Anamorphic Art: Distorting Images to Create Meaningful Reflections
By Harper Cryer
Introduction
Throughout the year in my AP Precalculus course at Concordia International School Shanghai, we studied different transformations. We learned how to simply move a graph up and down a few units or stretch and compress it as well as how to convert coordinates from cartesian to polar. When it came time for our final project, Dr. Tong wanted us to display these transformations in the form of physical art pieces. From the options given, I chose to pursue the route of anamorphic art in the form of distorting an image and using a reflective cylindrical object to show it as it should be. Dr. Tong encouraged me to distort the Shanghai skyline as it’s a beautiful scene and means a lot to people here in Shanghai. While I did that, I also decided to pursue other pieces I also had a connection to.
Process
Trial and Error
At first, I was unsure how to create a piece that would be correctly scaled and reflect properly onto a cylindrical object. However, upon doing some research I found tutorials on how to do so by hand which involved creating grids and hand sketching your picture. While this option looked plausible, I knew it’d be too complicated to try and sketch the skyline in the detail I wanted.
After more research, I found that it was possible to distort your image using photo editing. I mainly worked on Photopea.com which was extremely simple for me to use and let me do everything I wanted. At first, I didn’t know how the scaling would work or anything of that sort, so I quickly put together a first test as seen in figures 1–3.
Once my image was printed, I was able to test the distortion and see if it worked. But as you can see in figures 4 and 5, the reflection didn’t look exactly like what the original image looked like. I realized the issue was with the size or circumference of the objects I used. In the image with the silver beaker, you can see that it’s placed farther away from the image than with the gold plant pot. This is because when the beaker was placed too close, the image was almost unrecognizable. I think this had a lot to do with the lip or added curve to the beaker that the plant pot didn’t have. Luckily. I also had access to a mirror sheet that could fold into the needed shape and therefore reflect an image more accurately than the beaker and plant pot.
Shanghai Skyline
Once I felt confident in trying to overcome the scaling issues I experienced in the previous step, I decided to move on to transform the Shanghai skyline. When I chose the picture, I knew I wanted it to be at night since that’s when it’s lit up. And rather than search for a generic image online, I chose one I took on my prom night. My friends and I had gone to the bund after prom to take pictures together and I captured this one. I think by using a picture of something so beautiful that I took myself, made this project especially meaningful. In my first attempt, I yielded the result in Figure 6:
Although it’s accurate and reflected properly, I couldn’t see the entire skyline when looking at the mirrored object straight on. I had to rotate the piece to see it since it’s a closed circle. I realized this is because the picture I had put into Photopea took up the entire space as seen in Figure 7.
I was able to fix this and make it so that you could see most of the skyline from one direction. I did so by putting the image on a white background and scaling the image down, as shown in Figure 8.
This gave me the distorted image in Figure 9, which worked a lot better than the previously created distortion.
This entire process was really enjoyable and showed me a lot about how the smallest details can change a distortion greatly.
Additional Experimentation
Since we were given a lot of class time to experiment with our projects and test multiple outcomes, I decided I wanted to see if animation could be reflected after distortion. I applied my previous knowledge, such as scaling the image to not fill the entire space and making sure each frame is aligned to do so. I’ve recently stepped into the world of Studio Ghibli films and thought it’d be fun to transform a scene or short clip from a few of my favorite films. I chose to find a few GIFs since they are easier to download and access than full scenes and because they don’t tend to have too many frames. After choosing the GIF and downloading it, I uploaded it to Photopea and followed the same process I had established earlier (seen in figure 10).
Once I had distorted and downloaded all the frames, I needed a way to make the images into the GIFs again. I determined that the best way to do that would be to put each image into iMovie and shorten each to 0.1 seconds (pictured in figure 11).
Once I tested to ensure this worked and was reflected correctly, I decided to make a couple more. I ended up with distorted GIFs of Calcifer from “Howl’s Moving Castle”, Kiki from “Kiki’s Delivery Service”, and Ponyo from “Ponyo”. The reflected Calcifer and Ponyo GIFs can be viewed additionally through these links: Calcifer and Ponyo. Although this process was a bit tedious, having to transform each frame, it was worth it since I enjoy these movies and can see how my hard work paid off.
Explanation
What Anamorphic Art Is
Britannica defines anamorphosis or anamorphic art as a “perspective technique” where a distorted picture can only be correctly “viewed from a particular angle, or reflected in a curved mirror”. Britannica also cites works from Leonardo da Vinci’s notebooks as early examples of anamorphic art.
Although da Vinci was the first, he certainly wasn’t the last. You might see this technique in street art, at illusion museums, or even projects like this one. I’ve personally seen many clips of this art form online in the form of chalk drawings on sidewalks, small paper drawings made to look 3D, etc. I’ve even been to a museum solely dedicated to optical illusions here in Shanghai. Anamorphic art can take many different forms and has come a long way since da Vinci first made his sketches.
Mathematical Connection
Throughout the process, Dr. Tong encouraged students to approach their pieces first from an artistic and creative standpoint and then learn about mathematical connections. This is because he believes in the importance of hands-on learning and real-world applications rather than learning formulas and plugging in variables. I think this approach was very valuable in the sense that I wasn’t bored by this project, nor did I just follow directions without knowing why I was doing certain things.
Once my pieces were done, I was able to look more deeply into the math behind anamorphic art and found the following formulas from Kimberly Rausch’s paper, “The Mathematics Behind Anamorphic Art”
As you can see in Figure 12, the formulas Rausch brings up account for variables such as the viewer’s position and radius of the cylinder. This helped me understand why certain objects weren’t working in reflecting my distortions and why some just needed to be moved farther from the image.
I found another source titled “Anamorphic Art” which helped explain why cylindrical objects require different calculations. It explained that for flat mirrors “the angle of ‘entry’ is the same as the ‘exit angle’” (figure 13) whereas for curved mirrors, “the angles are measured against the tangent line” (figure 14). The same source provided the following diagrams which helped me visualize and fully understand what was happening with the curved mirror.
Final Reflection
Upon being assigned this project, I was unsure of how this related to our coursework or even math in general. But after being able to spend time with different transformations and distortions, and test how variables such as the image’s size/ positioning, my viewing point, the size of the mirrored object, etc. I feel confident in my understanding of how anamorphic art and precalculus connect.
Citations
Encyclopædia Britannica, inc. (n.d.). Anamorphosis. Encyclopædia Britannica. https://www.britannica.com/art/anamorphosis-art
The mathematics behind Anamorphic Art. (n.d.). https://archive.bridgesmathart.org/2012/bridges2012-513.pdf
Rausch, K. (n.d.). Anamorphic art — integral domain home. https://integral-domain.org/lwilliams/math110/Art/Fall2018Docs/AnamorphicArt.pdf
Turón, L. (2020, August 21). Anamorphic drawing workshop by Laura Turón. YouTube. https://www.youtube.com/watch?v=dq2oxN2l7gk
