Zebra Longwing Butterfly in Polar Equations

At Concordia International School Shanghai, a group of students taking the 22–23 Honors Precalculus class taught by Dr. Tong was assigned to create butterfly sketches using polar equations. This log tracks my progress in this project and how I created the zebra longwing butterfly (see Fig. 1).

Fig. 1: A superimposed version of the final product

The zebra longwing butterfly has long, narrow wings that are mostly black with white zebra-like stripes. My strategy going into the drawing of this butterfly is a big-to-small approach. I will begin with the outline of the butterfly and work my way inside, ending with the small dots throughout the butterfly's body.

To use polar equations and be able to translate them, I decided that I would use the coordinate strategy where my “equations” in Desmos were in (x,y) form where x and y were separate polar equations. Specifically, if r was my untranslated polar equations, (rcos(theta), rsin(theta)) would be the form in Desmos. If translation is needed, I would add or subtract specific values to the x and y values. For example, if I needed to shift the polar equation one unit to the right, I would add to rcos(theta).

However, before I began the graphing and drawing process, I decided that it would be easiest for me to mirror the image of the butterfly. This means I am effectively drawing both sides of the butterfly simultaneously, as all I need to do is reflect the graph across the y-axis (See Fig. 2).

Fig. 2: A mirrored image of the Zebra Longwing Butterfly

Stage 1: Outmost Outline

The outmost outline of the butterfly consists of mostly curved lines, which was perfect for polar equations. Immediately, I knew I had simple polar equations such as rose curves, circles, and lemniscates at my disposal.

However, it was obvious that the entire polar equation would rarely be used because many of the curved lines end abruptly or end at a sharp turn. One good example of this was the antenna of the butterfly. These antennas were just curved lines that stopped at two points. Of course, using the entirety of any polar equation would result in excess lines and curves on the graph. I decided that a rose curve would provide an accurate curved line that mimicked the one on the butterfly. I also decided on the proper trigonometric function (cosine or sin), r value, and a value for the rose curve. With the untranslated polar function created, I converted the equation into the coordinate form mentioned earlier and translated the equations to be lined up with the antennas of the photo. However, the question remains: how do I eliminate the excess lines? This brings me to the major takeaways of my first few equations of this project.

There are two main ways to restrict the equation to eliminate excess lines: by directly limiting the values of theta (the value of t on Desmos) or limiting either the x or y in the (x,y) form. The first way allows extracting certain rose petals from a rose curve. However, the second will be used more in this project as it is much more straightforward and can immediately restrict each equation to what I need for the butterfly. For each antenna, I first restricted the t value (theta) to isolate the only petal that is used, then I restricted the y equation so that the petal starts from the bottom of the antenna and ends at the top of the antenna.

Another takeaway from this beginning section was from drawing the wings. While these wings just seem like a curved line that could be modeled from one polar equation, that will not be the case. To have equations that closely mimicked the one in the photo, multiple polar equations must be used. As a result, there were multiple polar equations connected just to form one continuous line. In fact, I linked four consecutive rose curves together to form the upper wing.

The last takeaway was the rotation of the conic. Sometimes, I had the exact shape and size of the polar equation that I wanted but was aligned with the curves on the graph. This meant I had to either add or subtract angles from the t value in the equation to rotate the polar equation to certain degrees.

These takeaways all stemmed from the brute force trial and error method. However, these takeaways would be extremely important for the rest of the butterfly. With these basic understandings of the polar equations, I just had to deal with trial and error and finding the correct coefficients for the rest of the polar equations of the butterfly (Note: easier said than done), shown in Figure 3.

Fig. 3: Butterfly Outline Superimposed

Stage 2: Stripes

This was the toughest section of the butterfly. There were sharp turns and weird angles with the stripes. However, with the knowledge from doing stage 1, this section of the butterfly was easier. However, I still got more creative with my equation selection for this stage. I also decided that each stripe would have its own unique color.

For the outer two stripes, I noticed that the stripes were more curved and decided to stick with using the simple polar equations, such as a rose curve, to mimic the butterfly curve. And to combat the discontinued lines and sharp angles, the stripes consisted of multiple polar equations. I also used domain and range restrictions to ensure the polar equations start and end where the stripes are (See Fig. 4).

Fig. 4: The two outmost stripes superimposed.

On the other hand, for the remaining two stripes, I noticed that the strips consisted of relatively straight lines than curved ones. For equations to produce such straight lines, I knew that the r values of the simple polar equations would have to be extremely big. So, I turned the polar equations of conics. I knew that parabolas and hyperbolas could be made very flat and mimic the straight lines on the butterfly. After playing with the sliders, I found the correct hyperbolas that adhered to the stripes on the butterfly (See Fig. 5).

Fig. 5: Inner stripes superimposed

Stage 3: Circles and Ellipses

My next stage was working on the circles and ellipses lined up at the bottom of the butterfly, two red circles on the wings, and the ellipses on the head. The circles were all relatively easy as I had my base equation consisting of cosine to keep the circles easy and consistent. Then I just adjusted the coefficients and shifted the equation to adhere to the circles on the butterfly (See Fig. 6).

Fig. 6: Superimposed circles on the butterfly

However, for the remaining “circles,” I realized that they were more elongated in one direction. This resulted in my using ellipses. With my knowledge of polar equations with ellipses, I found the proper a and b values that made each ellipse the correct size according to the one it attempts to mimic (See Fig. 7).

Fig. 7: Superimposed ellipses on the butterfly

Stage 4: Body

The last stage is the body of the butterfly, down the middle. The body was very straightforward as there were short, sometimes curved lines. Using what I learned from the previous stages, simple polar equations drew the lines down the middle. There were also horizontal lines that were mimicked by linear functions (See Fig. 8).

Fig. 8: Superimposed body of the butterfly

With the entire butterfly done, here is the final product (See Fig. 9 and Fig. 10):

(Fig. 9: Polar equation butterfly on Desmos).

(Fig. 10: Superimposed final product)

Conclusion

This polar equation project required a LOT of a trial and error. Although this documented the process, it is unable to characterize the sheer amount of work that was required to make the butterfly come to life. If you want to see the equations that make up each individual part of the butterfly, you can do follow the link: Desmos Link.