Polar Project: How I drew a building with polar curves

At Concordia International School Shanghai, the Honors Precalculus class did a project based on our Chapter 10 content: Analytic Geometry. The objective of the project is to draw architecture out of polar curves.

For the polar project, I decided to work with Daniel for the architecture portion of the project. Since we were also project managers, we were looking for possible architectural buildings that our peers can use. While we were searching, Daniel randomly mentioned the word “observatory” as one of the possible ideas. That struck me — there’s a super awesome astronomy museum right here in Shanghai! The epiphany got me searching for pictures of the museum, and it was perfect for this project. It was made of two elliptical discs on top of each other, both with interesting ellipses and curves that are perfect to graph with polar equations.

Daniel and I decided to use this building for our project. We started with scaling and moving the graph on Desmos to see where it was the most appropriate. We decided that we will first graph an ellipse to serve as the “frame of reference” for the rest of the building. The first ellipse we wanted to graph was the outline of the top disc. First, we figured out the center point of the ellipse and how long the major axis/minor axis was. We graphed that in a rectangular equation to ensure it’s right, since transforming ellipses is easier with a rectangular equation. Then we used x=rcostheta and y=rsintheta to convert the rectangular equation into a polar equation. This was a lengthy process, and we ended up with a quadratic looking equation at the end so we had to utilize the quadratic formula in order to single out r. Then, we used r=whateverwegot as the polar equation for the graph.

That was when we ran into our first problem: rotating ellipses. We had the center point, major axis, and minor axis all ready to go, but the angle wasn’t quite right. At first, we tried to add the angle we needed to the theta value, but we realized not only does this rotate the ellipse itself, but it also moves the ellipse around the pole. We decided to think in reverse.

First, we tested out the value that is needed for the ellipse to rotate itself to the angle we want it to. Let’s say, for example, we want the ellipse itself to rotate pi/4 radians on the positive vertical axis, which is where it already is. We add +pi/4 but the ellipse ends up in quadrant 2, left of where it was originally. Then, we negated the pi/4, which is -pi/4, and added that to the theta value instead. This rotates the ellipse itself at -pi/4 radians and moves the ellipse to quadrant 1, the right of where it was originally. That gives us a new ellipse in quadrant 1. We took the center of that ellipse and combined it with the length of the major axis and minor axis of the original ellipse equation, giving us a new equation of an unrotated ellipse in quadrant 1. Then, to get the ellipse to rotate itself pi/4 radians, we add pi/4 to the theta of the NEW polar equation. This ellipse ends up at the positive vertical axis, rotated at pi/4 degrees itself. That is how we avoided the problem of the ellipses rotating around the pole.

We did that for all the curves that needed rotation on our graph. It was lengthy, but it works relatively well and it was accurate. Another important component of our graph lines. Similarly, we first graphed the line we need in the rectangular form and then converted it to polar form, so that it’s easier to adjust its location of it. Overall, the teamwork was great because we split up the work evenly, me taking the top disc and Daniel taking the bottom. The end result was super aesthetically pleasing and satisfying to see, too!