Skip to main content

Unexpected Level Curves

There will always be unexpected things in the exploration of mathematics, and mathematics will never only show in a single direction. We have spent a lot of time learning the graphics and applications of the equations of the plane and lament the changes of the equations on the plane. It is diverse. The three-dimensional equation graphics are even more bizarre. This is what I hope to show today, rather strange three-dimensional graphics, parameterized surfaces and level curves.

Applying mathematics in life is very extensive, and many things can be constructed with mathematical models. This is also the purpose of our learning. It is not only to calculate some simple numbers but to use it more widely. When we study mathematics, we will come into contact with different equations. For some simple equations, we will guess what it looks like, the linear equations in two unknowns, the shape of the parabola, and the radians. When we learn three-dimensional equations, imagination begins to become scarce gradually. For example, the equation: f(x,y)=(x*x-y*y)/(x*x+y*y) I want to show today. Can you imagine what it looks like when you see it for the first time?

If it is described, its shape is like placing a paper towel in the middle, and the thumb and index finger of the two hands clamp it diagonally against each other. Maybe what I described is not very clear, then let's look at the picture.

Does this seem to be unexpected? Its appearance is not so easy to describe. Then we make a small change to the equation, and both x and y in the denominator become half of the original, which is the equation f(x,y)=(x*x-y*y)/(0.5x*0.5x+0.5y*0.5y), in this case, we will enlarge its special part to make its fluctuation higher a little.

Then look at the level curves, we set f(x,y)=k. In this way, the curve of its level can be obtained by the change of k, If we extract x from the equation, then its equation will look like this
The reason why I don’t show the equation for extracting y is because that will produce imaginary number i, which will make the graph too troublesome. So we still use the original equation to construct the graph and convert f(x,y) to an integer, then In this way, a lot of burdens can be reduced, and our result is as shown in the figure below：

Aha, they all seem to be broken or bent close to the origin, so the actual situation is that they can be infinitely close to 0, but they cannot be 0.
Because of its overall height limitation, the value is in the range of -4 to 4 (if it is the original equation, the range is -1 to 1 if the x and y of the denominator are not reduced by half. so I change it).

It seems a bit peculiar, so according to its size, add 9 level curves. Then limit it to a square with a side length of 4 (x=-2, x=2, and y=-2, y=2), and its overall modeling looks like this (after adding level curves).
Only the middle part of the graph constructed by this equation is special. As x and y increase, it also becomes flatter. I want to show this because it’s not all of them. It’s like a curved shape, but there is a very steep special shape in the middle, but the surroundings are relatively smooth. Maybe it’s not very common in life or can be applied to it, but This example is very special.

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