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

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