Ambiguous Objects

Introduction
Ambiguous objects are very unique images or objects that create ambiguity. Ambiguity is a vague meaning of a phrase, statement, image, etc., is not explicitly defined. Ambiguous objects can be either two-dimensional or three-dimensional. For our topic today, we will be understanding how ambiguous objects work through understanding exploitation of graphical similarities between two functions of a curve. Ambiguous objects may be more commonly known as optical illusions. Some of the more commonly known optical illusions are the Rubin vase, the rabbit-duck illusion, and the Kanizsa Triangle.

The image above is an example of the Kanizsa Triangle optical illusion. The spatially separated fragments give the impression of a triangle! These are just some examples of optical illusions that have been created. However, those optical illusions are two-dimensional. Now that we have a brief introduction with some common forms of ambiguous objects, we can further understand just how much we can stretch the ambiguity.
Ambiguous Objects?
As stated before, ambiguous objects can be three-dimensional as well. Today, we will be looking at two curves that will "share" the space of the curve and create a rather interesting shape. Then, depending on how the single object is viewed, the two different curves will be viewed. To understand this better, imagine that you are standing on one side of the object and someone else is standing on the other side of the object. From your perspective, you may see a circle or a triangle to be blunt, while the other person may be seeing a pentagon or another shape. This is the main idea that we are going to be looking at today!
Ambiguous Object Example
If we want to create an ambiguous object, we first need to start with picking out two shapes that we want our individuals to see. For my example, I have chosen the first shape to be a heart and the second shape to be kind of like a rhombus. From the image, we now have the shapes that our individuals will see, but we need to create the respective curves in the same plane by first parameterizing our shapes. For the heart, it is parameterized by the following equations: \[x(t)=cos(x)^3\] \[y(t)=0.5sin(x)-cos(x)^2\]

For the rhombus it is parameterized by the following equations: \[x(t)=cos(x)^3\] \[y(t)=2sin(x)+cos(x)^3\].

Now that we have our parameterizations, we want our individuals to be on opposite ends of the shapes, so they can see either shape. The individuals will be looking at the object from a 45-degree angle. From this angle, there is a certain point of intersection on where the two individuals are viewing the object from. To find this intersection, we have to represent respective vectors for both the individuals. The individual viewing the heart has a vector of (0,1,1) and the individual viewing the rhombus has a vector of (0,-1,1). Now, when we find the equation through the parametrization from before with the respective vector, we get a new parameterization that we can then set equal to each other. For the heart, we get a paramertization of \([cos(x)^3,0.5sin(x)-cos(x)^2+s,s]\) and for the rhombus \([cos(x)^3,2sin(x)+cos(x)^3-t,t]\). Now, we can set the two equal to each other to find the intersection. By doing this, we can find where the points intersect. This will give us a new curve with the parametrization of \[[cos(x),\frac{1}{2}(cos(x)^3+cos(x)^2+1.5sin(x),-\frac{1}{2}(cos(x)^3+cos(x)^2+1.5sin(x)]\] Lastly, we will generate our final image and add a visual aid of where to view the object from as if you were one of the individuals from before.

Why This Example?
I chose this example for multiple reasons. First, I really liked how the object stretched the ambiguity and produced a rather interesting curve. I believe that the parameterization tied with the two shapes was rather easy to solve, thus making the example easy and understanding to someone just learning the topic. I really liked this object as it allowed me to further understand ambiguous objects myself.