How Does the Placement of a Reflective Cup on the Ridged Saucer Change the Resultant Reflection?
By Megan Wong, Zoe Gong, Sophie Yasunaga
Abstract
A new transformational technique has been discovered by Yul Cho and Sang-Ha Cho, using the distortion of reflective surfaces to create artwork. This technique may prove to be a useful tool in generating anamorphic art, allowing complicated designs to be creatively incorporated into objects.
Introduction
In Concordia International School Shanghai, Dr. Tong gave us the opportunity to showcase the subjects we learned in class as memorabilia. We wanted to create a piece that showed that on our math journey, one of the key teachers who had supported us was Doctor Tong. So, our project showed our faces, Zoe Gong, Megan Wong, and Sophie Yasunaga, reflected on Dr. Tong’s face, representing how the skills he taught us were carried on with us for the rest of our math career.
We decided on the project we wanted to do by looking at different forms of transformational art- these included shadow art, distortion art, reflective art, etc. Ultimately, we decided to create a pattern on a saucer and use a mirror cup to distort it into the picture we wanted. This form of art uses anamorphic techniques, transformations occurring from the distortion of the reflective cup. Through this experiment, we hope to explore how different ridges will change the way the same image is reflected.
Experiment Setup
The independent variable was the type of saucer used. In this experiment, we used two types of saucers- A saucer where the cup was to be placed in the center, and another where the cup was to be placed near the edge. The control variable was the images used- both saucers had the same distorted image, which was altered according to what would reflect perfectly on a flat, angled surface. Because of this, we knew neither saucer would have a perfect fit for the reflection, but it was the best compromise; it would have been a biased experiment had we tailored the image to a ridged surface, as then it would already be a better fit to one or the other saucer, thus rendering the experiment pointless. The dependent variable was the reflection we would record from both cups.
How We Made Our Mirror Cup and Saucer
Below is the finalized design for our saucer. The outward faces of the ridges will depict our three group members, and the inward faces our teachers. This will give the effect of our faces on the saucer, and our teacher’s face reflected within the mirror cup. The digital drawing seen in Figure 1 was made with Procreate and distorted with the Liquify tool.
The model of the saucer was made using Fusion 360. We created a flat ellipse plane and added curved ridges which would hide the pattern being reflected in the saucer. Two iterations were created, one where the cup would be placed in the center as shown in Figure 2, and one where the cup would be placed on the edge.
After the two models were complete, we 3D printed them using a Bambu Studio 3D Printing Machine. The final models were constructed of filament as seen in Figure 4.
Once we had the physical models of both saucers, we printed out the patterns on paper and cut them to fit into the ridges. Because the height of the ridges was two millimeters, the width of the paper also had to be two millimeters to avoid blocking any part of the pattern. Because the ridges were curved, we also had to cut the paper strips in curved segments, with the inner circles having the most curves and the outer circles having straighter strips. The process of placing and gluing the paper strips is depicted in Figure 5.
Mathematical Principles
Inner circle: 3mm
Height: 1 cm
Inner angle: 50 degrees, radius: 7mm
Outer angle: 55 degrees, radius: 7mm
Diameter options: 3 cm (7 cm, 10 cm, 13 cm)
Observer angle:
Total Height: 16
Total Length: 39.5
Tan: opposite/adjacent
Tan theta= 39.5/16=2.468750
Angle: arctan(2.468750) = 1.185933 rad
Rad convert to degrees
(1.185933*180)/pi
=67.948956 degrees (a little more than pi/3)
The angle of the inner circle, 50 degrees, is an important factor in the formation of the reflected image. The inner circle’s segments must align with each other in a mirror to create a clear, undistorted image. If the inner circle angle is not aligned with the reflection angle, the reflected image may appear distorted or fragmented.
The law of reflection states that the angle of reflection is equal to the angle of incidence, also the incident ray. The point source of light in front of the flat mirror sends rays in all directions. According to the law of reflection, these rays that strike the mirror will reflect. The “source” is the virtual image of the original point source. It is called “virtual” because no light emerges from it, but it appears as a source of light.
In Figure 8, the rays that emerge from the point source strike the mirror at different angles. The mirror’s reflection occurs when rays hit it at different angles and meet at a single point. This is demonstrated by the line from the point source perpendicular to the mirror. The bisector drawn perpendicularly from the point source to the mirror is the line that cuts the triangle formed by the incidence ray and the extended reflection ray in half. This reflects the rays at a specific point behind the mirror, ensuring they all meet at the same point.
Additionally, the angle which the observer views the reflection is also important. At 67.949 degrees, the observer will see a non-distorted image. This was calculated with the height and length of the angle of the eye. The viewing angle affects the visibility of a reflection, with shallow angles preventing light from reaching the observer’s eyes and steep angles allowing clear reflections, provided the surface is properly positioned.
Example Cup and Saucer Art
For the saucer, we decided to experiment with two different models- in order to explore the impact of the placement of the reflective object, we planned to create one saucer with the cup in the center and another saucer where the cup would be placed off center. By looking for references for both types of plates, we found the images below:
Conclusion
The reflected image was surprisingly legible in both reflections, especially considering the distortion was created according to a flat plane rather than a ridged surface. Interestingly, the reflection was slightly better in the saucer with the cup in the center (light blue) than in the saucer with the cup on the edge (dark blue). This could be because there were significantly more ridges on the dark blue saucer, vertically elongating the picture in a way we did not account for.
It’s also important to consider sources of human error, as we cut the paper strips by hand rather than by a machine. This accounts for some variety in the width and curve of the paper, meaning the picture will not be perfectly reflected. The parts of the paper that came in contact with wet glue also shrunk slightly when the glue dried, causing further distortion. If we repeat this experiment in the future, we will use a paper cutter or protractor to cut the strips evenly and take more care when gluing the strips onto the saucer.
Overall, the experiment was successful and effectively demonstrated that while the cup’s placement will have a slight influence on the reflection produced by it, both can produce similar results.
