Samantha Driskill 

Footnote 18 

5/7/08 

Families of Surfaces Part III - click here for the Maple 11 Version

Prompt : Members of the family of surfaces given in spherical coordinates by the equation have been suggested as models for tumors and have been called bumpy spheres and wrinkled spheres.  Use a computer to investigate this family of surfaces, assuming that m and n are positive integers.  What roles do the values of m and n play in the shape of the surface? 

Respone: In order to figure out what roles "m" and "n" play in determining the shape of the surface, I have to have a base function to compare different variable changes to.  So the first values of "m" and "n" that I am graphing are m = 1 and n=1.  Below is the resulting graph which resembles a slightly lopsided sphere: 

Now that I have my base function to compare all other changes in the value of "m" and "n" that I make with, I can begin to manipulate some values.  The first thing I decide to change is the value of "m" to see what sort of affect that change will have on the graph.  The resulting effect was that the sphere now seems to be made up of two bumps, much resembling two balls sqwished together and wrapped in tape. Below is the graph were m = 2 and n = 1: 

To continue to determine the exact affect "m" has on the spherical surface, I increased the value of "m" by one so that m = 3 and n =1.  The resulting effect was that now the surface was made up of three bumps, again like three balls had been sqwished and taped together. Below is the resulting graph: 

After graphing two variable changes and noting on the changes of the surface, I can theorize that as I increase the values of "m" and keep n =1, the surface will contain "m" number of bumps.  To check my theory, I graphed values of m = 4 and 5 while n = 1.  Below are the resulting graphs: 

Now that I have determined the affect of "m" onto the spherical surface, I need to determine the affect "n" has on the surface.  Therefore, I kept the value of "m" constant and equal to 1 and changed the values of "n" so that m = 1 and n = 2.  Below is the resulting graph which looks like the top and bottom of the lopsided sphere have been pulled in opposite directions, forming a kind of wrinkle effect: 

To continue on my search in determining how the value "n" affects the shape of the surface, I kept m =1 and increased "n" by one so that n = 3, 4, 5, 6, and 7.  In comparing these graphs to each other and to my original graph, I can say that as the numerical value of "n" increase, there will be "n" numbers of "wrinkles" in the spherical surface. Below is the resulting graph where m = 1 and n = 3, 4, 5, 6, and 7. 

In conclusion, I have determined that the value "m" changes the spherical surface to contain "m" number of bumps while the value "n" changes the spherical surface to contain "n" number of wrinkles.  However, to see how increasing both values of "m" and "n" affects the spherical surface, I have included some graphs with random values of "m" and "n".  Below are the resulting graphs which include "m" number of bumps and "n" number of wrinkles: