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Ambiguous Objects

Ambiguous objects are objects that create an optical illusion to the human eye by exploiting the similarities between two objects. Ambiguity can occur between two dimensional images or shapes but can also appear in three dimensional objects. For example, you've probably seen sometime of ambiguous, optical illusion as an image in a psychology class at some point. The image below (from an article on https://www.nature.com/articles/s41598-018-31129-7) shows you an old woman, as well as a young woman depending on how you view the image.
Do you see it? You are only able to see one image at a time because the artist draws one feature that doubles as a different feature in the other image. For example, in the above image, the jaw line of the young woman doubles as the chin of the old lady. Do you see it now? It's a pretty good trick for your mind.

Ambiguous objects in three dimensions can be even harder to spot sometimes, but they are constructed the same way. They use shared features of two separate parts and create a single object where you can see both individual parts depending on how you view it. We are using a 3D printer to print these objects so we actually have them to view and don't have to imagine using 2D images.

I have decided to combine the parameterized curve of \([cos(x), cos^3(x), 0]\)
with the parameterized curve of \([cos(x), sin(x), 0]\).
Once we build our ambiguous object, an oberver from either end of our object standing at a 45 degree angle above our object, looking down, should be able to see each of the two images pictured above. Since our observers will be viewing from this angle, we must determine where the two lines of vision intersect. To do this we must first find the lines by finding the equation through \([cos(x), cos^3(x), 0]\) and parallel to \((0, 1, 1)\) and the equation through \([cos(x), sin(x), 0]\) and parallel to \((0, -1, -1)\). Our resulting parameterizations are \([cos(x), cos^3(x)+s, s]\) and \([cos(x), sin(x)-t, t]\). Now, in order to determine what these two curves have in common, we must set these new parameterizations equal to each other and solve for their point of intersection. After setting \([cos(x), cos^3(x)+s, s] = [cos(x), sin(x)-t, t]\) , we find that \(s = t = 1/2(sin(x)-cos^3(x))\) which, after plugging s and t back in to our parameterization, we find that a new curve is formed using commonalities between the two curves we started with and that is as follows: \[[cos(x), 1/2(sin(x)+cos^3(x)), 1/2(sin(x)-cos^3(x))]\]
Now, all thats left to do is finish building our object and we can do that by extending down our new curve to form a cylinder-like shape shown below.
A rectanglar plate has been added so we know to look from both of the longer sides of the rectangle. Viewed from one side and our red dot we see our first curve of \([cos(x), cos^3(x), 0]\).
Viewed from the other side and our green dot we see our second curve of \([cos(x), sin(x), 0]\).
I chose this particular shape because once the ambiguous object was built, it was very easy to differentiate between the two curves I started with and even someone who has never seen an ambiguous object before would be able to see the different curves. My print should be able to print without supports and hopefully I won't have too many issues.

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