Graphing Basic Polar Equations

Key Questions

  • What are limacons and cardioids?

    Limacons are polar functions of the type:
    r=a+-bcos(theta)
    r=a+-bsin(theta)
    With |a/b|<1 or 1<|a/b|<2 or |a/b|>=2

    Consider, for example: r=2+3cos(theta)
    Graphically:
    enter image source here

    Cardioids are polar functions of the type:
    r=a+-bcos(theta)
    r=a+-bsin(theta)
    But with |a/b|=1

    Consider, for example: r=2+2cos(theta)
    Graphically:
    enter image source here

    in both cases:
    0<=theta<=2pi

    .....................................................................................................................
    I used Excel to plot the graphs and in both cases to obtain the values in the x and y columns you must remember the relationship between polar (first two columns) and rectangular (second two columns) coordinates:

    enter image source here

    Gió · 1 · Dec 28 2014
  • What is the general form of limacons and cardioids and how do you graph transformations?

    You can find a lot of information and easy explained stuff in "K. A. Stroud - Engineering Mathematics. MacMillan, p. 539, 1970", such as:
    K. A. Stroud - Engineering Mathematics. MacMillan
    enter image source here

    If you want to plot them in Cartesian coordinates remember the transformation:
    x=rcos(theta)
    y=rsin(theta)

    For example:
    in the first one: r=asin(theta) choose various values of the angle theta evaluate the corresponding r and plug them into the transformation equations for x and y. Try it with a program such as Excel... it is fun!!!

    Gió · 2 · Mar 14 2015
  • How do you graph basic polar equations?

    You consider a function of the type:
    r=f(theta)

    So you give values of the angle theta and the function gives you values of r.

    To graph polar functions you have to find points that lie at a distance r from the origin and form (the segment r) an angle theta with the x axis.
    enter image source here
    Take for example the polar function:
    r=3

    This function describes points that for every angle theta lie at a distance of 3 from the origin!!!

    Graphically:
    enter image source here
    The result is a circle of radius r=3.

    Now, the only complication is when r becomes NEGATIVE ...how do I plot this?
    We use a trick....we take the positive and flip it about the origin!!!!!!
    enter image source here
    Take for example the polar function:
    r=-3

    This function describes points that for every angle theta lie at a distance of...-3 from the origin????
    We use our trick!

    Graphically:
    enter image source here
    Every point of the old graph flipped about the origin!!!!
    It is a circle...again!!!!

    Now try by yourself with:
    r=2cos(theta)
    Build a table of theta and r and plot it...you should get another circle but with its center....on the x axis (in (1,0)) and radius =1.

    There are more complicated (and graphically beautiful) polar functions such as limacons, cardioids, roses, lemniscates, etc…try them!!!

    Gió · 1 · Dec 28 2014

Questions

The Polar System