Polar Equations Project: Strawberry

By Ryan Tang

Throughout online learning, Dr. Tong’s Honors PreCalculus class at Concordia International School Shanghai had the opportunity to expand their scope of understanding of unit 10 on polar equations by composing a project on various types of fruits by utilizing the Desmos online graphing calculator. I decided to use polar equations to represent a strawberry, since it is my favorite fruit, on a 2D plane.

To begin with, I explored the internet searching for a reference picture to base my project on. The final image I have decided on using is the picture below. The criteria for the reference picture were its image quality and positioning. Luckily, this image meets both the expectations and it can show both the inside and the outside of a strawberry.

Prototype:

After deciding on the reference picture, I began constructing my first prototype of this project. As I have learned in unit 9 of the Honors PreCalculus curriculum, a cardioid provides a shape similar to the body of a strawberry, hence, I tried to incorporate this design in my first prototype.

After various attempts at reorganizing the equation, I realized that it was impossible to create a representation of the body of the strawberry with a cardioid. I then gathered information from the internet and discovered a way to represent a heart shape on polar equations other than a cardioid.

As shown in the picture above, I found another way to represent the body of the strawberry. After testing out the values, this was the closest to the reference picture. Following this, I discovered a way to represent the leaves on the strawberry online using minimums.

Above was my very first prototype for this project, however, I soon realized the many insufficiencies of the representation. It only provided a general shape of the fruit but it was not detailed enough.

Right Strawberry:

Then, I changed gears and headed on to my second prototype where I approached this project from another perspective.

For my second prototype, I tried to manipulate parabolas, lines, ellipses, cardioids, and their restrictions to form the boundaries of the strawberry.

I used four cardioid graphs forming the boundaries of the upper strawberry and manipulated the restrictions for the angle theta. This way, it provided a much more accurate representation when compared to my prior prototype.

After this, I moved on and created the boundaries for the rest of the strawberry placed to the right of the axis using parabolas and line segments. But before I could graph the strawberry on the polar axis, I first had to plot the equations on the rectangular axis.

To convert these equations into their polar representations, I first substituted the x and y with their relative polar representations which are rcos θ and rsin θ. Then, to isolate the variable “r”, I would have to utilize the quadratic formula solving for “r”. Before I could input my values into the formula, I had to set up my equation in the standard form of a quadratic, 0=ax²+bx+c. Then, using the quadratic formula would result in the polar equation shown below.

For a quadratic equation, such as the y=(x-0.1)²–3.23, the converting process would be as follow:

For a line, such as the y=-1.98x-3.8, the converting process would be as follow:

After deriving the polar equations, I then restricted the various graphs so that they could best fit the boundaries of the strawberry using trial and error, leading to the product below.

I then repeated this process for the rest of this strawberry, including its leaves, and the stems inside. Part of the equations is shown below.

Left Strawberry:

After I completed the strawberry placed to the right of the axis, I then began to construct the one on the left that showed the skin of the strawberry.

I followed the same procedures as I did before by outlining the boundaries of the strawberry, which included the body and the leaves, and converted rectangular equations to their polar representations. Part of the equations is shown below.

After I have derived the polar representation of these equations, I then had to choose the sections of this graph that were useful to my project by using restrictions. The restriction {-1.48<rcosθ<-0.63}, as shown in the third equation above, successfully limits the x value of this graph to the section that would be most useful for the strawberry.

Now that I have constructed the outline of the strawberry, I then began to create the seeds on the skin. I decided to use small ellipses to represent these seeds.

To construct these ellipses, I used the same method as I did before, where I first arrived at the rectangular equation, then converted it to its polar representation using the concept of x=rcosθ, y=rsinθ, and the quadratic formula. I tried to keep the size of the ellipse a constant, making the converting process slightly easier since there would be fewer variables for me to edit.

For the first ellipse equation shown above, the conversion would be as follow:

For the second ellipse equation shown above, the conversion would be as follow:

As you can see, there are only 3 variables that I have to adjust for the different ellipses which are the sinθ in the b term, the cosθ in the term, and the c term where the a term was kept constant. This is because we are only shifting the ellipses while keeping the size of the ellipse constant. And because I am shifting most of the seeds by the interval of 0.3 in the x coordinate and 0.1 in the y coordinate, the change in variables is very predictable as the value moves changes by 0.006cosθ and 0.001 to the sinθ in the b to the b term. This way, it significantly shortened the amount of time needed for me to construct the seeds for this strawberry, making it more efficient.

By repeating the steps above to convert ellipses from rectangular to polar for the rest of the seeds, I completed the 31 seeds on this strawberry. Below is the completed strawberry with some of the equations shown on the side.

After I completed the graph, I then colored each of the lines on the graph according to the parts of the strawberry. The body will be colored red, the leaves will be green, the seeds will be colored black, and the stem will be colored orange. The stems are purposefully chosen as dotted lines to show that they are inside the strawberry.

I then repeated this process for the rest of the project and concluded with 144 polar equations, some of which are shown below.​​

This is the link to the full 144 equations: link

Reference Picture:

Final Strawberry Graph:

Final Superimposed Graph:

Conclusion:

This project enhanced my learning of the concept of polar equations taught in unit 10 of the Honors PreCalculus curriculum since I can now utilize equations and their restriction, convert between rectangular and polar, and create representations on a graph for any object with ease. This project effectively concludes my learning on “Topics in Analytic Geometry” as it not only incorporated a vast range of knowledge/concepts covered in this unit but also drew outside connections with the past units.

This is a project done by the Honors PreCalculus curriculum at Concordia International School Shanghai.