Part B: Polar Equation Graph of Changshai Meixihu Center

5 min readMay 27, 2022

Introduction to the Changsha Meixihu International Culture and Arts Center:

Changsha Meixihu International Culture & Arts Centre is a contemporary art museum (MICA) in Hunan, China; in fact, it is a 1,800-seat theatre with supporting facilities and a multipurpose hall. Its organic architectural language is defined by pedestrian routes that weave through the site to connect with neighbouring streets. It provides a scenic view of the Meixi Lake and is located on Line 2 of the metro system. It hosts a variety of functions in its sculpted lobbies, bars and hospitality suites, as well as the necessary ancillary functions including administration offices, rehearsal studios, backstage logistics, wardrobe and dressing rooms.

Introduction to the Changsha Meixihu International Culture and Arts Center:

Desmos Graph: Write-Up:

Graph: https://www.desmos.com/calculator/qbrzlrp9jm

In Honors Pre-Calculus at Concordia Shanghai International School, we started exploring the concept of polar equations, which are equations graphed based on angles and radius, not the x-y coordinates. It was challenging in the sense that we had learned about regular graphs our entire primary and secondary education up until now. Our instructor gave us a task: graph an architecture of your choice using equations that can replicate the object or shape in polar coordinates.

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.

(Picture of the Changsha Meixihu, from https://www.zaha-hadid.com/architecture/changsha-meixihu-international-culture-art-centre/)

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.

(Picture of supposed lemniscate tracing upper left curve of building; it didn’t work out in the end)

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/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. For example, we developed a system of secant polar equations that when manipulated, could yield all kinds of straight lines needed in the architecture.

(Trusty “stairs” secant equation family)

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).

(Conversion process of equations; this is a secant equation in the above picture)

(Polar hyperbola with origin at the center)

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 bolded 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.