Heath Fritillary — Polar Equation Portrayal
Henry Zhao — Concordia International School Shanghai
In my Honors Precalculus class taught by Dr. Tong at Concordia International School Shanghai, my classmates and I had a project that involved using polar curves to graph out a butterfly.
I had some experience with using graphs to create images on Desmos, as I had a project back in grade 8 that instructed us to create some type of pattern using what we’ve learned on Desmos. In addition, one of my friends is extremely interested in using Desmos to create depictions of real-world items, such as the periodic table of elements and the moving pattern of electrons around the nucleus of an atom. However, when this project was first announced, I still felt slightly unsure about what to do, as the shapes involved in the outline and patterns of a butterfly are usually intricate and irregular, and the lessons on conics and polar equations/graphs aren’t covered yet back then. However, as they become covered, I began to have an idea about what types of graphs should I use in my project.
Now, let’s take a look at how I created my project.
1. The Brainstorming Process
When selecting the butterfly for my project, I initially tried to find one with a color that I liked. However, with some advice from Dr. Tong, I later turned to find a butterfly with a more complicated pattern and ended up selecting the Heath Fritillary and a photo that matches up nearly symmetrically with the axes.
2. Beginning of Construction
To start off, I decided to begin constructing my butterfly at the less rigorous parts — The antennas and the outline of the wing. At first, I used parabolas for both the antennas and the upper outline of the wings. Then, I used several ellipses to create the outline for the rest of the wing outlines and the body of the butterfly. It’s also at this phase that I encountered an issue: The graphs that I create sometimes don’t match up perfectly. To make them do so, I have to use extremely precise numbers for the graphs’ domains, sometimes to the thousandths digit.
3. Details
After finishing up the outline construction, I proceeded to the most rigorous part of this construction: The patterns of the butterfly. At first, I had no clue how to construct the patterns, as they simply look like random curves at a glance. However, I noticed later that the thin patterns are regular curves that can be constructed through normal parabola/ellipse equations, and I, therefore, used them to create the patterns. However, the image is slightly asymmetrical, which means that I cannot use the same equation for both sides as I did for the outline in some places. In addition, during this phase I became unsatisfied with the antennas as they bend inward instead of outward as in the original image, therefore I used two separate equations to reconstruct them.
During this phase, I still need to be precise on the domains to ensure that the lines and curves end where they should be. It was also at this phase that I begin using the available colors on Desmos to distinguish the different parts of the butterfly, which is quite convenient as Desmos included both orange and black, the two main colors of my butterfly type.
4. Wrap-up Work
After finishing up with the patterns, I noticed that some of the graphs that I used for the patterns are not matching up with the thickness in the original image, and as a result, I used the “thickness” feature of Desmos to alter the thickness of the graph and replicate the stripes of the butterfly better.
