Calculus II
Lesson 27: Polar Graphs
___________________________________________________________________________________
A. Cardioids & Limacons
We're going to look at a variety of cardioids, which are graph of the form
y = a +- or y=a +-
and see how the relationship among the components effects the graph. COMPARING a AND b
In particular, there are three cases : |a| = |b|. |a| > |b|, and |a| < |b|. Each of these cases creates a distinctive version of the limacon.
When |a| = |b|, the graph passes through the origin.
This shape is known a cardioid, or heart shaped curve. Note the reference circles of radius 1 and 2.
> restart; with(plots):
Warning, the name changecoords has been redefined
> polarplot( {1,2, 1+sin(theta)}, theta = 0..2*Pi, scaling = constrained);
When |a| = |b|, the graph maintains some distance between it and the origin, resulting in a rounder, puffier plot.
> polarplot({1,3,5, 3+2*sin(theta)},theta = 0..2*Pi, scaling = constrained);
When |a| < |b|, the graph not only passes through the origin, but also part of it folds inside itself.
> polarplot({2,3,8, 3+5*sin(theta)},theta = 0..2*Pi, scaling = constrained);
To see all of these varieties in one glance, execute the next block of commands.
> display( polarplot( 8 + 8*cos(theta) , theta = 0..2*Pi, scaling = constrained, color = green, thickness = 3), polarplot({8 + a*cos(theta) $ a = 9..15}, theta = 0..2*Pi, color = blue), polarplot({ 8 + a*cos(theta) $ a = 1..7}, theta = 0..2*Pi, color = red));
CHOICE OF TRIG FUNCTION
There are four variations iin the format : sine, cosine, -sine, and -cosine. How does the choice of one of these effect the graph? Lets take a look at all four at once!
Can you decide which graph belongs to which? Think about what values of theta make the sine and cosine maxima!
> polarplot({ 8 + 7*sin(theta), 8 + 7*cos(theta), 8 - 7*sin(theta), 8 -7*cos(theta)}, theta = 0..2*Pi, scaling = constrained);
> polarplot( 10 + sin(2*Pi*theta), theta = 0..20*Pi, color = coral, scaling = constrained);
B. The Rose Garden
We're going to look polar functions of the form f = a sin(n ) and r = a cos(n ) which are sometimes called multi-petaled roses.
EVEN AND ODD NUMBER PETALS
The first distinction to be made is between when n is an even or odd number.
When n is an odd number, the resulting rose has exactly n petals
> polarplot( {9, 9*sin(5* theta)}, theta = 0..2*Pi, scaling = constrained);
However, when n is even, the rose has 2n petals.
> polarplot( {5, 5*sin(6*theta)} , theta = 0..2*Pi, scaling = constrained);
Try creating some other roses on your own with different numbers of petals to verify that the even/odd relationship holds.
What about a single-petaled rose?
Do you recognize the inner shaped of the "single petaled rose"?
> polarplot( {9, 9*sin(theta)}, theta = 0..2*Pi, scaling = constrained);
SINE AND COSINE
Although sin(x) and cos(x) will create an n-petaled roses inscribed in the unit circle, what is the difference between them?
The graph with the sine appears tangent to the positive x axis, while the cosine version has a petal centered at the positive x axis.
> polarplot( {sin(3*theta), cos(3*theta)}, theta = 0..2*Pi, scaling = constrained);
Here is an illustration of the same idea with even more petals.
> polarplot({sin(6*theta),cos(6*theta)}, theta = 0..2*Pi, scaling = constrained);
AMPLITUDE
In the formula above, how does the number a, which is the amplitude in effect the graph? Here we let a =1,2,3...,12 and see how the resulting graphs look
Each different color is a different graph. You can see that they are inscribed in circles of radius 1,2,3,...,12.
> polarplot( {a*cos(6*theta) $ a = 1..12}, theta = 0..2*Pi, scaling = constrained);
C. Valentine Curves
Valentine curves - there is really no such name but it seemed reasonable when you take a hybrid of rings, hearts(cardioids), and flowers(roses).
> polarplot( 4 + cos(6*theta) , theta = 0..2*Pi, scaling = constrained);
> polarplot( 4 + 3*sin(7*theta), theta = 0..2*Pi, scaling = constrained);
This one wraps in on itself
> polarplot( 3 + 7*sin(3*theta), theta = 0..2*Pi, scaling = constrained);
Here are whole families of similar curves
> polarplot( { 6 + a*cos(6*theta) $ a = 1..11}, theta = 0..2*Pi, scaling = constrained);
> polarplot( {12 + a*sin(7*theta) $ a = 1..12}, theta = 0..2*Pi, scaling = constrained);
D. Familiar Shapes Disguised In Polar Form
There are many familiar shapes such as lines, circles, parabolas, and ellipses which can be expressed in polar form.
In polar coordinates, the simplest function for r is r = constant, which makes a circle centered at the origin. Lets look at the graphs of r = 1, r = 2, ... , r = 20.
This draws concentric circles of radius 1,2,...,20
> polarplot( {k $ k = 1..20}, theta = 0..2*Pi, scaling = constrained);
We can also draw circles not centered at the origin.
> polarplot( cos(theta), theta = 0..2*Pi, scaling = constrained);
> polarplot( cos(theta - Pi/4), theta = 0..2*Pi, scaling = constrained);
...and ellipses and parabolas....
> polarplot( 1/(8 - 7*cos(theta)), theta = 0..2*Pi, scaling = constrained);
>
> polarplot( 1/(1 - cos(theta)), theta = 0..2*Pi);
> polarplot( 1/(3 + 2*sin(theta)), theta = 0..2*Pi, scaling = constrained);
...even horizontal and vertical lines
> polarplot( 2*csc(theta), theta = -2*Pi..2*Pi);
> polarplot(2*sec(theta), theta = -2*Pi..2*Pi);
E. Spiraling Graphs
A basic spiral is of the form r = theta.
> polarplot(theta,theta = 0..4*Pi, scaling = constrained);
> polarplot(theta, theta = 0..40*Pi, scaling = constrained);
Again, a larger range of values for theta gives more chance for the graph to wrap around.
Even more interesting graphs can be created using the product of theta and a trigonometric function. As theta increases there is some sort of spiraling effect.
> polarplot( theta*sin(theta), theta = 0..3*Pi, scaling = constrained);
> polarplot( theta*sin(theta), theta = 0..100*Pi, scaling = constrained);
As we increase the range of values for theta, we get even more of the same.
Here is another variation.
> polarplot( 2*cos(theta) + sqrt( abs( 4*cos(theta)^2 -3)), theta = 0..2*Pi, scaling = constrained, numpoints = 1000);
F. How To Build A Better Rose
The so-called 'roses' above, really bore more of a resemblance to daisies. Here is something that looks a little more rose-like.
> polarplot( theta + 2*sin(2*Pi*theta), theta = 0..12*Pi,color = red, thickness = 2 );
Here are some other beautiful botanicals.
> polarplot( theta + 3*sin(4*theta) - 5*cos(4*theta), theta = 0..12*Pi,color = violet, thickness = 2 );
> polarplot( theta + 2*sin(2*Pi*theta) + 4*cos(2*Pi*theta), theta = 0..12*Pi,color = green, thickness = 2 , numpoints = 1000);
> polarplot( 2*cos(theta) + sqrt( abs( 4*cos(theta)^2 -3)), theta = 0..2*Pi,scaling = constrained, numpoints = 1000 );
> polarplot( cos(.95*theta), theta = 0..40*Pi,scaling = constrained, color = brown);
