Global Graphing Through the Lens of AP Precalculus 🌎

Our year in Dr. Tong’s AP Precalculus course at Concordia International School Shanghai cumulates in an artwork that uses concepts of polar and rectangular coordinates to mirror images onto a cylindrical surface that reflects the image as is. Taking inspiration from my family’s love of travel, I decided to connect these concepts and combine pictures from 3 locations around the world my family visits and combine them into a collective rotative family portrait. Consisting of family pictures from Vancouver, Melbourne, and Shanghai, I tried to make a rotatable “gallery” of some of our memories and reflect it onto a reflective surface molded in a cylindrical glass, setup seen in Figure 1.

Figure 1. DIY setup of the cylindrical mirror used as a medium of reflection.

After this, I compiled a collection of 4 images of distinct skylines my family and I took pictures in front of and used that as the focal point of my project, cumulating our travels into a mathematically-based collection of portraits and using them all as separate entities, individually manipulating the images using the “warp” feature on Procreate while using the mirrored glass as a reference to alter the image appropriately so it would look like the original image, just reflected onto the glass. However, this approach was not as efficient and was more inaccurate than I had hoped because going off of a case-by-case basis with the images would not yield accurate results. Instead, I had to take on a more mathematical approach based on the effects that arc angle measures had on producing a minimally distorted image when it was reflected onto the glass.

Initially, I had a shape that made a straight edge on the bottom of the image, but not the top, as seen in Figure 2.

Figure 2. The first draft of an altered image to be reflected onto the mirror.

This is when I had the epiphany that if the bottom of the image was successful, and the top wasn’t, the arcs of the warped image would have to be consistent for the top to be straight as well because of the consistent angle of the cylindrical glass. After this realization, I traced the arc from the bottom and curve fit it to the top, elongating the drawn arc and erasing the necessary parts of the image, process seen in Figure 3.

Figure 3. An animation of the process used to maintain a consistent arc angle.

This turned out to be the necessary alteration to make not only the edges of the image straight, but also to create uniformity within the images so that the differences were not compromising the unity of the images. After this, the rest of the process was smooth, especially with the aid of digital technologies to easily help me manipulate the image opacities to gain a clearer view of what to keep and what to erase, the step seen in Figure 4 with a picture from Vancouver and Shanghai. I used the picture from Vancouver as the reference image, fitting all of the successive images onto that one to ensure consistency between the distorted images. I found this consistency to be incredibly important, especially when printing.

Figure 4. Image 2 (Shanghai) superimposed onto image 1 (Vancouver) for consistent curve-fitting, sizing, and shaping.

To understand how the angles of the original image offset the distortion introduced by the glass, it is essential to first understand distortions and the coordinate transformations involved. A cylindrical mirror distorts images because it is limited to one direction’s stretch which distorts it along the curvature of the mirror. When reflected onto a cylindrical mirror, the points are then stretched around the cylinder (much like polar coordinates), instead of flattened on an even surface (much like rectangular coordinates), which transforms the image along the curve of the cylinder, introducing distortion. To “undo” this effect, we need to take into account the nature of polar coordinates by mapping the original, rectangular, image points to polar coordinates to offset the distortion caused by the cylindrical mirror. When the arc angle 𝜃 remains consistent, each segment of the edge is transformed in the same way regardless of the arc length. This ensures that the transformation is uniform, preserving the straightness of the edge. This is why the arc size does not matter, but rather the angle characteristic it exhibits due to two main reasons: scaling and the preservation of geometric relationships. In terms of scaling, when the angle 𝜃 is consistent, the scaling effect from the cylindrical surface is uniform, and the edge of the image is reflected as a straight line because all the points are appropriately and proportionately spaced. The preservation of geometric relationships is also a defining characteristic because the maintenance of straight edges requires a consistent translation of angles which keeps the geometric relationships between points, limiting the distortion present.

After this process, the final product is consistent and finished, as seen in Figure 5! Although one of the images I printed out was different from the same size, keeping in mind the aforementioned concepts, the image remained unaltered and consistent when reflected onto the cylinder.

Figure 5. Final product with all of the images reflected!

However, no project comes without its challenges and limitations. The surface on which the images are projected is incredibly significant to the projection of an image onto the surface. In this specific case, I used a mirrored sheet that I rolled up and placed into a glass, reference in Figure 1. Because of this, one of the edges that overlapped introduced a line that produced an abrupt line that altered the images reflected onto it, producing a slight discrepancy in the image, outlined in Figure 6. The surface of the mirror is important in the transformed manifestation of the image because it has to be virtually flawless and perfectly smooth before the image can be seamlessly transferred onto the glass. In addition, dents on the mirror's surface can also alter the reflected image, where the divots on it can also produce flaws in the transformed image due to surface inconsistencies. In addition, the images also have to be positioned perfectly flat on the table with no curving of the paper. When there are unexpected curves in the image itself, shadows presented on the table are amplified on the mirror, limiting the full extent to which the images can be directly reflected.

Figure 6. An example of how the inconsistent surface generates flaws within the reflected image.

Finally, the last inconsistency that notably presented itself was the limitations of distance on the alteration of the image. When the image is too close or too far in terms of distance from the mirror, significant distortions are present mainly due to the curvature of the mirror. A highly distorted, magnified reflection where straight lines may appear curved and spatial proportions are significantly altered is present at these distances. Maintaining an appropriate distance is highly valuable because the angles and distances have a profound effect on not only the objects themselves but also the viewer’s perception of the images and the distortion present.

Overall, this project provided me with a lot of insight into the effects of reflection, angles, and perception on transferring images between surfaces and was highly valuable in its implications for architecture, art, and visual communication while providing a strong intersection between the overlapping realms of art and math.

Figure 7. A world map. Letts & Son 1883.

The ever-changing landscape of our Earth has been documented for centuries, leaving many enchanted with the limitless ways to depict it, specifically through the lens of cartography. For 5,000+ years, maps were the best way to show the development of our Earth’s cultural, political, physical, and geographical tapestries, and will continue to be an integral part of the fascinating world in which we live.

Figure 8. The Waterman Butterfly projection.

Long-debated topics of map projections are still studied even today to achieve the most accurate representation of our world, and there are some incredibly creative approaches, such as the Waterman Butterfly as seen in Figure 2. Translating a multi-faceted, 3-dimensional planet onto a 2-dimensional surface does not come without its limits, however, and limiting these inaccuracies is an incredibly difficult task. However, with the aid of polar coordinates, a future of accurate ways to depict our globe is getting closer and closer. The Azimuthal Equidistant projection, seen in Figure 3, is one of the most accurate projections when taken from a point of view from either the North or South Pole because it utilizes polar coordinates to gain a more accurate and holistic view of the globe due to its circular shape that mirrors the cross-section of our spherical Earth. Used by the United Nations, the Azimuthal projection has more mathematical implications and applications than one might expect.

Figure 9. The Azimuthal projection from different angles. https://gisgeography.com/azimuthal-projection-orthographic-stereographic-gnomonic/

Through the utilization of polar characteristics, the Azimuthal projection emphasizes the representation of spatial characteristics by preserving angular relationships (the countries depicted) around a central point (the given pole). In the polar coordinate system, specific points are placed by their distance from a central point and a reference angle much like how the Azimuthal projection represents countries and locations on their map. This provides a clearer overview of our Earth due to its direct application to a circular layout, unlike a rectangular layout, which is more susceptible to projection inaccuracies because of the discrepancy between the map’s depiction and the actual shape of the Earth. This overlapping use of radial coordinates makes the Azimuthal projection a more practical application of polar coordinate principles in cartography due to its high applicability and relative accuracy.

More common projections, such as the Mercator projection where the Earth’s surface is represented on a rectangular grid, can introduce distortions, especially near the poles due to the linear alteration of latitude and longitude lines used for pinpointing location. Similarly, rectangular forms of depicting points on a graph have limitations in the patterns it can exhibit, especially in the case of sinusoidal functions where overlapping values may not be clear. However, in polar-based projections, distances and angles from a central point are preserved, resulting in a circular layout that more accurately represents distances from the given center (or pole, in the case of cartography). Overall, polar coordinates often provide a more accurate depiction of spatial relationships on the Earth’s surface in specific contexts, particularly around a central point of the projection.

Figure 10. The Mercator projection. https://en.wikipedia.org/wiki/Mercator_projection

Ultimately, the strive for accurate map projections remains a challenge due to the unavoidable distortions in translating a 3-dimensional globe onto a 2-dimensional surface, however, polar coordinate-based projections like the Azimuthal projection offer a more accurate depiction by preserving distances and angular relationships around a central pole, providing a clearer and more accurate view of the Earth. In contrast, traditional rectangular projections such as the Mercator projection, seen in Figure 4, often introduce significant distortions, particularly near the poles, highlighting the advantages of circular layouts in cartography as well as exemplifying the significance polar coordinates have on the timeless depiction of the holy ground we live on.

References:

Azimuthal equidistant projections. (n.d.). https://gisweb.massey.ac.nz/topic/webreferencesites/Digital%20Maps/dean/src/warnercnr.colostate.edu/azimuthal.html

Azimuthal Equidistant — ARCGIS Pro | Documentation. (n.d.). https://pro.arcgis.com/en/pro-app/latest/help/mapping/properties/azimuthal-equidistant.htm#:~:text=Sources-,Description,any%20point%20on%20the%20globe.

Čučaković, A., & Paunović, M. (2015). Cylindrical mirror anamorphosis and Urban-Architectural ambience. Nexus Network Journal/Nexus Network Journal, 17(2), 605–622. https://doi.org/10.1007/s00004-015-0239-7

Imaging by cylindrical concave mirror. (2016, September 25). Physics Forums: Science Discussion, Homework Help, Articles. https://www.physicsforums.com/threads/imaging-by-cylindrical-concave-mirror.886551/