Ruled surface: Sombrero

What is a ruled surface? A ruled surface is a surface which can be made up of straight lines, and where for every point on the surface a line can be drawn through that is actually on the surface. How can you make shapes with straight lines? It's very easy actually, consider a stair case, DNA, or a pyramid made from logs. Then consider being able to twist and turn these straight lined shape and imagine the possibilities that could come from that! What is Plücker's conoid you may ask? Plücker's conoid is a ruled surface defined by the function of two variables: \[z=\frac{2xy}{x^2+y^2}\]. its polar parametization results in cylindrical coordinates and are \[x(r,\theta)=rcos\theta\] \[y(r,\theta)=rsin\theta\] \[z(r,\theta)=2cos\theta sin\theta\].
The generic form \[x(r,\theta)=rcos\theta\] \[y(r,\theta)=rsin\theta\] \[z(r,\theta)=c sin(n\theta)\] where n lets you pick the number of folds in the surface, and the oscilating horizontal line along the z-axis would have a period of \(2π/n\).

(3-D models depicting Plücker's conoid, from https://mathworld.wolfram.com/PlueckersConoid.html)
This is what the original conoid looks like.

This is what the surface looks like before we add any fillers or ruled lines. Its just a ring. A ring made by what? Why, the function \[z(r,\theta)=2cos\theta sin\theta\] of course! So, all we do is add lines or ruled lines that create and are therefore a part of the surface of this object, and also are straight, and also do not intersect. Straight lines connecting this? And they form the surface? Yes! Follow along below to see it from different angles. How does it look when we change different variables? How does it look when we add less lines? All these and more are answered below. Now, we want to stretch it out a little bit so we can make better sense of the behavior of the lines on the top and bottom of the surface. Are they really straight? By changing the coefficients of the variables in the functions producing the surface, we can see more clearly what is happening albeit with a now new surface. Plus it looks like a sombrero. As of now the parametrization is \[x(r,\theta)=rcos\theta\] \[y(r,\theta)=rsin\theta\] \[z(r,\theta)=2cos\theta sin\theta\], but when we change it to \[x(r,\theta)=6rcos\theta\] \[y(r,\theta)=rsin\theta\] \[z(r,\theta)=2cos\theta sin\theta\] we see that the surface gets skinny and long turning from an almost fortune cookie shape into a snowshoe shape.

When we change just the y parametrization we get \[x(r,\theta)=rcos\theta\] \[y(r,\theta)=6rsin\theta\] \[z(r,\theta)=2cos\theta sin\theta\], resulting in

which looks the same as when we change the x parametrization scale, but along the y-axis now.
To show that these are really straight lines, I looked at different angles with a different interval for which the lines were being drawn at so it was easier to see. These are when the x parameter is changed but it will look the exact same but with respect to the y axis when the y parameter is changed.

So, what happens when we change the z parametierzation scale from a factor of 2 to 6? We get \[x(r,\theta)=rcos\theta\] \[y(r,\theta)=rsin\theta\] \[z(r,\theta)=5cos\theta sin\theta\],

resulting in a taller surface. Below we see that these lines are in fact straight by again reducing the frequency at which these lines are made to see the lines more clearly.

When we combine all of these aspects we get \[x(r,\theta)=6rcos\theta\] \[y(r,\theta)=6rsin\theta\] \[z(r,\theta)=5cos\theta sin\theta\] resulting in

In my opinion it is easier to see that these are in fact straight lines as you can follow them from the outside all the way in. When we divide these coefficients by 4 we get the following surface.

I chose this shape because I love sombreros and I want to see what this looks like in 3-D. This looks somwaht similar to the original shape we are working with, where the lines meet the middle very quickly and it is hard to see what is going on. I think "rolling out" the Plücker's conoid is much better for depicting ruled surfaces and results in a different surface too. I would argue that changing just the z parametierzation as above that gave us the TIE fighte/bowtie looking shape is just as important and different from the original as is changing the x and y parametierzation giving us a sombrero shape. It's more than just scaling it to see the appearence of the ruled lines, its changing the dimensions of the shape while keeping certain interesting aspects of its integrity to better understand what's really going on and express creativity in the same go. I love this surface and everything about it. It has been my favorite topic to cover and think about in the real world. I plan on using this print out as a fun textile object to keep at my desk to feel when I am working. The openscad model is about 30x30 units long and wide and about 5 units high. This will be equally scaled down for the print to be 1/20 units that it is currently and result in a print of 1.5x1.5x0.15 in.