Polar Equations Project: The Ringlet
Introduction
Nature is full of beauty and wonder, and one of the most captivating sights is that of a butterfly gracefully fluttering through the air. As the inhabitants of our beautiful planet for more than 200 million years, these delicate creatures come in an array of vibrant colors and patterns based on each species’ distinctive characteristics. This year in Dr. Tong’s Honors Precalculus class at Concordia International School Shanghai, we were given the opportunity to utilize our knowledge of polar equations and draw out different types of butterflies. After doing some research on butterfly species, I selected the Ringlet and found a reference picture of it for my polar equations project as shown below:
After observing the different parts of this butterfly, I decided to map it out with lines, ellipses, and parabolas. Although there are no special shapes that require the use of more complex conics, there are many details in the wings and middle body of the butterfly that I had to pay extra attention to in order to depict an accurate representation of it.
Stage 1: Background
I started building my polar curve project by outlining the stems in the background. They consist of straight lines broken down into shorter segments, so I used the y=mx+b formula to align each segment of the stem with a line.
An important takeaway from this section is how to manipulate the length of each line to match the length in the background by setting restrictions on the range of the x variable when it is in rectangular form, θ (directed angle from pole), and r (directed distance from pole) when it is in polar form. By substituting the x and y with their relative polar representations, rcosθ and rsinθ, based on the definitions of r, cosθ, and sinθ I moved all the terms to one side of the equation to combine like terms and isolate r using algebraic principles. This allowed me to successfully derive the polar form of each equation. Although it was a time-consuming process, the outcomes were very helpful in helping me construct the boundaries of the stem as I used trial and error to find the correct θ and r range for each equation. The final equations for the background are as below:
Stage 2: Spots on the Wings
Next, I turned my attention to the spots on the Ringlet’s wings. I noticed a pattern of a smaller ellipse or circle embodied in a larger one for each spot. Therefore, I decided to use the ellipse formula, (x-h)²/a² + (y-k)²/b² = 1, to first find the rectangular equation for each of the two ellipses in one spot. Since a and b determined the width and height of the ellipse respectively and (h,k) is the coordinate at which the ellipse’s center is located, I used trial and error to locate each ellipse that overlaid the spots on the reference picture and repeated this process until 19 ellipses were created to represent each spot.
Since each spot of the butterfly varied in size and length, it took me some time to find the most suitable values for a, b, h, and k for each equation before moving on to the next step. Then, I used the same conversion method from stage 1 to convert each equation to its polar representation by substituting x and y with rcosθ and rsinθ and isolating r. Since the spots are full ellipses, I didn’t have to set any restrictions on r or θ. Below is an example of a rectangular equation and its polar form for one spot on the wing:
Stage 3: Middle Body
After finishing the relatively straightforward parts, I worked on the middle body of the butterfly by using a combination of parabolas, ellipses, and lines. Since the butterfly I selected is not symmetrical, I decided to illustrate every line or curve that is visible on the reference picture to present as many details as possible. Since I was already familiar with the construction of lines and ellipses from stages 1 and 2, the biggest challenge I faced at this stage was the construction of parabolas, which were needed for the butterfly’s antennae, eyes, and lower part of its middle body.
Feeling clueless about where to begin, I visited YouTube and found an extremely helpful formula from a tutorial about Desmos Art: y=a √b+(x-h)² + k. Instead of only being able to adjust the h and k values from the standard equation (y-k)²=4p(x-h), this up-down curve formula allowed me to adjust four values: a, b, h, and k. It was convenient because I could manipulate the direction and width of the curve’s opening (a), the location of the curve’s vertex (b), the location of the curve horizontally (h), and the location of the curve vertically (k) easily. Furthermore, I recognized the importance of identifying a specific portion of each parabola by regularly comparing the parabola I currently have to the designated curve on the reference picture and setting a restriction on the range of the x variable of the parabola to perfectly overlay the curve.
However, I soon realized a problem: one formula is not enough for all the parabolas when it only creates parabolas that open upwards or downwards. Many curves are easier to construct with parabolas that open leftwards or rightwards, so I decided to use the vertex form for quadratic equations: x=a(y-k)²+h. Similar to the previous formula but with one less adjustable value and a curve that opens leftwards with a is negative and rightwards when a is positive, I completed the rest of the parabolas for this section. My biggest takeaway from this stage is the proficiency I developed after creating 9 lines, 4 ellipses, and 45 parabolas to create the full middle body. At last, I repeated the steps of converting a rectangular equation into its polar form and finalized my equations, of which some are shown below:
Stage 4: Left Wing and Right Wing
Finally, I worked on the wings of the Ringlet. I started with the left wing and used a total of 30 parabolas and 7 lines to construct the bits and pieces of it. Since I have practiced building rectangular equations of lines, ellipses, and parabolas in each of the previous stages, I thought that this would be an easy process at first.
However, I soon realized that constructing only one parabola for the entire outline of the upper wing is inadequate: short segments of multiple parabolas are necessary to form an accurate boundary of the butterfly’s wings. Therefore, I used a total of 8 parabolas just for the upper left wing to trace every curve with precision. Then, I repeated the process of constructing parabolas by inputting different values into the two formulas in stage 3 and restricted the range of the x variable to present a coherent and well-connected outline of the wings. Following that, I observed the curves inside the boundaries of the wings and used a total of 16 parabolas with dashed lines to distinguish them from the rest of the boundaries. I repeated this process for the right wing of the Ringlet, which turned out to be more efficient to construct because many parabolas from the left wing can be reused in the right wing after adjusting its h and k values and selecting a different range from the modified parabola.
In total, I used 161 polar equations to construct this project with a combination of lines, parabolas, and ellipses to portray the Ringlet butterfly. The hours of hard work paid off in the end when I felt nothing but pride and delight in the final product I created. This project has undoubtedly reinforced my knowledge of polar equations, especially the conversions between rectangular and polar forms even when I realized that some of the equations cannot be converted to polar form due to the presence of a square root in the derived formula I used. Although it was time-consuming, I enjoyed the process of creating equations and adjusting their values to create a piece of artwork when it seemed impossible to achieve at the beginning. The full 161 equations are available in the Desmos link below:
https://www.desmos.com/calculator/bh12pywd1p
Final Graph
Final Superimposed Graph
