Utilizing Forced Perspective and Visual Angles to Create Shadow Art of Doctor Tong and Astro Boy
This year, in Dr. Tong’s AP Precalculus class at Concordia International School Shanghai, the cumulative project was a 3D-based challenge to either demonstrate the relationship between the cartesian and polar axes by reflecting images on a round reflective surface or to create shadow art to demonstrate forced perspective and angles. Astro Boy is a popular Japanese anime character, a character we see very often because Doctor Tong uses a 3D figurine of him to hold rubber bands. As we were sitting at our desks one Wednesday afternoon, an enlightening thought crossed our minds as we looked at our math teacher sitting at his desk: “What would it look like if we carved his face into cardboard and shone a light on it to cast a Dr. Tong-shaped shadow onto a random wall? It was this thought that eventually blossomed into a shadow art piece of the iconic astronaut robot and our even more iconic math teacher.
Cardboard Carving Process
First, we sketched out the shapes onto cardboard. We split the two images we were projecting. For Astro Boy, we first sketched out the character, rocket, and stars onto a piece of cardboard, and then carved out the shapes by following our guiding lines, as seen in Figure 2. We carved out the parts that we wanted shaded, such as his hair, eyes, clothes, the window in the rocket, the star, etc. For the picture of Dr. Tong, the process was a little more complicated than that of Astro Boy.
Firstly, we found a picture of him from Google and drew over it, using only black and white to block out the parts we were to cut and keep on the cardboard. Secondly, we printed out the picture and placed it over a piece of cardboard, using an exacto knife to carve the picture into the cardboard. Upon removing the picture, there was a faint outline of Dr. Tong’s face that was still visible. From there, it was only a process of carving out the main facial features — the eyes, eyebrows, nose, mouth, and of course, glasses — as well as parts we had previously blocked out like the hair, tie, and arm as seen in Figure 3. After anywhere from 45 minutes to 2 hours — time sure flies by when you’re having fun! — all that was left was to refine the cardboard cutout with sandpaper.
There were a few unfortunate instances where Dr. Tong was dismembered and decapitated, but aside from that and a few first-degree burns, we achieved a cutout that fellow peers viewed as a replica of Doctor Tong that captured his essence wholly and definitively. Unfortunately, Doctor Tong viewed it as “pieces of scrap.” Nonetheless, the main components of our shadow art piece were completed.
Assembly Process
After carving out the pieces, we began gluing them onto a cardboard base. We intended for one side to have Astro Boy, the rocket, and stars, and the other to have Doctor Tong and the Concordia building. We achieved this by utilizing forced perspective and visual angles. For Astro Boy, we decided to bisect him from the shoulder down to his legs, cutting him in half. We then placed the half with his head in front and the half with the remainder of his body behind. However, we had failed to account for the fact that the distance between the two halves of his body would make it so that they would not align in a shadow. Therefore, we had to make slight adjustments to half of Astro Boy’s body with his head attached by shifting his pants higher up through the use of carving and pasting more cardboard on. Having accomplished the most difficult part of this side, all we had left to do was to stick down the rocket and star for the finishing touch, as evidenced in Figure 4.
For Dr. Tong, we separated Dr. Tong’s arm from his body — on purpose, this time — and stuck it to one side of the Astro Boy’s detached arm and body while the rest of his face was attached to the other side of the rocket. To block Astro Boy’s head from showing on Dr. Tong’s side, we carved out a Concordia building and attached it to the side of Astro Boy’s hair. To make adjustments for the distance of Dr. Tong’s shoulder, we simply had to remove 2 centimeters off the bottom and it once again fit with his body. The only problem we encountered during this process was that wires were required to hold up Dr. Tong’s head (figure 5）, as the previously mentioned decapitation had significantly weakened the structural integrity of the cut-out. Besides that, this side was mostly smooth sailing as well.
The Science Behind Shadows
Shadows are the absence of light formed when an object blocks space from the light hitting it. Depending on the intensity, angle, and distance of the light source, the shadows will transform accordingly. For example, similar to forced perspective for objects, the closer a light source is to the object it shines light on, the larger the shadow will be. Because shadows are formed due to the lack of light, they do not have values like what we naturally see. Instead, images formed from shadows are composed of black (the space the object with light shining on takes up) and the background color (space where the light shines through naturally). Each shadow has three parts, the umbra, penumbra, and antumbra, written in the order of the components of a shadow from darkest to lightest.
The Psychological Aspect of Perspective and Perception
The reason objects in the distance look smaller is due to our eyes’ binocular cues, which is information received from both of our eyes that helps the brain to perceive depth and distance. Retinal disparity refers to the way we view the world from two slightly different angles due to the (approximately) 2.5-inch space between our eyes. Our brain accommodates the space by filling it in and making sure our vision doesn’t overlap too much. Therefore, the carveouts and shadows of the project will look slightly different for different people depending on their positioning and degree of retinal disparity. Objects growing smaller and closer together in the distance happen because our eyes move further apart to accommodate for the increased distance, hence making parallel lines ‘converge’ in the distance. This ties into the relative size of an object which determines the object’s size at several distances — smaller when farther and larger when closer. Closure is a psychological concept to explain positive and negative space, where our brain automatically fills in the gaps in a picture when we see incomplete objects. This can be seen especially in the carveout of Dr Tong, where the brain automatically fills in the gaps for the carved-out shirt, eyes, nose, and mouth to see Dr. Tong instead of random scraps of cardboard.
The Mathematical Aspect of Perception and Perspective
The visual angle or angular size of the object is how much of our field of view said object takes up. When we move closer to the object, the visual angle increases, making the object look larger. When we are further away, the visual angle decreases, making the object look smaller （figure 6). Hence, to accommodate for the object’s angular size, different parts of the cardboard cutouts must be different sizes so that from a distance, they all look to fit together and the shadow will be a complete whole when light is shined on the cutout. Stereoscopic vision is the ability to see the world in a 3 dimensional way using both eyes. Triangulation, therefore, is a trigonometric solving method where we solve for the length of the triangle using D as the distance between our eyes and the angle of our image:
Force perspective is a technique that utilizes optical illusions to alter the distance or size of an object as opposed to the actual distance/distance. People use force perspectively most often when they pretend to hold up the Leaning Tower of Pisa in pictures despite the tower being much further and larger than normal. With h representing height and D representing distance from the lens, the forced perspective formula is:
Conclusion
Using these formulas as a basic guideline when carving the shapes, such as adjusting the size of the cutouts based on their distance to the light source, we created a shadow art piece that projects different images when rotated. With the tools and knowledge that our math teacher had given us, we were able to apply these skills in a real-world situation to capture not only our teacher but also his alter ego — Astro Boy.
Citations:
The Science of Forced Perspective at Disney Parks. (2020). United States. Youtube.com.
Meyer’s, D. G. (2018). Meyer’s Psychology for AP (3rd ed.). Worth Publishers.
Michael A. Seeds, (2010). Stars and Galaxies (7 ed.). Brooks Cole. p. 39.
Pogge, Richard. “Lecture 9: Eclipses of the Sun & Moon”. Astronomy 161: An Introduction to Solar System Astronomy. Ohio State University.
