Polar Dragon Fruit
In my freshman year, my Honors Precalculus class at Concordia International School Shanghai was taught on quite a few topics — one of them being the polar coordinate system. I found the class quite difficult due to a change from trimesters to semesters when I moved schools in January, which resulted in missing a few of the units entirely.
During class, we learned about the differences between our usual Cartesian coordinate system and the polar coordinate system — something we see and use more than we realize. One of the best examples of polar coordinates are radar scanners, which scan in one direction at a time and records how far away any objects are. The polar coordinate system relies on these two values, 𝜃 and r, just like how the Cartesian system relies on x and y, the horizontal and vertical values.
But I wanted to take my knowledge of the topic further. In class, we learned about some of the simple polar equations: limaçons, rose curves, circles, and lemniscates, to name a few. We also learned the polar versions of conic sections, and I was surprised at how simple they were compared to their Cartesian counterparts. Soon after starting the subsequent homework assignment, I realized I wasn’t too familiar with how the polar functions were graphed.
And so, I taught myself. In the form of a fruit.
The first thing I did was give the dragon fruit an elliptical base to build around, using a simple rotated ellipse focused around the pole. I then added two other ellipses to represent the two halves of dragon fruit, but they were also focused on the origin and not where they should have been. After running around in circles for a while, I converted the polar equation into a matching parametric equation so I could offset the graph and place it where it needed to go.
I then spent some time thinking about what type of equation to use as a petal, and ended up using a very special looking rose curve — a pointy one that I was able to achieve with absolute value. Afterwards, I gave the petals a little more detail with some parabolic green dividers to represent the green tips of the petals. I also added some green, leafy bits on the exterior.
After adding those leaves and petals, the only thing that remained were the little black seeds found inside the two halves. Instead of creating dozens of seeds, one by one, I created a table of r values and 𝜃 values that would become the point of a single elliptical seed’s focus. By creating two tables, one for the top half and one for the bottom, I was able to place a small ellipse for every (r, 𝜃) value in the table and create exactly sixty-two seeds.
That brought me to the end of my polar journey! This little project taught me so much about manipulating polar equations. It’s always fun to express a little creativity, no matter the subject, and this dragon fruit project really helped me out with finding ways to do so.
Here are all the equations used for this project:
