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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).

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