Polar Project — Epiphyllum

Honors Precalculus Class B2

Yiran Zhao

Flowers blossom and wither, leaving their most appealing time to the world. The idea of depicting a flower using polar equations first came to me when studying rose curves. It eventually became true when the Polar Graphing Project was assigned by Dr. Tong, my Honors Precalculus teacher in Concordia International School Shanghai. Among all the varieties of flowers, I chose the Epiphyllum for its appearance and its symbolic meaning that good things don’t last long. And because of its short blossom, it’s also a sign of luckiness for people who see it bloom.

Below is the final individual graph and the final superimposed graph for the project.

figure 1.1 {final graph}

figure 1.2 {final superimposed graph}

Preliminary Aspects:

Flower: Epiphyllum

The whole process for epiphyllum flower to change from a bud to blossom and finally perish is very quick. Epiphyllum bloom only lasts a few hours, around 4–6 hours, or less, at night. (Because the flower is white and it only blooms at night, I chose to use a black background & white lines for the final graph, shown as the first image) Also, because it is hard to have continuous graphs of the whole growing and dying process of the Epiphyllum, I decided to have different petals as different frames and complete the process of it blooming.

Frames:

I guess I will need about 15–20 frames, including 5 frames at first, 5 frames for its growth, and 5 frames for the final flower. I did 17 frames in total, with 8 frames for bud, 6 frames for the development from the bud to the flower, and the last three frames for the spreading petals.

Progress:

Stage 1: Sketching on the photo of the flower (numbering each petal)

figure 2.1 2.2 2.3

In this stage, since this step is rather simple, the only difficulty in it is to distinguish all the lines I need for the graph from the vague photo. The solution for this problem was to edit the picture to make the lines more obvious and use bright colors to outline the petals. Also, I had an initial design on how to organize my equations, for instance, how many equations I will need for different frames.

Stage 2: Trying and graphing the equations

fugure 3{Screenshot of the beginning of the project}

In this Stage there are mainly two problems, converting some rectangular equations to polar equations and finding the domain takes a long time, especially when sometimes it’s in a quadrant where its opposite. Because it’s more accurate and easier for me to find the equations that match the flower when they are in rectangular form. For converting equations, my solution was to utilize some technology (random rectangular to polar equation calculator) and try to find the pattern. And for finding the domain, my method was to first estimate the domain then try out in different quadrants if it’s hard to determine which quadrants it should be.

Stage 3: Planning the order of the frames & put the screenshots into a video

In this stage I first screenshot every frame of my graph, then used a gif website (https://ezgif.com/maker) to connect the screenshots into a gif. At last, use the same website to convert the gif into a video.

figure 4{Screenshot of the website}

Stage 4: Concluding steps

When I first turned the background black and all the equations white by using the “Reverse Contrast” function in Desmos, it didn’t look as I expected. So, I decided to change the color of the center parts. I looked up the RGB numbers of the colors I wanted and put them into equations to create the color in Desmos. Then, I will be able to change some of the equations to colors that Desmos doesn’t have initially.

figure 5.1{“Reverse Contrast” in Desmos}

figure 5.2{rgb equation I used}

figure 6 {another version, with a different color in the middle, of my graph}

figure 7{Some examples of my equations}