Wavy Skirt Surface by khorstm

Last crawled date: 3 years, 3 months ago

Kim Horstman
George Mason University Math 401: Mathematics through 3D Printing
11/8/2020
In multivariable calculus, we learned about partial derivatives and their applications. The most notable of those was the use to locate critical points for a two-variable function. A critical point is a point in the domain of the function where the function is either not differentiable or the derivative is equal to zero. The second derivative test is used to identify a critical point as a local maximum, local minimum, or saddle point. A saddle point is a point on the surface of the graph of a function where the slopes/derivatives in orthogonal directions are all zero (critical point), but which is not a local extremum of the function. The name comes from the fact that the example in two dimensions is a surface that curves up in one direction, and curves down in a different direction, resembling a saddle or a mountain pass between two peaks.
There are many types of saddles ranging from the simple saddle to the monkey saddle, starfish, and octopus saddles (names given from the animals they resemble). These saddles often occur in real topography. Traversing a drainage divide involves repeatedly moving from a peak down to a saddle point or valley and then up to another peak. A variant of the starfish saddle, called the wavy skirt saddle, is when the center point is a peak, not a saddle point.
The equation for this surface is f(r, t) = a r[cos(n t) - b] where the function has linear r-dependence and a shifted function of t, while n determines the number of waves the skirt has. I did my project of order 6.
A real topographical example of this type of saddle is the west peak of Mount Sopris in Colorado. An examination of the peak shows that elevations decrease linearly from the peak along any of the three ridges or three valleys that meet at the peak. The modified saddle surface called a wavy skirt surface was introduced in order to capture these features.