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