Curved mirrors distort images because they send pieces of the image off in different directions that look strange to your eye; this is the basic principle behind funhouse mirrors. But if you send a distorted image into a mirror, then from a certain perspective the image will look correct to your eye! This kind of art is generically called "anamorphic art" and is commonly seen in the from of dramatic street art that looks three dimensional from certain perspectives.
You can play with "cylindrical anamorphoses" pretty easily at home. Constructed using a cylindrical mirror, cylindrical anamorphoses were a popular attraction at parties during the Victorian era, but you can still use them to impress your friends at dinner parties here in the 21st Century!
There are two ways to produce the reflected artwork --- one way is by hand (the classic way), and the other is with software like Photoshop. I'll outline how I do it by hand here, and in the last step describe how to do it with Photoshop.
Step 1: Materials
(1) Cylindrical Mirror -- I used a chromed sink extension tube I picked up at the hardware store
(2) Paper, Pencil, Ruler, Compass, Protractor
(3) Image for your art
(4) Computer (optional)
Step 2: Reference Art
The anamorphosis requires a piece of distorted art to reflect. To make the distorted art, print out the image you want to use, so you can work and write on it. The mirror reverses images left to right, so if you want your image to have the expected left-right appearance when reflected (for instance, words or recognizable symbols), flip the image horizontally before you print it.
Step 3: Making Your Drawing Grid
The art is transferred to a distorted grid (the "drawing grid") that is made up of concentric circles and radial lines.
(a) In the center of a blank piece of paper, trace an outline of your cylindrical mirror.
(b) Using your compass, draw concentric circles around the mirror location, spaced by 1/4 inch.
(c) From the center of the mirror circle, draw radial lines outward, separated by an angle of 22.5 degrees.
(d) The resulting drawing grid is shown here. You can also make one up on the computer, so you can quickly print them off if you want to make a new piece of art. The disadvantage of the computer drawing grids is you cannot erase the grid lines once your art is transferred.
Step 4: Check Your Drawing Grid
If you're worried your drawing grid didn't get drawn right, you can check it by putting your mirror down in the center.
You should see a perfect, square grid reflected in the mirror! This is the mirror correcting for the distortion of the curved mirror -- the same physics that will turn your distorted image into a something to look at!
There is still visible distortion where the mirror sits on the piece of paper; this is because the concentric circle and radial line grid is not exactly correct (mathematically) down near the contact point with the mirror. We will keep our art far from this distortion region.
Step 5: Art Reference Grid
To transfer the art, we will use a modified version of the "grid transfer method" that many of us are taught in grade school.
(a) On the piece of art you would like to display in the mirror, draw a perfect square grid (the "art grid"). The number of vertical grid lines should match the number of radial lines you drew for the reflection, and the number of horizontal lines should match the number of circles you drew outside the mirror circle.
(b) It is helpful to label the drawing grid and the art grid. I use letters for the vertical lines (radial lines), and numbers for the horizontal lines (concentric circles). On the art grid, label the lines left to right, and bottom to top. On the drawing grid, label the radial lines clockwise, and the concentric circles from the center outward.
Step 6: Transfer Sketching
To transfer your image, focus your attention on one square of the art grid at a time.
What is in it? Note where lines touch the edge of the square, and where details appear in the square (just left of center, etc.).
Note the letter and number of the square you are working with, and find the corresponding letter and number on the drawing grid.
The corresponding "square" on the drawing grid is not square at all! You must transfer what you see in the square grid to the cylinder grid by noting where details appear, and drawing them in their distorted locations.
You can check your progress at any time by simply putting the mirror in place. Some cylindrical mirror artists create their distorted images by simply sketching freehand while watching the reflection not their hands!
Step 7: Finishing Up the Transfer
I sketch initially in pencil, so I can make adjustments and erase mistakes, then I go back over it with something more bold, like a Sharpie.
When you are done, clean up what you don't want to see with an eraser, and display your new art!
Step 8: Making Distorted Art With a Computer
Modern graphics software often has the ability to distort images properly for display in a cylindrical mirror. This is particularly useful if you want to display a complex picture, or a picture in color.
In Photoshop, load your image up, flip it horizontally, and apply the filter called "Polar Coordinates" (Filter > Distort > Polar Coordinates).
This Superman logo was done in Photoshop. To get the distortion exactly the way you like, you may have to play around with the whitespace on the canvas, moving the image around before applying the filter.
Step 9: SCIENCE 1: Flat Mirror Reflection
One of the fundamental laws in optics is the "law of reflection." It says that when light reflects off a surface, it bounces off at the same angle it entered.
To see how this works for a flat mirror, imagine a simple object, like a row of colored dice. Physicists think of light as travelling along straight lines called "rays" -- think of them as arrows that point the direction the light is going. Each of the dice has rays of light that your eye receives, telling you the color, location, and shape of the dice.
When a ray hits a mirrored surface, the law of reflection says if you measure the angle between the mirror and the incoming ray, the outgoing ray has exactly the same angle between it and the surface.
The consequence of this is that someone looking at the mirror, like Selene in our image, sees all the light from the row of dice arriving together and lined up just like it left the dice. The only oddity is that the image is reversed; if Abigail had looked directly at the row of dice, without seeing the reflection in the mirror, she sees the order reversed compared to Selene.
Step 10: SCIENCE 2: Curved Mirror Reflection
Because our cylindrical mirror surface is curved, incoming rays from the row of dice get reflected off in different directions. That means light from an image doesn't stay together to reconstruct a compact image for your eye to see.
Each bit of light obeys the law of reflection, bouncing off at the same angle as the angle it hit the mirror. But each bit of light gets reflected in a slightly different direction because the surface is curved.
This distorts the image to an observer like Kate, who is looking at the mirror surface.
Step 11: SCIENCE 3: Anamorphic Reflection
The cylindrical anamorphosis takes advantage of the law of reflection by thinking about the distortion in reverse.
The distorted image is drawn in such a way that when each bit of light hits the mirror, the result is the light is brought together so a viewer like Calvin sees an undistorted image.
By spreading the image out on the cylindrical grid, you are using the law of reflection backward. The mirror takes widely distributed bits of light and reassembles them into a compact form that your eye sees as an undistorted picture.