As we go through the semester, one can see our gradual distance from some calculus topics. We have explored calculus three, and now we will make one last stop in this realm. The idea for this week, we looked into different rules surfaces and their creation.
Now, you may be thinking to yourself "What is a ruled surface?". Ruled surfaces are something that may not have been discussed in-depthly in an average classroom, but is actually an object that appears commonly in nature. First the more technical definition: A ruled surface is a set of points that are being swept by a moving straight line. What this means is that, if you have a line and another line, if you can connect each point of one line to the other with a straight line, that surface created is called ruled.
Now, where would we come across something like this? Well, it is fairly common in your day to day objects! For example, the shape of a nuclear power plant cooling plant, which is the shape of a hyperboloid, is a ruled surface! Along the edges of the circles on the top and bottom, you can draw some straight lines from one circle to the other! Another example would be your cylindrical water bottom, just for the same reason as above! More interesting is that you are able to change the angle at which the lines are at so you get what would appear to be a twisting shape that is actually composed of only straight lines!
Personally, I think these objects are very cool when you have these twists because it makes the object look so different depending on how tightly you twist it. Imagine you have some clay in your hand and you roll it out into a line. By twisting this line with your hand, you make it so the center becomes what appears to be a single point. In ruled surfaces, this is also the case when you make what we call the “Phase” 180, which is where the lines are going at a degree of 180. You get exactly a cone when you start with a cylindrical shape and make the phase 180!
As we have gone through in previous explorations, there is a clear connection between creating these 3D surfaces and using parameterization. This will allow us to use only 2 variables rather than 3. In particular, we want to describe our normal $x, y, z$ variables in the form of $r, \theta$, for radiis and angle measures respectively.
Our Problem:
Once our discussion began, the first thing to come to mind was the Leaning Tower of Pisa. This tower is famously known for its leaning (quite literally in the name) and still standing today. While it has been given some support over the years, this tower still stands, where it leans with an angle of 4 degrees. My goal is to reconstruct this object, showing it off as the ruled surface that it is, along with keeping many of the original aspects true to the model. Most importantly, the object created should be angled at 4 degrees.
Here is a photo of the tower if you are unfamiliar with it!
Lets get into it! Starting off, we know we need some circles, creating the base and the top, with one being shifted from the other. We can begin with our normal parameterization, setting $x_1 = r * cos({\theta}_1)$ and $y_1 = r * sin({\theta}_1)$. This is the foundation for our tower, being our first function. Moreover, since this is the base, the $z_1$ component would just be zero, since we want this to be laying on the floor of the space we are dealing with. Thus, our first function will be in the form of: $f = (r * sin({\theta}_1), \ r * sin({\theta}_1), \ 0)$.
By the same construction, we can generate the function for the top of the tower. A couple of problems arise here. Firstly, how tall is this tower supposed to be? Secondly, how off putting the top has to be from the base? Since we know we want it to be angled off at 4 degrees, we can construct what would be a similar triangle to find dimensions best suited for our model. In particular, if we offset the top by exactly the shared radius of the two circles, we can find the height needed! So, for both of our constructions, the $r$ value will be the same!
Let's write down what we know so far. We know from our discussion above that we should have something in the form of $g = (r * sin({\theta}_2) + r, \ r * sin({\theta}_2) + r, \ z)$. Along with this, remember the radius is the same as above, and we need to know the new angle measures and the height. Since we know the angle would be 4 degrees, we know: \\ $\cos(4) = \frac{r}{\sqrt{r^2+z^2}}$. Let's finally assign some length to the radius, so $r = 20$. This will be the radius of the trig functions from before. But, we want to shift the top circle by half of the radius, so to solve for the height we will be using 10 instead. So, $\cos(4) = \frac{10}{\sqrt{10^2+z^2}} \Rightarrow 10^2 + z^2 = \frac{10}{cos(4)}^2 \Rightarrow z = \sqrt{(\frac{10}{cos(4)})^2-100}$
This simplifies down to: $z \approx$ 143.
One final discussion to be had is over the angle measures. These new angle measures will be the original $\theta$, adding on 90, where this is the Phase discussed at the beginning. Thus, the second function is in the form: $g =(20 * sin({\theta}_1 + 90) + 10, \ 20 * sin({\theta}_1+90) + 10, \ 143) $.
Now we have the two functions : $f = (20 * sin({\theta}_1), \ 20 * sin({\theta}_1), \ 0) $ and $g =(20 * sin({\theta}_1 + 90) + 10, \ 20 * sin({\theta}_1+90) + 10, \ 143)$. As discussed with our construction, this shape is 20 by 20 by 143 (being our $x , y, z$ respectively. Below is the openscad screenshots of our code we have generated through this and the shape generated!
I chose this example for a couple of reasons. Firstly, it plays along with my food research shapes from two blogs ago with the taco stand. Secondly, this shape is a nice example of an object you would see in everyday life as well as being a famous monument.
Now, you may be thinking to yourself "What is a ruled surface?". Ruled surfaces are something that may not have been discussed in-depthly in an average classroom, but is actually an object that appears commonly in nature. First the more technical definition: A ruled surface is a set of points that are being swept by a moving straight line. What this means is that, if you have a line and another line, if you can connect each point of one line to the other with a straight line, that surface created is called ruled.
Now, where would we come across something like this? Well, it is fairly common in your day to day objects! For example, the shape of a nuclear power plant cooling plant, which is the shape of a hyperboloid, is a ruled surface! Along the edges of the circles on the top and bottom, you can draw some straight lines from one circle to the other! Another example would be your cylindrical water bottom, just for the same reason as above! More interesting is that you are able to change the angle at which the lines are at so you get what would appear to be a twisting shape that is actually composed of only straight lines!
Personally, I think these objects are very cool when you have these twists because it makes the object look so different depending on how tightly you twist it. Imagine you have some clay in your hand and you roll it out into a line. By twisting this line with your hand, you make it so the center becomes what appears to be a single point. In ruled surfaces, this is also the case when you make what we call the “Phase” 180, which is where the lines are going at a degree of 180. You get exactly a cone when you start with a cylindrical shape and make the phase 180!
As we have gone through in previous explorations, there is a clear connection between creating these 3D surfaces and using parameterization. This will allow us to use only 2 variables rather than 3. In particular, we want to describe our normal $x, y, z$ variables in the form of $r, \theta$, for radiis and angle measures respectively.
Our Problem:
Once our discussion began, the first thing to come to mind was the Leaning Tower of Pisa. This tower is famously known for its leaning (quite literally in the name) and still standing today. While it has been given some support over the years, this tower still stands, where it leans with an angle of 4 degrees. My goal is to reconstruct this object, showing it off as the ruled surface that it is, along with keeping many of the original aspects true to the model. Most importantly, the object created should be angled at 4 degrees.
Here is a photo of the tower if you are unfamiliar with it!
Lets get into it! Starting off, we know we need some circles, creating the base and the top, with one being shifted from the other. We can begin with our normal parameterization, setting $x_1 = r * cos({\theta}_1)$ and $y_1 = r * sin({\theta}_1)$. This is the foundation for our tower, being our first function. Moreover, since this is the base, the $z_1$ component would just be zero, since we want this to be laying on the floor of the space we are dealing with. Thus, our first function will be in the form of: $f = (r * sin({\theta}_1), \ r * sin({\theta}_1), \ 0)$.
By the same construction, we can generate the function for the top of the tower. A couple of problems arise here. Firstly, how tall is this tower supposed to be? Secondly, how off putting the top has to be from the base? Since we know we want it to be angled off at 4 degrees, we can construct what would be a similar triangle to find dimensions best suited for our model. In particular, if we offset the top by exactly the shared radius of the two circles, we can find the height needed! So, for both of our constructions, the $r$ value will be the same!
Let's write down what we know so far. We know from our discussion above that we should have something in the form of $g = (r * sin({\theta}_2) + r, \ r * sin({\theta}_2) + r, \ z)$. Along with this, remember the radius is the same as above, and we need to know the new angle measures and the height. Since we know the angle would be 4 degrees, we know: \\ $\cos(4) = \frac{r}{\sqrt{r^2+z^2}}$. Let's finally assign some length to the radius, so $r = 20$. This will be the radius of the trig functions from before. But, we want to shift the top circle by half of the radius, so to solve for the height we will be using 10 instead. So, $\cos(4) = \frac{10}{\sqrt{10^2+z^2}} \Rightarrow 10^2 + z^2 = \frac{10}{cos(4)}^2 \Rightarrow z = \sqrt{(\frac{10}{cos(4)})^2-100}$
This simplifies down to: $z \approx$ 143.
One final discussion to be had is over the angle measures. These new angle measures will be the original $\theta$, adding on 90, where this is the Phase discussed at the beginning. Thus, the second function is in the form: $g =(20 * sin({\theta}_1 + 90) + 10, \ 20 * sin({\theta}_1+90) + 10, \ 143) $.
Now we have the two functions : $f = (20 * sin({\theta}_1), \ 20 * sin({\theta}_1), \ 0) $ and $g =(20 * sin({\theta}_1 + 90) + 10, \ 20 * sin({\theta}_1+90) + 10, \ 143)$. As discussed with our construction, this shape is 20 by 20 by 143 (being our $x , y, z$ respectively. Below is the openscad screenshots of our code we have generated through this and the shape generated!
I chose this example for a couple of reasons. Firstly, it plays along with my food research shapes from two blogs ago with the taco stand. Secondly, this shape is a nice example of an object you would see in everyday life as well as being a famous monument.
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