Anamorphic Art: Effect of Distortion on Pictures
By Sari Zhongye and Sieun In
Introduction
Our project was inspired by Dr. Peter Tong’s AP Precalculus class at Concordia International School Shanghai. The project we created explores how changing the number of images affects their distortion when transforming between rectangular and polar coordinate systems. We experimented with varying the number of images from 2 to 9 and created a null hypothesis about the relationship. There is no significant difference between the distortion created by polar coordinates and the number of pictures present.
Fundamentals of distortions
The transformation between rectangular and polar coordinate systems can lead to significant distortions in how images and objects are represented. In the rectangular coordinate system, the x and y axes form a 2D grid according to its horizontal and vertical displacement, in polar coordinates, the position is defined by a radial distance (r) from the origin and an angular coordinate (θ). When converting an image from rectangular to polar coordinates, the straight lines and uniform grid structure from the original image become distorted. Straight lines in the rectangular frame are curved in the polar representation, and the spacing between grid points varies with the radial distance from the center. This non-linear scaling and warping effect is known as polar distortion. The degree of distortion increases, or is stretched more, the further away it is from the center point. Conversely, converting an image from polar to rectangular coordinates also introduces different types of distortions, such as the expansion of the image towards the edges.
Fundamentals of reflection
The curvature of a mirror’s surface significantly impacts how light reflects off and the resulting appearance of reflected objects or images. On a flat, planar mirror, light rays reflect at equal angles to the surface normal, preserving the overall shape and proportions of the original object. However, curved mirrors cause the reflected light rays to converge or diverge in complex ways. Depending on how the mirror is curved, it will reflect the picture in different ways. Convex mirrors curve outward and act as diverging lenses. They create a reduced, upright, virtual image of the original object. The degree of curvature, or the radius of curvature, directly determines the amount of distortion in the reflected image. Mirrors with higher curvature, or a smaller radius, exhibit more extreme magnification or minification effects. In our project, after the pictures are converted into polar coordinates, it is reflected onto a curved mirror to view. The curved mirror will counteract the distortion created and thus create an image without distortion.
Creating a distorted image
The process started by distorting 2 images.
However, we realized that we wanted to create a white circle in the center of the distorted image to place the mirror cylinder. To do this, we started experimenting with different formats.
After several tries, we learned that leaving a certain white space on the top of the image created a circle at the center after the distortion (see Figure 2). Moreover, the dimensions had to be a square for the middle space to be a perfect circle, not an ellipse.
To distort more images, we first planned out the dimensions so that the size of the circle formed in the middle could stay constant (See Figure 3). All the images had a 3:5 width-to-height ratio. The same process repeated for all the number of images from 2–8.
A pattern we noticed was that as the number of images increased, the size of the outer circle decreased (see Figure 4). This was due to the leftover white space on the bottom after formatting the pictures (see Figure 3). The white space surrounding the distorted images also increased when the number of images increased. Moreover, since all the images had to have a 3:5 width-to-height ratio, the size also had to get smaller as the number of images increased. Therefore, the size of the outer circle decreased as the number of images increased.
Reflected Photos:
Comparing pictures
We can see that all of the pictures can create a “perfect reflection”, the only difference is that as the number of pictures increases, the width decreases. This is because there is less space that can be occupied, as there are more pictures and a constant space. There is no significant difference between the reflected images, meaning the null hypothesis should be accepted. This is further supported by the law of reflection, which states that the angle of incidence equals the angle of reflection. On a convex curved mirror, which curves outward, the varying surfaces along the curvature cause the reflected light rays to remain linear and without distortion. As a result, when an image is reflected off a convex mirror and then transformed into polar coordinates, the linear nature of the reflections ensures there is no additional distortion introduced by the coordinate system change. This property makes convex mirrors useful for accurate reflections, regardless of the inherent minification effects (see Figure 12).
Result & conclusion
In conclusion, the change in the number of pictures affects the size of the reflection from the mirror. However, the level of distortion and how it is reflected stays the same due to the law of reflection and the counteraction between the curved mirror and polar distortions. A curved mirror, in this case, the mirror cylinder, distorts a picture in a way that is similar to polar distortion. And since the picture was already distorted, from rectangular to polar coordinates, the reflected image from the curved mirror depicts the original photo before the distortion.
