Project 9: Parametric & Polar Equations

Instructions

Use Maxima to work this project. Each group should email an annotated Maxima file to the instructor.

Parametric Artwork

Design a graph that can be broken into multiple parametric curves and then use Maxima to draw it. Look at problems 9.1.29-30 for examples. Use only parametric curves in your graph.

You may that drawing the graph on grid paper and then finding the equation of each curve will speed up the process immensely. You may also wish to play around in Winplot first and then transfer the graph over to Maxima.

Here is an example graph. You do not need to including lettering.

coffee:makelist(parametric(r*cos(t),1.5+0.25*r*sin(t),t,0,2*%pi),r,0,1.9,0.1)$draw2d( xrange = [-2,2], yrange = [-2,2], dimensions = [600,600], xaxis = false, xtics = false, yaxis = false, ytics = false, axis_left = false, axis_right = false, axis_bottom = false, axis_top = false, line_width = 3, parametric(2*cos(t),1.5+0.5*sin(t),t,0,2*%pi), parametric(1.5*cos(t),-1.5+0.325*sin(t),t,%pi,2*%pi), parametric(1.5+0.5*t,-1.5+3*t,t,0,1), parametric(-1.5-0.5*t,-1.5+3*t,t,0,1), color = brown, line_width = 1, coffee, color = red, line_width = 10, parametric(-1+0.4*cos(t),-0.2+sin(t),t,%pi/3,5*%pi/3), parametric(0.4*cos(t),-0.4+sin(t),t,%pi,2*%pi), parametric(-0.4,-0.4+t,t,0,1), parametric(0.4,-0.4+t,t,0,1), parametric(0.8,-1.2+2*t,t,0,1), parametric(1+0.4*cos(t),0.4+0.4*sin(t),t,-%pi/2,%pi/2), parametric(0.8+0.2*t,0,t,0,1), parametric(0.8+0.2*t,0.8,t,0,1) )$

Polar Graphs

For each of the following polar graphs, draw the curve, find the length of the curve, and find the area of the region enclosed by the curve.

1. The odd petaled rose $$r = 5 \cos 3\theta$$
2. The even petaled rose $$r = 5 \sin 4\theta$$
3. The cardiod $$r = 2 + 2 \cos \theta$$