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Parametric Surfaces & Level Curves : Elevations of Horsetooth Rock

Parametric Surfaces & Level Curves : Elevations of Horsetooth Rock

Parametric surfaces are surfaces that exist in \( ℝ^3 \), or in 3D, which is defined by two parameters in \( ℝ^2 \). Thus we have a 3D surface that is given by a function of two parameters, usually \(x\) and \(y\), that gives us a \(z\)—typically thought of as the vertical direction, or the one that "comes towards you." This can be written in a somewhat familiar form(s) as:

\[ f(x,y)=z, \\ f(x,z)=y, \\ f(y,z)=x.\]

Now that we have established the basics of what a parametric equation is we can move on to a discussion of level curves. A level curve of a two variable function \(f\) is a curve of the form \( f(x,y)=k\), where \(k\) is a constant in the range of \(f\). So, what we are really doing is setting \(k\)—which is a coordinate of a point in relation to the \(z\)-axis—to a constant "height" such that we can draw a curve on our surface that essentially represents/demarcates where that height occurs on our surface.

You have probably seen level curves before on topographic maps(an example is pictured below). These maps show natural features and terrain, like most non-political maps do, but also show topographic lines. These lines show elevation, which is useful for determining the elevation of a location, or the slope and subsequent "steepness" of a location—whether it be a building site, hike route, etc..

The example topographic map above shows Horsetooth Mountain or Horsetooth Rock and includes topographic lines. Particularly, you can see that elevations of 6,566 feet and 7,255 feet are marked clearly with dark lines. What these lines represent are constant elevations, this is the same as setting \(k\) as a constant, they are level curves of a parametric equation. Furthermore, it is possible—though difficult—to create a parametric equation that actually maps the surface of surface of Horsetooth Rock and then we could calculate the level curves for those elevations.

Horsetooth Rock is an actual natural feature located in Larimer County, Colorado, and is contained within a state park and municipal open space. There is also a trail to the top that is approximately 5 miles long and has a change in elevation of about 1,500 feet. Over the summer I hiked this trail and as such Horsetooth Rock was my inspiration of starting point in developing a parametric equation and some level curves. Horsetooth Rock pictured below along with a link to Larimer County's website:

https://www.larimer.org/naturalresources/parks/horsetooth-mountain

As previously stated, constructing an equation that would exactly recreate the surface of this natural landmark would be quite difficult so, and approximation of the surface of the actual rock itself has been constructed. That equation is shown below: \[ f(x,y) = -0.02(x^4-4x^2)-2y^2+5\] Which maps a surface that is shown in the following image:

To plot our level curves we can solve this equation, using a constant \(k\) as our elevation, and solve our equation for \(y\)—changing it into slope-intercept form, that will allow us to plot some level curves—that equation is shown below: \[ y= \sqrt{ \frac{5-k-0.02(x^4-4x^2)}{2}} \] and will plot curves that resemble the following image when viewed from above.

We can combine our 3D graph of our approximation of Horsetooth Rock with our level curves plot to give a surface with level curves that looks like the following

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