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Level Curves

    Have you ever driven in a mountain range and wondered how you would make it to the top? A car can’t drive straight up the mountain, it would be too steep. Instead, they must slowly progress up the mountain by driving around the mountain at a flat rate, gain some elevation slowly, then drive on a constant elevation around again, and continue this process until they’ve reached the top. 

    Or have you ever noticed the water level changes in a man-made lake? If the lake is beginning to flood, they may open the dam to release some of the water. After this you would be able to see a line on the shore or sea wall where the previous water level was compared to the current water level. If this were to continue to happen until the lake was drained completely, you would be able to see different rings around the lake representing the different levels of water. 

    The different sections of flat roads at constant elevation on a mountain and the marks from the different water levels both represent level curves. Level curves represent points at a constant elevation and how they progress in the z direction. 

    Level curves are often used to make a topographic map. They take features of the earth at different elevations and project it onto a 2D map. Each constant change of elevation on a mountain (where you would drive a car on) would project downward on to map as a circle. As the mountain increases in height, the level curve decreases in radius representing the different elevations. Same for the lake, each water level (the different rings) would project onto a graph as a level curve by decreasing the radius as it gets closer to the bottom of the lake. After doing this for each critical point, we will see a level curve graph form.

An Example

For my example, I wanted to choose an object that would produce a visual that follows along with the mountain and lakes example because level curves are most commonly used in topographic maps. Also I’ve grown up traveling to different mountain and lake destinations, so I felt this was very applicable to my own life and would be easy to follow along with for others.

So let’s look at the function

 

This function is going to form a 3D curve in each quadrant of a graph; therefore, we will be able to see level curves in each quadrant. In quadrant 1 and 3, there will be a curve in a positive z direction and in the negative direction for quadrants 2 and 4.  

These will represent the mountains (positive curves) and lakes (negative curves) referenced in the beginning. Therefore, to visualize these level curves, we think about where a car could drive on this curve and where each water level could be seen at and mark these as level curves. Then transferring to a 2D graph to look similar to this.

We can also calculate where these level curves could be.

First chose a value for the z direction, then we will determine in terms of y the function to use at that given height. 

For example, the z-value will be 1. 

 

If we continue this with other z values, we will see the level curves appear.  

We then set the entire equation equal to that z value and solve for a function to graph. Then graph the level curve function.

Something important to notice and include for this function is that eventually all these points will meet at the origin and flatten out which is why we also see a vertical and horizontal line at x = 0 and y = 0.

Now that we have the level curves, we can see how this object will look in 3D. The following model represents f(x,y) with 10 level curves traced on the object. (The circular rings around the curves are the level curves)



The domain of the model is [-2,2] x [ -2, 2] and will be 4in x 4in. 

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