Making Art with Polar Equations

Emma Feng and Ashley Yu

Emma’s Fruit Project

Emma’s Statement

It is incredible the amount of effort and dedication it takes to draw using math. I have always enjoyed math, but never tried to do anything creative with it. When our teacher, Dr. Peter Tong, offered a chance to showcase what we have throughout the Honors Precalculus, at Concordia International School Shanghai, I decided to exceed his expectations by illustrating a fruit on Desmos, using nothing but polar equations.

I had chosen a Mangosteen specifically for its mesmerizing pin wheel of fruit, and the curves that would suit polar equations well. By inserting a high quality Mangosteen picture into the Desmos document, I could begin putting equations on top of the picture in an attempt to align the curves.

Polar Graphs of Mangosteen

In the beginning, I started by experimenting with rose curves to try and fit the petals inside the fruit. However, I came to realize that the insides were not symmetrical, as there were 7 petals all of different sizes and ambiguities. After several attempts of skewing the rose petal curve, I decided I would have to do each petal individually.

Sadly, this was not the end of my trials, as the petals were not smooth- the edges would lump, fitting unevenly with the curve. Granted, I had underestimated the difficulty of this task. So, for each individual petal, I used multiple rose curves to better conform to the indents and imperfections of the fruit, then limited each equation so that they would connect seamlessly. Again, this was difficult as well, as when working with rose curves, isolating a specific, small portion takes a lot of trial and error to seclude that piece perfectly. When the slope is high, it is difficult to approximate the limit for it, as it easily fluctuates.

When I had finished the insides of the fruit, I also was faced with the daunting task of learning how to offset polar equations from the origin to form the shell of the fruit; however, this particular problem paled in comparison to the leaves. The leaves of the Mangosteen were definitely the hardest part of the project; they curved at many different angles, faced away from the center, and introduced points.

From here, I had to learn other types of polar equations besides rose curves and ellipses- lines, parabolas, hyperbolas- to fit these crevices. After experimenting with rotation of polar curves and limits, I had to use every and any variation of equations I could, as the leaves included all kinds of shapes and sizes.

The hardest part of the leaves was when curves would face away from the origin, as it was difficult to find hyperbola equations that weren’t too narrow to fit properly. Occasionally, I would use linear polar equations to construct the tips of the leaves, as there was no other way to create that sharp look. After I had finished the leaves, I was finished with my Mangosteen project. For this superimposed picture, I lowered the opacity to 0.3 and took a screenshot of the equations outlining the shape.

As of now, many students only push through high school math class simply to pass and move on in life. However, no matter the subject I choose to pursue for college, I always invest 100% effort in any topic of interest. In Concordia International School Shanghai, every student is taught to try their hardest in all that they do, and I am no different. It is my pride to expand beyond the textbooks and lectures, but rather to challenge myself to apply the knowledge I have acquired into creative fields. As every student should, we should not only focus in class, but seize the opportunities that are presented to us.

Ashley’s Architecture Project

Ashley’s Statement

In a regular Honors Pre-calculus course at Concordia International School Shanghai, students are introduced to the basics of polar equations and graphs, including rose curves, limaçons, and various comics. However, the issue that lies within many curricula is that students are not encouraged to further explore the topic. As the Chinese proverb states, “读万卷书，不如行万里路” — traveling ten thousand miles is better than reading ten thousand books. Rather than dropping students with countless new equations to memorize, students should be provided with the opportunity to explore the concepts themselves — to travel thousands of miles to acquire a in-depth understanding. Fortunately, I was provided with the opportunity to make my own journey in learning polar graphs at Concordia International School Shanghai.

Polar graphs are often known for their unique curves. While buildings are typically known to have rigid edges, some possess more distinctive wavelike structures — such as the Changsha Meixihu International Culture and Arts Center. Our goal was to create a detailed outline of our chosen architecture with the basic knowledge we were given about polar equations.

Polar Graphs of Changsha Meixihu International Culture and Arts Center

To create a replica of the building using polar equations, I first had to find the most prominent curves in the architecture. The Changsha Meixihu International Culture and Arts Center possesses many similar curves, so my partner’s and my goal was to find the equations for only a few most noticeable lines. Naturally, I decided to start with the outline of the architecture, as the rest of the building follows a similar pattern. I started with equations that I am more familiar with, looking for patterns among the architecture. Mayhap the most difficult part of my chosen architecture is that it contains a great variety of curves.

Rose curves provided some the lines closer to the origin. However, it was a challenge for my partner and me to refine the curve to better suit the architecture. Through the exploration process, I was able to find more concepts that deepened my knowledge in polar equations. For instance, adding a number to theta will rotate the graph by a certain degree around the origin. This really helped better fit the rose curves to our building.

We also found limaçons and circles to be of little help, considering that they are more round in nature. The Changsha Meixihu Museum tends to have more gentle and gradual curves instead. We attempted to use the polar equations for common conics, such as parabolas, ellipses, and hyperbolas. These graphs tended to work out better. However, the most difficult part may be that the majority of the polar curves are centered around the origin. Our chosen architecture did not share that characteristic with polar graphs — instead, the building was more spaced out, possessing a similar shape as a sine curve.

Since we only had some foundational knowledge from class, the beginning of our project was basically trial and error. This helped me reinforce concepts that we have briefly touched upon in class — for instance, the unique characteristics of cosecant and secant graphs that set them apart from other polar graphs.

After some time, I began utilizing more concepts that were taught in class and applying them to our project. We were taught how to convert rectangular coordinates to polar coordinates, but we did not learn as much about converting graphs. Nevertheless, having this knowledge is already enough for me. After finding curves of the buildings using parabolas in rectangular form, I was able to transform them to polar form by replacing the x and y variables with rcos(theta) and rsin(theta) respectively. After employing the quadratic formula to the newly derived equations, I was able to freely convert the equations between rectangular and polar form. This freed me from the constraints of the polar equations that I had previously used and the parabola no longer had to center around the origin.

Of course, I faced more challenges during the project. For instance, the domains of my newly derived quadratic equations functioned significantly differently from the other polar equations. I later figured out that I only had to switch the plus and minus sign in the quadratic equation. However, being able to learn on my own — even just through trial and error — helped me better grasp the concept. By traveling thousands of miles in pursuit of greater knowledge, I was able to come closer to the final destination than I would have be able to through merely reading the textbooks.