Level Curves

 

How to think about parameterized surfaces and level curves.
When you are working with three dimensions you have an x, y, and z-plane. The z-plane coordinates of functions are found by an equation containing both x's and y's that together form a number or coordinate to be plotted in the z-plane. In the 2-D plane, you would see a function like \(y=x\) or \(x=y\) that gives you points that form lines on the 2-D plane from a given relationship between x and y coordinates. In the 3-D plane, we would see a function that looks something like \(f(x,y)=x^3y+2x\) that assigns an x and y value into the f(x,y) function giving you a z value. For example, \(f(x,y)=x^3y+2x\) assigns to (x=2,y=3) the number \(2^3*3+4=f(x,y)=z=28\) .
The set \({(x,y)|f(x,y)=c=const}\) is called a contour curve or level curve of f. If you draw several contour curves or level curves at different constants, \({f(x,y)=c}\) you produce a contour map of the function f. This contour map made up of level curves at differnt constant values is really cool because it tells us the behavior along the shape or function we are looking at along those constant lines. We can use this to determine elevation, dips, drops, ridges, smootheness, behavior and more of shapes, functions, models, and even land! How smoothe are the varying heights and changes of the the airfoil of a plane? Simply produce a 3-D model of it, and overlay contour lines and you will see! How aerodynamic is a car? Map the paths of air going over it, treat sections of streams as a level curve, and create a controur map to see! Using level curves tells us lots of information about the behavior of the lines and coordinates that make up a function or a model. They are most often used to see the gradient or slope of the shape by looking at the steepness between constants/level curves as I will discuss later on.
Why I chose the surfaces and parametrization I did. I chose the function \[f(x,y)=x^2+y^2\] .
wolframalpha 3-D graph

openscad 3-D model (this is a 1:1units scale model, z[0,3]units, x[-1,1]units, y[-1,1]units)

I chose this function because I wanted to see the equation of a circle in 3-D as opposed to the usual 2-D. I also chose this function because it is somewhat easier in my opnion than other functions to see the comparison of level curves drawn using arbitrary constants in 2-D and seeing how the change in their spaces relates to height in 3-D. Some functions are really cool to look at and have an interesting contour map but it is hard to explain how to see it in 2-D using contour lines or how to see the similarities in the spacing between level curves in 2-D into height/depth in 3-D. This example looks almost like a foldable cup or a foldable bowl in 2-D and thus easier to compare to a 3-D model before even seeing it in 3-D. You can make any level curves you want on a function by replacing \(f(x,y)\) with a constant. The result will show you how a line or level curve behaves at that point in 3-D.
This desmos graph shows 8 level curves just like my 3-D model from openscad has. They are equally spaced out so we can parametize the function and see how the level curves act at constant intervals giving us more information about what's going on. For this desmos graph example I simply divided 1 by 8 to get eight level curves. Replacing f(x,y) with k values k=0,0.125,0.25,0.375...1. I get eight level curves at evenly spaced intervals. We are essentially seeing how a curve/line behaves in the z-plane along a k-surface. Choosing evenly spaces intervals in a given parameter gives us a parameterization that helps us identify what's going on between the level curves on a function/model. Below we see 8 level curves, we are seeing how many curves and what the curves look like when we have radius of 0 all the way to a radius of 1 as that is what we have modeled on wolfram and openscad.

This tells us how dramatic the height difference will be between that level curve and the next by how far away they are from one another. The farther away the lines are, the more gradual the difference in height is. This is because it has more space to cover that distance in height and results in a more gentle height acceleration. The closer the lines are from one another, the steeper the slope is, or the faster it is for a change in height between two lines in a shorter space. This is easy to see when we notice that the constants are increasing by the same amount each time, and looking at how the distance covered in the xy direction is decreasing as the z direction is increasing the same ammount.

We see from the graph on wolframalpha and desmos that the circles get closer together as we choose higher numbers as constants. That means as we gain height, or add height in the z-plane, we are getting a more steeper slope and the level curves represent a steeper climb. This is compared to constants closer to 0 or the base height, which represent slower climbs in eleveation and change in height for the start of the shape.