Skip to main content

Do Over! Strange Ruled surfaces

For this time. We can choose a topic we have done before, and I hope to improve on the original basis. So I chose the one that failed to print the most among all my models. That is, Ruled surfaces. Due to lack of experience, the printing of the model defaulted to be able to show what I really wanted, but it was obvious that the last printed product did not make me happy. After printing many times, I learned that if you want to have a clear line and be successful, the bigger it is, the easier it is to show what I want. So first review the definition of Ruled surfaces: In geometry, if at least one straight line passes at any point on a curved surface, the curved surface is called a ruled surface. Another common saying is that if a curved surface can be formed by a straight line through continuous motion, it can be called a Ruled surfaces.

The last time I chose the combination of closed curve and non-closed curve, then this time I decided to try an even stranger combination: a combination of multiple line segments and curves. So how to realize multiple line segments?

The answer is simple, the tan function. The image of the tan function is divided into segments of curves due to its special properties. This is exactly what I want.
The choice for the interval is from 0° to 360°, which is 0 to 2π. In this case, this curve will have obvious breaks at 90° and 270°, because it approaches positive infinity and negative infinity, respectively. In other words, within the interval, there are three curves (oh yeah). In order to highlight the tan function, I am not adding too many elements, the equation is [tan(x),0,.04*x]. To be honest, I didn’t know what it would look like before it was made.
The second curve is still a straight line temporarily, which allows me to see the first curve clearly. If the second one is just a straight line, I cannot accept it. When I thought that since I used the tan function, the other one would use a spirally ascending curve. That would be great. The equation is [4*sin(x),4*cos(x),.04*x].
From the upper angle, you can see the spiraling and rising curve, which looks good, just like DNA spiraling and rising, there is nothing strange.

But if we look at it from another angle, it looks... easy to break
Because the first curve is divided into three parts, the whole is connected by the second curve, which makes it look a little strange. The three line segments are connected to the second curve respectively. Both the upper and lower parts form a quarter circle, and the longer line segment in the middle and the second curve form a half circle (viewed from above). I like this. Personally, this one is stranger than the last one, and it looks cooler.
Because it seemed untenable, I added a base to it. The size of its final model should be: x: 58.02 y: 50 z: 78.19

Comments

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...

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...

Ruled Surfaces : Trefoil

Ruled Surfaces : Trefoil A ruled surface is a surface that consists straight lines, called rulings, which lie upon the surface. These surfaces are formed of a set of points that are "swept" by a straight line. This is relatively intuitive once you see a good visual, but can be a bit abstract without that concrete example. A very basic example of a ruled surface is a cylinder; if we have a straight line and move it in a circle we create a cylinder made entirely of straight line. Note that the surface will only be a cylinder if all the lines are parallel. If the lines are not parallel we can create hyperboloids and cones depending on how much we have rotated. The rotation we are describing here is not a simple turning action, but more of a twisting motion—less like rotating a can by turning it and more like wringing out a washcloth by twisting it. Specifically, a cylinder is essentially two circles connected by rulings, if we keep one of the circles...