The Comma in Polar Equations
Every butterfly has its unique outline, patterns, and set of colors. However, they share a commonality — symmetry. Whatever image is portrayed on one wing is reflected onto the other side. Such intricacy of butterflies incentivized our Honors Precalculus class at Concordia International School Shanghai to create drawings of butterflies with polar equations. Our teacher Dr. Peter Tong decided for each student to create a different butterfly species.
I begin my project by searching for a butterfly on Butterfly Conservation. There was a variety of butterflies with intriguing patterns and designs, but what caught my attention was the Comma. Unlike many of the other species, the Comma possessed a simple pattern on the surface of its wings but an extremely complex outline. This is due to its camouflaging nature, as it attempts to resemble a dead leaf. The following Figure 1 is the image I utilized:
I first insert the image above into Desmos. I rotate and position the image so that the line of approximate symmetry between the two wings lies on the y-axis. I reduce the opacity, as shown in Figure 2, so that any lines I create above the image would be more visible to the eye.
Before beginning any tracing, I created some folders, as labelled in Figure 3, to ensure that I stay organized.
Simply by first glance, my guess is that the majority of my equations will either be parabolas (mostly for the outline) or ellipses (mostly for the patterns within the outline). I write out the general formulas in Figure 4 and place them in separate folders for convenient access throughout my project.
While completing the step above, I notice that every time a new variable is included, “add slider: _” appears, as shown in Figure 5.
Clicking on the variable provides me with a slider like the one below in Figure 6:
I create a slider for every variable that appears in my general formulas. I add them to a new folder called “Variables,” as indicated in Figure 7.
I begin tracing the outline of the right wing by copying and pasting the general formula of the parabola. I play around with the toggles in Figure 8 based on my prior knowledge of how each of the variables controls or alters the graph of the equation.
Once I ensure that the graph is approximately tracing the line I would like to draw, I plug in the variables’ values from the sliders into the equation I copied and pasted, as shown in Figure 9. This fixes the first equation and opens the sliders up to the next one.
In Figure 10, I create the next equation with the same method. I ensure that the intersecting point between the first and the second equations’ graphs is as close as possible to where the lines meet in the actual image.
I notice that the current length of the parabolas is limitless because no special restrictions have been placed on their domains or ranges. Such restrictions are written with inequalities in the brackets {} after each of the equations. I also take note that it becomes much more efficient to write the complete domain or range of a graph after writing the equation for the next line because the intersection point already provides one of the limitations. It also decreases the likelihood of visible gaps appearing between the different lines. For the example below, because I see that the intersecting coordinate is approximately (7.342,3.315), I set the domain of 𝑥 to {0.5≤𝑥≤7.342} in Figure 11.
I continue repeating the process above by adding new equations, domains, and ranges. Whether I choose to restrict the domain or the range depends on which general formula I choose to use and which segments of each equation’s graph I need. Sometimes both the domain and the range must be restricted, to ensure that there are no floating lines outside the design.
I realize that some lines do not match well with any of the general formulas. I reflect back on what we learned in the later half of Chapter 10 of the textbook Precalculus with Limits: 3rd Edition. This brought me to the rotation of conics. Rotating a conic requires substituting 𝑥=𝑥′cos𝑡−𝑦′sin𝑡 and 𝑦=𝑥′sin𝑡+𝑦′cos𝑡 into the equation of the initial conic one would like to rotate. In these formulas, 𝑡 represents the angle (in radians) with which the graphs are rotated around the origin. I create new folders for the general expressions of the rotation of each variable and add a slider for the value of 𝑡. As indicated in Figure 12, I change the possible values of 𝑡 to 0≤𝑡<2𝜋 to ensure that the graph can reach all possible rotations.
When I see that tracing a line may require rotations of general formulas, I plug 𝑥′cos𝑡−𝑦′sin𝑡 and 𝑥′sin𝑡+𝑦′cos𝑡 into 𝑥 and 𝑦, respectively. I repeat the same process I used above of sliding the different variables but include the extra variable of 𝑡 to rotate the graphs I create. The domains and ranges also become slightly more difficult, as it often requires combining the two types of restrictions. Furthermore, I notice that while {𝑚,𝑛} represents when either 𝑚 or 𝑛 are true, {𝑚} {𝑛} represents when both 𝑚 or 𝑛 are true. Below is an example of a rotated conic (in black). The complete equation, as written in Figure 13, is (xsin(0.6)+ycos(0.6)–4.3)²=4(0.3)(xcos(0.6)–ysin(0.6)–4.5), and its restrictions are {𝑦≥0.22} {6.02≤𝑥≤7.438}.
I continue repeating the processes above, sometimes utilizing rotated conics when seemingly necessary. I complete the outline of the right wing in Figure 14.
The nature of the butterfly is that each wing’s design is nearly identical to the other wing’s. Therefore, I begin trying to find a way to reflect the outline of the right wing onto the left side of the 𝑦-axis. This means that if the right wing is based on the function 𝑓(𝑥), the left wing must be based on the function 𝑓(−𝑥) so that all of the original 𝑥-values are reflected onto the other side of the 𝑦-axis. For instance, the first equation of the outline of the right wing is (x–6.5)²=4(–5)(y–3.35). To create the first equation of the outline of the left wing, we must copy and paste the equation into the “Left Wing Outline” folder and alter the equation into (–x–6.5)²=4(–5)(y–3.35), as shown in Figure 15.
I notice that restrictions on the range of 𝑦 can all be reused because they remain constant on each side of the 𝑦-axis. However, when I attempt to copy and paste over the same domain of 𝑥, the graphs would either disappear altogether or appear in areas that were undesirable. This is when I realize that the domains must also be the opposite of those on the right wing. For instance, the domain of the first equation for the outline of the right wing is {0.5≤𝑥≤7.342}. This implies that the domain of the first equation for the outline of the left wing is {−7.342≤𝑥≤−0.5}, as we can see in Figure 16.
I find that for the rotated conics, simply changing the 𝑥-value to −𝑥 is not sufficient. This is because, after the reflection, the graph must also be rotated in the opposite direction. Therefore, for all values of 𝑥′𝑐𝑜𝑠𝑡−𝑦′𝑠𝑖𝑛𝑡 and 𝑥′𝑠𝑖𝑛𝑡+𝑦′𝑐𝑜𝑠𝑡, the expressions must become 𝑥′𝑐𝑜𝑠(2𝜋−𝑡)−𝑦′𝑠𝑖𝑛(2𝜋−𝑡) and 𝑥′𝑠𝑖𝑛(2𝜋−𝑡)+𝑦′𝑐𝑜𝑠(2𝜋−𝑡), respectively.
I repeat the step above over and over until every equation in the folder “Right Wing Outline” has a symmetrical equation in the folder “Left Wing Outline,” as shown in Figure 17.
I then work on the folders “Upper Right Wing Patterns” in Figure 18 and “Lower Right Wing Patterns” in Figure 19 with the same mechanism. However, I begin using the general formulas of ellipses as well, as some of the patterns fit such equations better. The domains and ranges become slightly more complicated, as the distinction between {𝑚,𝑛} and {𝑚} {𝑛} (as mentioned above) becomes more useful. Sometimes, different sections of an ellipse need to be removed.
Similar to previous processes, I reflect the equations from the folders “Upper Right Wing Patterns” and “Lower Right Wing Patterns” across the 𝑦-axis and place the new equations into the folders “Upper Left Wing Patterns” in Figure 20 and “Lower Left Wing Patterns” in Figure 21, respectively.
With different combinations of parabolas and ellipses, I also create the center of the butterfly, including its body, its head, its eyes, and its antennas, as demonstrated in Figure 22.
I reach the final step of altering the colors of the outline and the patterns of the butterfly. Although simply using a combination of orange and black would be sufficient, I decide to do some extensive research on how to create more customized colors in Desmos. I come across a very useful YouTube video — Desmos How to Create a Custom Color. I learn that I can create custom colors with self-designated RGB coordinates (Figure 23). The following are a couple of examples of colors I created and utilized in my design:
The colors appear in the sidebar as new coordinates of 𝑐 are created, as shown in Figure 24.
I designate these colors to the equations I formulated for the butterfly design in Figure 25.
I remove the reference image to reveal the final design.
The following Figure 26 is the final design:
The following is the link to the Desmos file: https://www.desmos.com/calculator/3qqlf3zbrr
