Transformations: Exploring Mathematics through Shadow Art and Mirror Anamorphosis
Introduction
This year, Dr. Tong challenged his AP Precalculus students to take on a project exploring mathematical transformations through art. Drawing inspiration from the concepts of shadows, perspective, as well as our lessons on polar and rectangular coordinates, I set out to create a unique blend of shadow and reflective art for my project.
Finding images to incorporate into my project proved to be quite challenging. My objective was to combine elements that resonated with me on a personal level while producing a captivating and imaginative piece. During my search for inspiration, my attention landed on a keychain featuring one of our school mascots, Spark. However, a lingering question remained: how could I connect this to something creative?
Amid despair, inspiration struck as I noticed a can of chickpeas on my kitchen table. It became clear that I could use the can as the base for the reflective surface, as I needed a cylinder mirror. My mind immediately jumped to the can of spinach that was beloved by my favorite childhood cartoon character, Popeye. Thus, my vision took shape — my artwork would showcase a shadow of Spark on one side, while on the other, it would portray Popeye alongside a reflective can projecting an image of a can of spinach.
Creating the Shadows
I started my project by sketching and cutting out the images of Spark and Popeye. However, I encountered my first problem when I tried to see how the cutouts took shape as shadows. Initially, I had divided the images into several smaller pieces, intending to arrange them at varying distances to create an anamorphic shadow art. Unfortunately, I had not taken into account that differences in the distances of the cutouts would impact the projection of the shadow, as depicted in Figure 1.
Realizing the direct relationship between shadow sizes and the distance of the light source, I had to redo certain sections by proportioning them accordingly to ensure that the sizes aligned correctly in the shadow. To achieve this, I scaled the pieces, making those closer to the light source smaller and those further away larger. This was done because objects in closer proximity to the light cast larger shadows, while greater distance results in smaller shadows. Despite some trial and error, I successfully aligned the pieces, resulting in a cohesive shadow, as illustrated in Figure 2 and Figure 3.
Reflecting Spinach!
Moving to the reflective aspect of my project, I began by drawing out my image and started using the “warp” function on Procreate. This function enabled me to manipulate the picture in various ways, such as enlarging or elongating specific parts of my image as desired, as evidenced in Figure 4.
As a guide, I started my distortion process by using my knowledge of polar and Cartesian planes as a framework.
A Cartesian, or rectangular, coordinate system is likely the most familiar to us. It consists of a coordinate plane with an x-axis and y-axis, both having the same unit length, as seen in Figure 5.
This type of coordinate system solely measures the horizontal and vertical distances of a coordinate from the origin point, which is expressed as (x, y). On the other hand, the polar coordinate system depicted in Figure 6 measures the angle of a point from a reference direction and the distance of a coordinate point from the origin, also known as the pole. The polar coordinates are expressed as (r, θ), with r representing the radius and θ being the angle.
But how does this connect to reflection? Because I am using a cylinder reflective surface, the curved nature of the surface makes it so that incoming rays from the image are reflected in various directions, as depicted in Figure 7.
Therefore, in order to view an undistorted image on a cylindrical mirror, the original image must be distorted in a way that ensures the convergence of each ray of light when it hits the mirror. This convergence ultimately leads to an undistorted image for the viewer, as depicted in Figure 8.
One way to distort the image so that the image reflected could appear normal is by converting Cartesian coordinates to polar coordinates. When a cylindrical mirror is positioned at the correct distance, polar coordinates can be transformed into Cartesian coordinates. In Figure 9, the half-circle represented in polar coordinates is straightened into lines on the reflective surface of the cylinder, while the radial lines become perpendicular to the lines formed by the half-circles.
Although it required experimentation and refinement, I was ultimately able to apply these concepts and create a distorted image of a spinach can that appeared normal when reflected on a cylindrical mirror.
Video Outcome!
References:
“Polar Coordinate System — Definition, Formula and Solved Examples.” BYJUS, BYJU’S, 11 Dec. 2019, byjus.com/maths/polar-coordinates/.
“Exploring Anamorphosis: Revealing Hidden Images with Mirrors.” Science in School, 4 June 2024, www.scienceinschool.org/article/2024/exploring-anamorphosis/.
“Cylindrical Mirror Art.” Instructables, 22 Mar. 2018, www.instructables.com/Cylindrical-Mirror-Art/.
Godiner, Jane. “What Is Anamorphic Art? Unveiling Hidden Dimensions in Reality.” Invaluable, 1 Nov. 2023, www.invaluable.com/blog/what-is-anamorphic-art-unveiling-hidden-dimensions-in-reality/.
“Precalculus.” Lumen, courses.lumenlearning.com/csn-precalculus/chapter/rectangular-coordinate-system-and-graphs/. Accessed 6 June 2024.
