Anamorphic Arts~Cats and more Cats
Have you ever seen an artwork that may look wierd at first sight, only to find out that it looks correct in another angle? Maybe you have seen these type of images on the internet, a book, or even possibly in a famous artwork. But what does this have to do with math?
In this year’s AP Precalculus, led by Dr. Tong, we explore these techniques and the underlying math.
Figure 1 is one image of my project made and we can see the cat obviously on the ipad screen showing it upside down and it has a curved line in front of it’s body. But then again it looks strange because of it’s wide head, on the mirror cylinder though, the reflection of the picture shows the originally curved line to be straight, and the cat’s head to be rounder.
Why is it like this and how is the “distorted picture” made? This will be revealed in the next few paragraphs.
~ Brainstorming and Failures ~
When I first started on the project, I had three options that I could do: An topographic 3D art, shadow art, and Anamorphic Art. I choose anamorphic arts because I was only good in drawing 2D items, and I didn’t know how to make a 3D model.
I brainstormed about what to draw and found out that I could actually just change a picture to make it anamorphic. But, I wanted to do art and it would be more fun if i could draw a sticker on wechat that I liked a lot, called “小豆泥”
When I first drew my artwork, I put the cylinder infront of me and tried to draw it just by looking at the cylinder. Although I have searched on line for examples, I still did not get the technique and ways of doing this type of art. So, I ended up with an distorted picture and a distorted reflection like this:
After all the trials and errors, I finally decided to search up an article about anamorphic art, looked into the technique, and made some conclusions.
~ What technique is used? ~
The technique used to make an deformed image straight on a mirrored reflection is called Anamorphic Art. It is an perspective technique used in art that gives a distorted image of the object represented in a picture when seen from the usual viewpoint “but so executed that if viewed from a particular angle, or reflected in a curved mirror, the distortion disappears and the image in the picture” (Britannica).
In Figure 2, the straight line on the Ipad is reflected to be curved on the cylinder. We know this because the star next to the straight line on the Ipad is next to a curved line in an mirrored image. Vice versa, the curved line on the Ipad is reflected to be straight on the cylinder.
From this we get Tip I:
- Straight in normal viewpoint → Curved in reflection
- Curved in normal viewpoint → Straight in reflection
This is because when we view a reflection on a flat mirror, the angle of reflection is the same at every point. But, if we view a reflection on a curved mirror, the angle of reflection is different at every point, making parts of the reflections seem streched. Shown in Figure 3 below, is the reflection on a flat surface on the left and a curved surface on the right.
We can also notice that in an Anamorphic Artwork, the further the item was located, the more it was stretched (in normal viewpoint)!
From this we get Tip 2:
- The further the part of the item is, the more stretched it should be in a normal viewpoint.
When I started to draw my animation using Tip 1 and 2, Ifound another problem. My question mark and the finger of the cat’s was different. They seemed mirrored on the y-axis. So, to solve this, I horizontally flipped what I drew to get a correct question mark
In Figure 4, we see that the question mark is facing the wrong direction.
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From this we get Tip 3:
- Horizontal flip anything that looks normal when looking at it from an normal view upside down
~ The Process ~
From Tip 1, “Curved in normal viewpoint → Straight in reflection”, we can draw conclusions that if we want a desk drawn infront of the cat, the line on the Ipad must be drawn curved:
From Tip 2, “The further the part of the item is, the more stretched it should be in a normal viewpoint.”, we know that if we drew a normal cat, it would have a small face and big body. To aviod that, the cat’s face should be drawn wide and the body should be drawn smaller. Look at Figure 6 to get an idea of a wrong head:body ratio.
So, in my artwork, the cat’s head looked wide because of this, while the body, the hands, and the cat beside it looks normal.
In Figure 7, I flipped the whole drawing to get a correct questionmark on the left. The picture shows a wrong reflection because of an unflipped drawing on the right. I did this because of Tip 3, “Horizontal flip anything that looks normal when looking at it from an normal view upside down”.
Now, with a correct drawing like Figure 1, I thought it was not enough and not interesting. So, I started making an animation using Procreate. First I made small changes on each frame and checked based on the cylinder to see if it was correct.
I ended up with 11 Frames shown in Figure 8–18:
With the frames done, Lets see the full animation!! (Figure 19 without cylinder reflection, Figure 20 with cylinder reflection)
