Skip to main content

A Very Fancy Basket

I don't know if any of you are avid basket weavers (disclaimer: I am not), but the thought of ruled surfaces made me think of the process in which basket weavers weave. The beginning steps in basket weaving often creates a ruled surface. Say the basket is going to be a round basket, the weaver will create two shapes, the base shape and the shape of the opening. From there, the weaver will take single strands of wicker or straw or whatever the basket is made out of and they will connect the base to the opening as tightly as possible. They will continue this process until the strands have made it all the way around. See picture for reference.
Imagine the strands are perfectly straight lines that go through both the top and bottom functions. This is essentially a ruled surface. A ruled surface occurs when a straight line can be drawn from one point on the surface to another while staying wholly on the surface. While the beginning stages of basket weaving create a ruled surface "esk" look, the completed basket is not a ruled surface because of the curved lines that fill in the gaps. For my surface, I chose to stick with the idea of a basket, but I wanted it to be a more complex basket than the one pictured earlier, so chose to edit my top and bottom functions from a simple circle. The parameterized functions that I chose to use are as follows: \[ f(t) = [10cos(t),10sin(t),10cos(2t)] \] \[ g(t) = [cos(t+80),5*sin(t+80),20] \] For my \(f(t)\) function, I took a circle of radius 10 in the xy-plane and translated it at the height of \(cos(2t)\), giving it a waved circle look as shown below.
For my \(g(t)\) function, I took an ellipse of radius 5 in the y-direction and radius 1 in the x-direction, in the xy-plane and translated it up to a height of 20 in the z-direction. Then, in order to give it a twist (haha), we rotated the surface by 80 degrees in the xy-plane, giving it a rotated oval shape as shown below.
Now, in order to create our ruled surface, we must draw lines connecting our two functions. While technically our ruled surface could extend on forever, we want to start weaving a basket and if the basket is going to fit in a home, we'll have to restrict our connecting lines to just our two functions and not extend them. By doing this, we can create the following ruled surface:
This lowkey makes our basket look like a seashell, so we'll want to view it from the opposite direction and add some thicker lines to not only make it look more like a basket, but to also make it printable. Now, it looks more like a cool basket.
While there are still some gaps between my lines, I think it looks cooler like this so I am going to attempt this first and if it fails, I will thicken my connecting lines until the print is successful. One other thing to note is how curved the surface looks even though it is made up of straight lines. This is largely due to the phase choice, the rotation of our \(g(t)\) function in the xy-plane. We chose a phase of 80, but had we chosen a much larger phase, our lines would have intersected, and it would not have been pretty.

If our initial print is successful, our object's dimensions should be about 47mm by 47mm by 69mm and we should be able to print with minimal supports (stay tuned).

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