Honors Precalculus Polar Project

Honors Precalculus 2022~2023

Class B2
Created & Logged by: Sussi Zhu

Figure 1: Superimposed image of polar graph on flower picture.

The beauty of flowers is not merely presented in their most blossomed season, rather it is the process of growth, bloom, and even the final wither. The charm of the animateness of flowers is painted by the Honors Precalculus B2 class with the guidance of our amazing teacher Dr. Tong at Concordia International School Shanghai. This year, we’ve taken this long-preserved polar project tradition in this course to a new realm. The B2 class produced an animation of the growth of different species of flowers, bringing a unique video of an exquisite flower garden.

Preparatory Processes:

After researching and visiting numerous flower websites on the internet, I finally landed on a violet/viola flower. ~Jan. 6

The reason I chose this specific flower is not only because of its graceful appeal and the elegant violet color it reveals but also because I had a bit of a history with this flower, not necessarily a pleasant one. When I was little, I used to be terrified of these flowers because they looked like butterflies with their wings spread. For some unexplainable reason, I was extremely terrified of butterflies at that age, so I would not even go near these flowers. But now, I am starting to see the beauty in them and beginning to appreciate and admire their beauty. Hence, I decided to dedicate this project to this particular flower in remembrance of my weird childhood fending off butterflies and violets.

Growth of Flower:

Bud:

Figure 2.1: Bud stage of violet

Growing…

Figure 2.2: Blossoming stage

Bloomed:

Figure 2.3: Fully bloomed

Initial Stages:

1. I had to decide how many frames I wanted to do and how long I wanted each to be. I initially decided on 10 frames from the bud growing to the fully bloomed flower. 1 second per frame would leave me with around 10 seconds of animation.

PROBLEM!!! I initially couldn’t find a full video of the violet flower blooming. But after a lot of searching on YouTube, google, and TikTok, I finally found one and it was perfect. Here is the link to the YouTube video: https://www.youtube.com/watch?v=KEovNQmIQBA

I screenshotted 10 photos from the YouTube video throughout the growth of the flower. ~Jan. 9~20

2. PROBLEM!!! How do I distinguish the different lines I want to make equations for from the irregular flower?

I decided to trace every frame on my iPad using an APP called Sketchbook. This way, I am able to make each line distinct and a whole lot simpler for the sake of using Desmos. ~Feb. 14 and throughout the holiday, the finished sketches are shown in Figure 2.

Figure 3: Traced sketches of different stages of violet’s growth

Graphing Stage:

1. I began by uploading my sketches to Desmos as a guide for when I graph, as shown in Figure 3.

Figure 4: Sketch uploaded to Desmos as a guide

The difficult part was figuring out how to write the polar equations. I sought help from the internet and my friends, who were super patient with me. Eventually, I was able to get the polar forms of the equations. And when I got the jist of it, it became quite easy, though it still took quite a while to plot every single equation.

1. Another PROBLEM I experienced was with the domains. For a while, I couldn’t figure out how to work out the domains for the equations, most significantly those equations where the domains are in several different quadrants. But I figured that if I copy and paste the equation several times and plot out the domain in sections, it would clear up most of the confusion (this was only for some lines, not all!).
2. As I was working, I noticed how much work and time needs to be put into my flower, so I decided to reduce the number of frames from 10 to 7, just to make sure that I have time to work on perfecting every petal and every line.

Final Steps:

1. Last minute, I decided that it would make much more sense to add a stem and leaves to the flower because without those features the flower looks quite bald. So I added an animated stem and 2 leaves, and it looked much prettier in my opinion. ~Apr 4–6

1. I debated on whether I would leave the flower colorful, giving each line a different color, or stick to the realistic aspects and make the flower purple. I finally settled on the second option. I just thought that it would make the flower look more realistic and make it easier to distinguish the type of flower I made. PROBLEM!!! I couldn’t figure out how to add more colors to Desmos, so with the limited shades of colors, the violet looked boring and plain. I asked for help from classmates after seeing that they knew how to add customized colors (Figure 4). This gave the flower more life and spirit. ~Apr 6
2. Lastly, I screen-recorded and screenshotted my frames to put them together into an animation.

Figure 5: Adjusting colors