Art Through Curved Reflections

Introduction

This project is created by Steve Nie and Andrew Wang from Dr. Peter Tong’s AP- Precalculus class at Concordia International School Shanghai.

Logic of Cylindrical Anamorphs

Curved mirrors can manipulate our perception, twisting the size and form of shapes through reflection. This phenomenon can be modeled and explained through reflection principles and geometric transformations.

Figure 1: The cylindrical mirror will “curve” the lines

1. Reflection on Cylindrical Surfaces

Now imagine a simple shape lying in front of a cylinder with a reflective surface, such as a horizontal and vertical cross line. Its reflection deviates from the original form.

- Horizontal Line

The horizontal line reflects inwards, conforming to the curvature of the cylinder. Mathematically, this transformation can be described as a barrel distortion, where the center of the line remains fixed, and the outer points are progressively shifted inwards as they move away from the center.

- Vertical Line

While the vertical line appears to maintain its straightness, a closer look reveals a change in its perceived height. This is due to the perspective introduced by the viewing angle. Depending on the angle of depression (the angle below the horizontal plane at which we observe the reflection), the reflected vertical line can appear compressed, undistorted, or stretched.

2. Angle of Perspective

- Angle of Depression

The angle looking down at the reflection determines the extent of distortion in the reflected image. Looking down at the reflected picture from various perspectives provides a variety of unique views, as the photographs above demonstrate. Due to the pictures’ limited quality, I highlighted the differences by adding a red line next to the original designs. Let us now designate θ as the angle of depression:

Figure 2:θ ≈ 10°

θ ≈ 10°

Looking at the reflection with a low viewing angle, the reflected vertical line appears compressed. This can be modeled by a vertical scaling factor of less than 1, effectively squeezing the line’s height to approximately 1 centimeter.

Figure 3:θ ≈ 50°

θ ≈ 50°

As shown in figure t a mid-range viewing angle, the reflected vertical line appears undistorted, similar to the original line. In this case, the vertical scaling factor becomes approximately 1, preserving the original height of around 7 centimeters

Figure 4:θ ≈ 75°

θ ≈ 75°

At a high viewing angle, the reflected vertical line appears stretched. This can be represented by a vertical scaling factor greater than 1, elongating the line’s height to nearly 11 centimeters.

Law of Reflection

The familiar experience of looking into a mirror and seeing a near-perfect replica of ourselves arises from the law of reflection. This law dictates that upon encountering a flat surface, the exit angle of the reflected ray is equal to the entry angle of the incident ray. This one-to-one correspondence between incoming and outgoing rays preserves the image geometry, resulting in an undistorted reflection.

Figure 5: Distances between the red, grey, and black rays are constant before and after the reflection, forming an exact picture as before.

However, reflection off of curved surfaces deviates significantly from this simplicity. While the law of reflection still applies to each individual light ray, the interactions between the curvature and reflection angles lead to a significant modification of the overall image.

Figure 6:A drawing of possible reflections on curved surfaces

To reflect the same image as a flat surface would do on a curved surface, the original image must be scattered. This scattering process would, essentially, distort and rearrange the image data to account for the varying curvature at different points. Relating back to the cross placed under the cylinder, imagine the vertical and horizontal lines as x and y axes; the grid lines would distort and bend as they conform to the curve, creating a warped reflection that cannot be achieved with a simple one-to-one reflection.

Figure 7:Scattered reflection on curved surface

Despite that, the bend reflection off a curved surface still conforms to the law of reflection in essence, the difference is that the angle of entry and exit is no longer determined by a vertical line shared by all rays. Instead, the reference surface where the angle of entry and exit of the rays bouncing off a curved surface is determined by a distinct tangent line to the point where the ray contacts the mirror and reflects off it. The tangent line, unique to each point of contact between the light ray and the curved surface, dictates the angle of reflection. With infinitely many such tangent lines possible on a curved surface, each incoming light ray undergoes a unique reflection process.

Implementation

With the concept mentioned earlier in mind, creating anamorphic art using a reflective cylinder simply requires distorting the image you found from a rectangular to a polar grid style. There are two major ways to distort the image: using the “grid transfer method” or software, such as Photoshop.

Transforming to polar coordinates by hand

The most direct way to transform to polar coordinates is to transform each point in rectangular coordinates to polar coordinates using the mathematical equations provided below:

Figure 8:calculations for transformations

But there is a much simpler way; you first draw a rectangular grid with labels, then draw a circular grid where the base of the square grid corresponds to the side of the circular grid closest to the mirror. Remember that images in mirrors are reversed back to front, you must take account of this in the labeling.

Design or draw the picture on the square grid, then use the labels on the two grids to transform the drawing into polar coordinates, creating the anamorphic version.

Figure 9:using grids to transform coordinates

Transforming to polar coordinates using Photoshop

For this project, since we are dealing with more complicated images, we used Adobe Photoshop. Here is the detailed process of how we created the image:

Figure 10: This is the picture we used.

1. Make the Image Square

• Open your image in an image editing software (Photoshop).
• Go to Image > Canvas Size.
• Set the width equal to the height of your image to make it a square. This might crop parts of your image if the original was not square

If the image is cropped, then:

• Select the entire image (Ctrl+A or Command+A on Mac).
• Press Ctrl+T (or Command+T on Mac) to transform the selection.
• Compress the image horizontally to fit within the square canvas. This makes sure the entire image is visible within the new square dimensions.

Figure 11:the squared version of the image

2. Adjust for Half-side Visibility

• Since the cylindrical mirror only shows half of the image (π radians or 180 degrees), you need to adjust the image accordingly.
• Add white space to each side of the image. The total width of the image with white spaces should be double the width of the compressed image, effectively placing the compressed image in the middle half of the canvas. This ensures the compressed image occupies the central 180 degrees of the cylindrical mirror’s reflection.
• You can do this by increasing the canvas size again and aligning the compressed image in the center, or by adding white rectangles to either side.

Figure 12:adjusted image

3. Convert to Polar Coordinates:

• With the adjusted image, go to Filter > Distort > Polar Coordinates in your image editing software.
• Choose the option to convert from “Rectangular to Polar”.
• This will wrap the image around a circle, mimicking how it will appear when reflected in a cylindrical mirror.

Figure 13:polar version of image

The resulting image will look distorted, but when viewed in a cylindrical mirror, it will reconstruct into the original, undistorted image.

Final result!!!

Figure 14:reflection in cylinder