Anamorphic Reflection Art
Introduction:
This project is created by Jerry Liang and Reece Whyman from Dr. Peter Tong’s AP- Precalculus class in Concordia International School Shanghai.
We decided to do this project because we are interested in how the reflection work. Through making this project, we’ve been through a lot of challenges. For example, finding the right concentration of the chemicals we’re using to make the silver coat coat on the beaker perminantely. We also learned how to design object using 3D printing website and how to use the 3D printer.
Silver Mirror reaction
The reflective mirror is done through an experiment called Tollens Reaction to create a silver coating on the wall of the beaker. To make this reaction we need to mix glucose (dextrose) with Tollens Reagent
Tollens Reagent: 2(Ag(NH3)2)OH
1. Add 150mL of 0.1 M AgNO3 aqueous solution to a beaker
2. Then add concentrated NH3 into the solution drop by drop until a brown precipitate is formed (seen in figure 2)
3. After seeing the brown precipitate keep on adding NH3 until the solution turns clear again (seen in figure 3)
The first precipitate we see comes is silver oxide from this reaction. As more NH3 is added the Ag ion would go react with NH3 and form Ag(NH3)2+ ion. According to the solubility rule, anything with NH3 is soluble so the solution turns clear again.
4. Measure 75 mL of 0.8M of KOH and pour it in the solution and a brown precipitate will be formed (seen in figure 4)
5. Add NH3 into the solution drop by drop until the solution turns clear and Tollens Reagent is made and pour it in the container you want the reaction to occur in
6. Add 100 mL of the Tollens Reagent and add around 1mL of 0.25 M glucose (dextrose) solution in the beaker (seen in figure 5)
7. Seal the top of the beaker fully and shake the container for around two minutes and the silver will be coated on the container (seen in figure 6)
Adding KOH into the solution can create a basic environment for the final reaction to happen because Glucose can be oxidized easily in basic environment. Adding KOH into the solution would make it react with the silver ions which would create a precipitate again so NH3 is added again to dissolve that precipitate. Glucose has an aldehydic group that can be easily oxidized and reduce the Tollens reagent. This can push the Ag+ from 2(Ag(NH3)2)OH out and make the silver ion into silver. Shaking the container can make the silver being pushed out to stick on every part of the container evenly.
The reflection
The physics behind out project is the Law of Reflection. The law states the incidence angle of the object is equal to the reflection angle of the object. In Figure 7, we can see the angle measured is between the ray and the Normal. Since this figure is a flat mirror, the normal would be perpendicular to the mirror.
For a normal undistorted image to reflect on a convex mirror, the law still applies.
For a convex mirror, when the object, in this case is the image, one type of incidence ray would hit the surface. We can draw a tangent line (purple line shown in Figure 8) of the convex mirror at the point that hits the mirror. The perpendicular line of that tangent line will be the normal (red line shown in figure 8). The Normal would pass through the centroid on the principal axis that is double the length of the focus (f). The normal would equally split the incidence angle and the reflection angle. Now we can identify the angle of the incidence ray to the normal which is the incidence angle. The reflection angle has to be the same as the incidence angle, this makes the reflection ray would pass through the focus point on the principal axis.
We can see from Figure 9, that no matter where the incidence ray of the image is interacted to the mirror, as long as the incidence ray is parallel to the principal axis, the normal would all pass through the centroid and all reflection ray would all pass through the focus. This makes the reflection ray distorted from the normal image so the image we see that is being reflected from this mirror will be distorted.
In this project, when we put a distorted image beside the mirror the image we see will be undistorted. This occurs because the incidence ray of an undistorted image will be the reflection ray of a distorted image (seen in Figure 10). This makes the image we see will be undistorted.
When we are looking at the image at other angles the picture is not being undistorted. We have to look at the beaker at a specific angle to make the image undistorted. This is because we are transferring a 2d image to a 3d beaker cylinder mirror. The reflection can’t only correlates to the theory above it also correlates to the height of the beaker.
From figure 11, if we look at the beaker at different angle, the ray of the image coming to our eye will have different length which would affect the final image we see. So, we need to find the correct angle to look at the beaker that gives the right length of the ray of the image.
To find the angle that is best to watch from is 42 degrees. We measured the length between the beaker and the viewer which is 29 cm (adjacent) and the distance between my eye to the beaker which is 32 cm (hypotenuse). Using these two data we can use cos (32/29) which around 42 degrees.
Putting them together
The way to make this happen is to convert the picture we want that is undistorted into a distorted image. The way to distort it is to convert the picture from rectangular grid to polar grid.
1. Find the image you want to distort
2. Since the reflecting mirror is a beaker, you have to restrict the size of the distorted image, so it won’t be distorted on a full polar grid scale. It needs to be distorted into a fan- shaped image.
3. The reflection of the image we can see would only happen on one side of the beaker because we can only see one side of the beaker (seen in Figure 12). We will have to make the image only fit in half a polar grid. So, the whole distorted picture can be reflected.
If the image completely surrounds the beaker the image would be reflected to two sides, but we are only able to see from one side.
4. Since we need the whole image to fill up half a polar grid which mean from 0 radian to pi radian. This means the length of the original picture will fill half of the polar grid. This means we need white space at each side of the original picture that each has half the length of the original picture. In this case, our original picture has a length of 6 centimeters. This means the white rectangular space on the left of the image should have a length of 3 centimeters and the white rectangular space on the right of the image should also have a length of 3 centimeters (seen in Figure 13).
Note: There will be a margin of error due to human errors when measuring.
5. Put this photo with the two strips of white rectangular space into photoshop and flip it vertically. Then press filter, then distort and lastly polar coordinate. The final image we use for reflection will be done (seen in Figure 14).
6. Use 3D printer to print a plate that for putting beaker in the middle and that can stick the distorted image on the side.
7. Cut the distorted image off and further cut the background of the distorted image because we only want the face. Then stick it on the side of the plate and put the beaker in the middle. Observe the side of the beaker that face the image, the image will be undistorted and become the original image.
To prove the distortion is transferring rectangular grid to polar grid we can add the grids on the distorted image and see the difference when reflected.
In Figure 15, we added a polar grid onto the image. After the image getting reflected, we can see the polar grid is distorted into a rectangular grid. Same with Figure 16, when we added a rectangular grid onto the image, we see the reflected image distorted the grid into half a polar grid. It’s only half because we cut the other half of the image away since it was white space. This shows to make the image distorted to the proportion that can be undistorted through reflection, we can simply convert the image that is on a rectangular grid to polar grid. This distortion can make the reflected image undistorted.
