Exploring Anamorphic Reflections: Reflecting on a Year of AP Precalc

Figure 1 — my finished anamorphic reflection with the reflective spherical surface

Introduction:

Art and mathematics have a long-standing relationship, with mathematical principles often serving as the foundation for artistic creation. This interplay can be seen vividly in optical illusions, which play tricks on our perception and challenge our understanding of reality. For my final project in Dr. Tong’s AP Precalculus course at Concordia International School Shanghai, I decided to explore this fascinating intersection by creating an anamorphic reflection illusion using a sphere. This project not only allowed me to delve into the beauty of mathematics but also served as a tribute to Dr. Tong, whose guidance has been instrumental throughout the year.

Drawing inspiration from the natural world, I wanted to create a piece that combined mathematical precision with artistic beauty. Anamorphic illusions, which appear distorted until viewed from a specific angle or in a reflective surface, provided the perfect medium. My project specifically focused on a spherical reflection, where a distorted image becomes coherent and recognizable when viewed in the reflection of a sphere.

The aim was to produce a visually engaging piece that also embodied the mathematical concepts we learned, such as polar and rectangular coordinates. Through this project, I hoped to create a memorable artwork that Dr. Tong could cherish, reflecting the impact he has had on our class and celebrating our shared journey in mathematics.

Figure 2 — anamorphic art by istván orosz

Understanding Anamorphic Illusions:

Definition

Anamorphic illusions are images that are purposefully distorted in such a way that they look correct only from a certain perspective or when viewed in a reflective surface.

History

This art form dates back to the Renaissance. One of the most famous examples is Hans Holbein’s “The Ambassadors,” which features a distorted skull that comes into proper perspective when viewed from the side.

qqqqFigure 3 & 4 — anamorphic illusions as shown in ‘The Ambassadors’ by Hans Holbein

Types

There are different types of anamorphic illusions:

1. Perspective-based: These illusions depend on the viewer’s position. The image appears distorted when viewed from other angles.
2. Mirror/reflection-based: These illusions use reflective surfaces, such as spheres or cylinders, to correct the distorted image when viewed in the reflection.

Figure 5 & 6 — examples of anamorphic reflections on the surface of a cup, created by Yul Cho and Sang-Ha Cho.

My Project: Anamorphic Reflection with a Sphere:

For our final project in AP Precalculus, we were encouraged to apply the knowledge we’ve acquired throughout the year to real-world applications. Inspired by the fascinating world of anamorphic illusions explored in class and influenced by notable artists like Yul Cho and Sang-Ha Cho, I set out to create an anamorphic reflection illusion with my own personal twist.

Rather than opting for conventional methods like printing an image onto a surface, I decided to paint the image on a canvas. Additionally, while anamorphic illusions traditionally utilize cylindrical surfaces to achieve their optical transformations due to their predictable reflective properties, I sought to challenge myself by using a sphere as the reflective surface for my artwork. This decision stemmed from a deeper exploration into the complexities of reflection and geometry; I aimed to capture the spherical symmetry and dynamic visual effects that a sphere could offer, rather than the more commonly used cylindrical forms. Through preliminary research, I quickly realized the inherent difficulties posed by the spherical surface’s complex curvature.

The Mathematics Behind Anamorphic Illusions:

Anamorphic illusions captivate us by transforming distorted images into coherent visuals when viewed from a specific angle or reflected in a curved surface. This fascinating intersection of art and mathematics can be achieved using various shapes, with cylindrical and spherical surfaces being the most common.

Cylindrical Anamorphic Reflection

Understanding the Geometry:

Imagine a vertical cylinder with a consistent radius r, centered at the origin along the y-axis. The observer’s viewpoint is located at

and the image we want to reflect is placed on a flat image plane, with points denoted as

Mathematical Principles:

Angle Calculation: The key to cylindrical reflection lies in calculating the angle θ around the cylinder’s axis, which determines where the point P will appear on the cylinder’s surface. This angle is given by:

Reflected Point Coordinates: Using this angle, we can find the coordinates of the reflected point P′ on the cylinder’s surface:

Here, θ is the angle around the cylinder’s axis from the observer’s viewpoint to the point on the image.

Creating the Illusion:

When the observer looks at the cylinder, they perceive the point P′ on its surface. The transformation makes the distorted image on the plane appear normal and undistorted, creating an anamorphic illusion. The uniform curvature of the cylinder simplifies the reflection process. The primary transformation involves calculating angles in a two-dimensional plane.

Spherical Anamorphic Reflection

Understanding the Geometry:

Now, consider a sphere centered at the origin with a radius r. The observer’s eye is located at

and the image is placed on the base plane (z = 0), with points denoted as

Mathematical Principles:

Intersection of View Line and Sphere: To find the point Q where the line of sight from V intersects the sphere, we use the parametric line equation:

Here, t is a parameter, and 𝐼⃗ is the perceived image point within the view disk.

Solving for Intersection: The intersection condition with the sphere is:

Substituting the line equation into this condition, we solve the resulting quadratic equation for t to find Q.

Law of Reflection: The law of reflection states that the angle of incidence equals the angle of reflection. Mathematically, this is expressed as:

where 𝑛⃗ is the normal vector at Q, given by:

Calculating the Reflected Point: The point S on the base plane is found by extending the line from Q through the reflection of V off Q:

This line is then intersected with the base plane (z = 0) to locate S.

Creating the Illusion:

The observer perceives the point I within the view disk. The complex curvature of the sphere causes significant distortion, making the reflected image appear dramatically different from its original form. This heightened distortion creates a striking anamorphic effect. The sphere’s varying curvature adds complexity, requiring three-dimensional geometric calculations and trigonometric transformations. The distortion varies with curvature, leading to more complex and less predictable patterns. Points near the edges of the sphere’s reflection appear highly stretched.

The Process of Crafting Anamorphic Reflections:

Creating anamorphic illusions is a blend of mathematical precision and artistic experimentation. My project involved reflecting images on both cylindrical and spherical surfaces to observe how these shapes distort images and find the correct transformations to achieve the desired undistorted view. Here’s a step-by-step outline of my process:

1. Selecting Images and Initial Setup:

The first step was selecting images that I wanted to transform. I chose a variety of scenes to understand how different types of images would behave when reflected.

Figure 7 — an example of one of the images I tested with distortion.

2. Initial Reflection Trials:

I started by projecting these images onto the spherical surface using various angles. The aim was to see how each image would distort and what angles would yield the best results. This involved:

• Positioning the image at different points on the base plane.
• Adjusting the observer’s viewpoint (the angles from which the image would be viewed).

Figure 8 — testing a distorted scene to see how it would appear when reflected in the sphere.

3. Trial and Error:

This phase was heavily reliant on trial and error. I would apply an initial transformation, reflect the image on the sphere, and then assess the result. If the image appeared distorted, I would:

• Recalculate the angles.
• Adjust the position of the image on the base plane.
• Repeat the reflection process until the image appeared undistorted in the sphere.

4. Refining the Transformation:

Once I found transformations that worked, I applied these transformations to the images I intended to use in my final artwork. This refinement ensured that the images would appear as intended when reflected on the cylindrical and spherical surfaces.

5. Painting the Transformed Scenes:

With the correct transformations in hand, I began the artistic phase of the project.

The final step was to transfer the transformed scene onto my canvas. I meticulously painted the distorted version of the image, knowing that when viewed in the sphere, it would appear as the intended, undistorted scene. I meticulously painted the transformed scenes onto my canvas, ensuring that each point and angle matched the calculated transformations. This step required careful attention to detail to preserve the integrity of the anamorphic illusion.

Figure 9— the final artwork which features two anamorphic illusions — a starry night sky with a lake and a vast green field with rivers, clouds, and mountains.

6. Final Adjustments and Observations:

After completing the paintings, I viewed the reflections from the intended viewpoints. Any minor distortions were corrected through further adjustments to the painting, ensuring the final reflected image appeared coherent and undistorted.

Figure 10 — one side of the canvas showing a starry night sky with a lake when reflected in the sphere.

Figure 11 — the other side of the canvas featuring a vast green field with flowers, rivers, clouds, and mountains when reflected.

My final rendition for this project included two anamorphic illusions on one canvas: a starry night sky with a lake, and a vast green field with rivers, clouds, and mountains. Both scenes transform when reflected in a spherical mirror surface, becoming undistorted when viewed from the center.

Figure 12 — the final rendition, thank you Dr. Tong for a year of AP Precalculus.

I thoroughly enjoyed the overall experience of working on this project. As I delved into the world of anamorphic illusions, I discovered how mathematics seamlessly integrates into our daily lives. By experimenting with angles and transformations, I was able to witness firsthand the mathematical and scientific principles that make anamorphic reflections possible.

This project led me to a deeper understanding of the fundamentals of math and provided valuable insights into its practical applications. Math extends beyond numbers and calculations; it serves as a powerful tool for exploring and understanding the world around us. My final artwork, featuring two distinct illusions of a starry night sky and a serene landscape, embodies the beautiful intersection of art and mathematics. This journey has reinforced my appreciation for the creative potential that arises when these two disciplines converge.