametric equations that generate any sort of curve (but not one from the video or lab). Plot the curve with Matlab to make sure it looks like you expect.

Relevant ideas or questions from the video

• Parametric equations can describe curves that aren’t functions
• Write x and y in terms of new variable t
• Plug in t values to get x and y values, then plot and connect the dots
• The initial point is at t = 0 (or whatever other t value you choose to start the curve at)
• You can sometimes eliminate the parameter — but since Matlab plotting is fundamentally parametric, this may not be that useful for Matlab.

Equations: x = sin(2t), y = sin(3t)

Code to plot this curve:

%% Plot x(t) = sin(2t), y(t) = sin(3t)
t = [ -2*pi : pi/50 : 2*pi ];
x = sin( 2 * t );
y = sin( 3 * t );
plot( x, y )

Here’s what it looks like:



If you’re interested, you can also plot curves in 3-dimensional space, using the plot3 function, like this:

%% Demonstrate a 3D curve
t = linspace( -2*pi, 2*pi, 6000 );
x = sin( 8 * t );
y = 4 * cos( 8 * t );
z = tan( 8 * t );
plot3( x, y, z )

Parametrics and Explicit Functions: express the function y = x² over the interval -2 ≤ x ≤ 2 as a pair of parametric equations, and try plotting them with Matlab.

The basic idea for parameterizing any explicit function: let the independent variable equal t, then replace occurrences of that variable with t in the equation for the dependent variable. For the above example, this would be

• x(t) = t
• y(t) = x² = t²

Now t ranges over the same values x would have, i.e., -2 ≤ t ≤ 2.

Start Plotting Lab

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