Skip to main content

Level Curve

If hikers walk along rugged trails, they might use a topographical map showing how steep the trails change. A topographical map contains curved lines, which are contour lines. Each contour line corresponds to the points on the map that have equal elevation. A level curve of a function of two variables\(f(x,y)\) is completely analogous to a contour line on a topographical map.
In mathematical, the contour line is known as level curves. A level curve of a function \(f(x,y)\) is the curve of points \((x,y)\) where \(f(x,y)\) is some constant value. A level curve is simply a cross section of the graph of \(z=f(x,y)\) taken at a constant value, say \(z=c\). A function has many level curves, as one obtains a different level curve for each value of \(c\) in the range of \(f(x,y)\). We can plot the level curves for different constants \(c\) together in a level curve plot, which is sometimes called a contour plot or contour line.

Let's work on a example to explore the curve line. Returning the function \(f(x,y)=x^2+y^2\). For some constant c, the level curve f(x,y)=c is the graph of \(c=x^2+y^2\). As long as \(c > 0\), this graph is a circle, as one can rewrite the equation for the level curve as\[\frac{x^2}{c}+\frac{y^2}{c} = 1\] For example, if \(c=1\), the level curve is the graph of \(x^2+y^2=1\). In the level curve plot of \(f(x,y)\) shown below, the smallest ellipse in the center is when \(c=1\). Working outward, the level curves are for \(c=2,3,\dots,6\) and bounded by \(R=[-2,2]\times[-2,2]\).
If we show this function in 3-Dimension, we will have an elliptic paraboloid, and the level curve will vary across this elliptic paraboloid. We can see the 3 D model in the following graph.
We can define the level curve:
If \(f(x,y)\) is a function of two variables \(x\) and \(y\), then the curve in an \((x,y)\)-coordinate system with points such that the \(x-\) and \(y-\) coordinate satisfy the equation\(f(x,y)=c\) is called the level curve with a (function) value equal to \(c\).

Notice that:
1.Level curves of a function of two variables can be drawn in an \((x,y)\)-coordinate system; the graph itself is drawn in an \((x,y,z)\)-coordinate system.
2. Level curves of the same function with different values cannot intersect.

Fig 1.Staff, POB Editorial. “A Construction Stakeout at Devils Tower National Monument.” Point of Beginning RSS, Point of Beginning, 12 Sept. 2020, https://www.pobonline.com/articles/102177-a-construction-stakeout-at-devils-tower-national-monument.
Fig 2. “Geology and Physiography of Devils Tower.” Pine Ridge, October 7, 2021. https://serc.carleton.edu/research_education/nativelands/pineridge/geology3.html. Staff, POB Editorial. “A Construction Stakeout at Devils Tower National Monument.” Point of Beginning RSS. Point of Beginning, September 12, 2020. https://www.pobonline.com/articles/102177-a-construction-stakeout-at-devils-tower-national-monument.

Author: Yueqi Li (Nicole)

Comments

Post a Comment

Popular posts from this blog

Do Over: Integration Over a Region in a Plane

Throughout the semester we have covered a variety of topics and how their mathematical orientation applies to real world scenarios. One topic we discussed, and I would like to revisit, is integration over a region in a plane which involves calculating a double integral. Integrating functions of two variables allows us to calculate the volume under the function in a 3D space. You can see a more in depth description and my previous example in my blog post, https://ukyma391.blogspot.com/2021/09/integration-for-over-regions-in-plane_27.html . I want to revisit this topic because in my previous attempt my volume calculations were incorrect, and my print lacked structural stability. I believed this print and calculation was the topic I could most improve on and wanted to give it another chance. What needed Improvement? The function used previously was f(x) = cos(xy) bounded on [-3,3] x [-1,3]. After solving for the estimated and actual volume, it was difficult to represent in a print...
Post a Comment
Read more

Minimal Surfaces

Minimum surfaces can be described in many equivalent ways. Today, we are going to focus on minimum surfaces by defining it using curvature. A surface is a minimum surface if and only if the mean curvature at every point is zero. This means that every point on the surface is a saddle point with equal and opposite curvature allowing the smallest surface area possible to form. Curvature helps define a minimal surface by looking at the normal vector. For a surface in R 3 , there is a tangent plane at each point. At each point in the surface, there is a normal vector perpendicular to the tangent plane. Then, we can intersect any plane that contains the normal vector with the surface to get a curve. Therefore, the mean curvature of a surface is defined by the following equation. Where theta is an angle from a starting plane that contains the normal vector. For this week’s project, we will be demonstrating minimum surfaces with a frame and soap bubbles! How It Works Minimum surfac...
Post a Comment
Read more

Do Over: Ruled Surfaces

Why to choose this project to repeat For the do over project, I would like to choose the ruled surfaces. I don't think my last project was creative, and the 3D printed effect was not very satisfactory. In the previous attempts, all the lines are connected between a straight line and a circle. This connection structure is relatively uncomplicated. The printed model has too many lines, resulting in too dense line arrangement. The gaps between lines are too small, and the final effect is that all the lines are connected into a curved surface, which is far from the effect I expected. What to be improved In this do over project, I would like to improve in two aspects. Firstly, a different ruled surface is chosen. In the previous model, one curve is a unit circle on the \(x-y\) plane, and the ruled surface is a right circular conoid. In this do over project, it is replaced by two border lines. Each borderline is in the shape of an isosceles right triangl...
Post a Comment
Read more