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September 27, 2022

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Exploring the challenging concept of polar equations

In Honors Pre-Calculus at Concordia International School Shanghai, we started exploring the concept of polar equations, which are graphed based on angles and radius, not simple x-y coordinates.

It was challenging, in the sense that we had learned about regular graphs in our primary and secondary education until now. Our instructor, Dr Peter Tong, gave us a task: graph an architecture of your choice using your own equations.

My partner and I chose the Changsha Meixihu International Culture and Arts Center, a wavelike structure with rounded domes, arched windows and an overall dystopian theme. It seemed like a very daunting task at the time.

We started by editing the photo into shapes that we learned about in Unit 10.8 Polar Equations of Analytic Geometry: limaçons, rose curves, circles, lemniscates, etc.

Then, we tried to graph those shapes by using the format of the shapes we desired. This was mostly a trial and error process, given the fundamentals we learned in class were needed to be expanded on.

We learned to change the variables if we wanted bigger or smaller polar shapes, and we toyed with adding exponents or taking the square root of the equations. We also learned to change the angle of θ to rotate the curve in a certain direction. Additionally, we learned to input appropriate domains to only graph the bit of the equation we wanted to show.

Eventually, we managed to find “trusty” equations for a family of polar shapes that we wanted. We only had to manipulate a few variables to find polar shapes similar to each other.

We also developed a system of secant polar equations that when manipulated, could yield all kinds of straight lines needed in the architecture.

My partner and I also converted several rectangular equations (x-y) into polar equations (rcosθ and rsinθ), most notably the parabolas, when we couldn’t find the polar alternative we wanted. We also used conic polar equations that had the origin lying at the center (see parabolas, circles, ellipses, hyperbolas).

After we got the outline done, we added in explicit details, such as the lines of the windows, the sunken arches above the windows, and the “netting” system interspersed with viewing windows at the bottom left of the structure.

For these details, we often used dotted lines in contrast to the bold black we used for the framework. This gave the entire architecture plot a more organized feeling.

We went back a couple of times to smooth out the curvatures, connect the lines and tamper with the variables as well as domains. Afterwards, the project was pretty much completed.

Fun fact: I watched a few YouTube tutorials about shading using inequalities, and I learned how to add color to the windows in our earlier drafts.

However, this technique wasn’t perfect as many of the windows on the Changsha Meixihu aren’t simple geometric shapes. Moreover, Desmos didn’t allow me to convert my rectangular (x-y) equations to polar, so I had to scrap the idea.




 

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