Skip to main content

Parameterized Surface with Level Curves

Parameterized surfaces and level curves

Let's consider a continuous surface \(z=f(x, y)\) defined for points \((x, y)\) in a domain \(D\) in the \(x-y\) plane, we can use level curves to visualize the function of two variables \(f(x, y)\) without leaving the plane. The level curves can be regarded as an implicit description of a surface.
For a given value of \(z=k\) in the value domain of \(z=f(x, y)\), the corresponding level curve is simply the cross section of the graph of \(z=f(x,y)\) and the plane \(z=k\). The equation of the corresponding level curve is defined as follows: \[f(x, y) -k =0\] We can select a series of suitable values of k in the value domain of \(f(x,y)\) for the equation of the level curve, then obtain a series of level curves to describe the surface.

The surface to be analyzed

The function of parameterized surface that we chose is defined as \[z=f(x,y)\] \[=-2x^2+2y^2 +(x^2+y^2)^2\] \[-1.5 \le x \le 1.5, -0.6 \le y \le 0.6\] The surface defined by the function is shown in the following picture:
We can choose 10 equally spaced values \(z=f(x,y)=k\), as \(k=-0.8, -0.4, 0, 0.4, 0.8, 1.2, 1.6, 2.0, 2.4, 2.8\), and get a series of level curves shown in the following picture:
For the value \(k=0\), we can get a special level curve which is called the lemniscate of Bernoulli. The function of the lemniscate of Bernoulli in the Cartesian coordinates can be defined as \[-2x^2+2y^2 +(x^2+y^2)^2 = 0\]
For the 3D print model, we add a base at the bottom to make the print process stable.

Why to use this example

This example of parameterized surface with level curves was chosen for the following reasons:
The function of the surface has a concise form, so that we can focus on the concept of visualizing the surface by level curves. For each \(k \lt 0\), the level curves for \(f(x,y)=k\) are two closed circle-like curves. For \(k = 0\), the level curve for \(f(x,y)=0\) is the lemniscate of Bernoulli. For each \(k \gt 0\), the level curve for \(f(x,y)=k\) is actually a closed curve. However, due to the constraints in the domain of the function \(f(x,y)\), \(-1.5 \le x \le 1.5, -0.6 \le y \le 0.6\), each level curve might be broken into four parts.
The parameterized surface can be better understand by using polar coordinates. For \(k = 0\), we get the lemniscate of Bernoulli as the level curve. Its function in polar coordinates can be defined as \[-2\rho^2cos(2\theta)+\rho^4=0\] \[-\pi \le \theta \le \pi\] The function of the lemniscate of Bernoulli in the polar coordinated has a more concise form than that in the Cartesian coordinates.

In geometry, the lemniscate of Bernoulli is an \(x-y\) plane curve defined from two given points \(F_1\) and \(F_2\), known as foci, at distance \(2c\) from each other as the locus of points. For a given point \(P\) on the curve, the distance between \(P\) and \(F_1\) is \(PF_1\), and the distance between \(P\) and \(F_2\) is \(PF_2\). We have \(PF_1PF_2 = c^2\).

This lemniscate was first described in 1694 by Jacob Bernoulli as a modification of an ellipse. The curve of the lemniscate of Bernoulli has a shape similar to the numeral 8 and to the ∞ symbol. The mechanism behind the lemniscate of Bernoulli can be applied to machines in the textile industry, to mechanical toys, etc.

In the above example, the function we chose made a surface with different kinds of level curves, and can be defined in different coordinate systems. This helps us to better understand the concept of parameterized surfaces and level curves.
Labels:

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

Knot 9-31

Knot 9-31 A knot is mathematics is defined as a closed, non-self-intersecting curve that is placed in three dimensions and cannot be the "unknot". The main difference between a knot in the real world and a known in mathematics is that a knot in mathematics does not contain any extra strands. The example following will help visualize this. Today, I have specifically chosen knot 9-31 from the Knot Atlas. This knot is very unique and contains some very interesting properties that we are going to look into. The Crossing Number For my knot today, I chose knot 9-31 from the Knot Atlas. This knot contains 9 crossings! The Unknotting Number The unknotting number is exactly what is sounds like. This is the minimum number of times the knot must be passed through itself to untie it. Luckily, the Knot Atlas is super useful and provides the unknotting number for us, but I still...
Post a Comment
Read more

Do Over: Integration for Over Regions in the Plane

Introduction Earlier in the semester, we visited the topic of integration for over region regions in the plane. This was our first experience with taking integration in the third dimension. To do this, we had to use double integrals to calculate the volume of a region between two surfaces. This topic seems relatively straightforward and anyone who has taken a higher calculus class knows of double integrals. Today, we are going to revisit this topic for a couple of reasons. Why Double Integrals Again? As mentioned before, I am revisiting this topic for a couple of reasons. One of the main reasons is due to how the 3D print turned out. Due to the function, I chose, the final rectangular prism was stand alone. When printing, this resulted in a sloppy prism that lacked structural integrity. The image below is a reference to my original design that includes the prism that had the issue. The main cause of ...
Post a Comment
Read more