Introduction
You have probably seen an optical illusion before. They are known to play tricks with our brain. We will be using ambiguous objects to create an optical illusion. In our illusion we will create an object that looks different from two different angles. In the picture below we can see a quick example of an optical illusion. We can see there are two different shapes that are created. The one on the bottom is made up of squares, but that same model has a reflection of circles. From the single model we are able to see two very different shapes.
Ambiguous Object Example
To create an optical illusion in 3-dimensions we can use almost any shape we want. For this optical illusion the two shapes that I have chosen are a heart and an infinity sign. To create the heart, I have used the equations below and can be seen in the image beside it.
\begin{align*} x(t)&= \sin(t)^3 \\ y(t)&= \cos(t)+\cos(t)^4 \\ z(t)&= 0 \end{align*}
For the infinity sign we can again see the equations below and the how it looks like in the image.
\begin{align*} x(t)&= \sin(t)^3 \\ y(t)&= \cos(t)-\cos(t)^3 \\ z(t)&= 0 \end{align*}
Now we can move on to making our model that would show both of our shapes from different positions. We know that our observers are looking down onto our shapes at a 45-degree angle from opposite ends. To create a single shape, we have to find where their line of sight intersects when they look at their shape from their respective side. As this is hard to visualize, we can look at the image below. In the image we can see that the red line represents the person who is looking at the heart and the green line represents the person looking at the infinity sign. We can also see if we follow the two lines they intersect. This is where our observers sights intersects.
This has to be applied for every point to make our single model. To do this we know the red line is the vector (0,1,1) and the green line is the vector (0,-1,1). Using these vectors, we get new parametrizations the top is for the heart and the bottom is for the infinity sign.
\[(\sin(x)^3, \cos(x)+\cos(x)^4+s,s)\] \[(\sin(x)^3, \cos(x)-\cos(x)^3-t,t)\]
We now have to set these two parametrizations equal to each other and solve.
\begin{align*}
\sin(x)^3&=\sin(x)^3
\\
\cos(x)+\cos(x)^4+s&=\cos(x)-\cos(x)^3-t
\\
s&=t
\end{align*}
We can see that our x-points are the exact same. We can now solve for s and t, which are equal to each other, as we can see from our z-points.
\begin{align*}
\cos(x)+\cos(x)^4+s&=\cos(x)-\cos(x)^3-s
\\
s+s&=\cos(x)-\cos(x)^3-\cos(x)-\cos(x)^4
\\
2*s&=-\cos(x)^3-\cos(x)^4
\\
t=s&=\frac{-\cos(x)^3-\cos(x)^4}{2}
\end{align*}
We can now plug the s and t values back into our two parametrized equations which would give us
\[\left(\sin(x)^3, \cos(x)+\frac{\cos(x)^4}{2}-\frac{cos(x)^3}{2},\frac{-\cos(x)^3-\cos(x)^4}{2}\right)\]
The parametrization we made above is of our two shapes combined and can be seen in the image on the above. In the image below we can see both of our shapes beside each other allowing us to see the optical illusion we made.
Why this example
I wanted to choose two shapes that look different from each other. By doing this we are able to see how the math works and helps build our optical illusion. The heart was something I also used for my ruled surface, so it was easier to work with on openscad. I also like how my model turned out to be because it looks like a heart, but we can still see both shapes. When printed this should be 2.7 X 2 X 1.5 inches.
You have probably seen an optical illusion before. They are known to play tricks with our brain. We will be using ambiguous objects to create an optical illusion. In our illusion we will create an object that looks different from two different angles. In the picture below we can see a quick example of an optical illusion. We can see there are two different shapes that are created. The one on the bottom is made up of squares, but that same model has a reflection of circles. From the single model we are able to see two very different shapes.
Ambiguous Object Example
To create an optical illusion in 3-dimensions we can use almost any shape we want. For this optical illusion the two shapes that I have chosen are a heart and an infinity sign. To create the heart, I have used the equations below and can be seen in the image beside it.
\begin{align*} x(t)&= \sin(t)^3 \\ y(t)&= \cos(t)+\cos(t)^4 \\ z(t)&= 0 \end{align*}
For the infinity sign we can again see the equations below and the how it looks like in the image.
\begin{align*} x(t)&= \sin(t)^3 \\ y(t)&= \cos(t)-\cos(t)^3 \\ z(t)&= 0 \end{align*}
Now we can move on to making our model that would show both of our shapes from different positions. We know that our observers are looking down onto our shapes at a 45-degree angle from opposite ends. To create a single shape, we have to find where their line of sight intersects when they look at their shape from their respective side. As this is hard to visualize, we can look at the image below. In the image we can see that the red line represents the person who is looking at the heart and the green line represents the person looking at the infinity sign. We can also see if we follow the two lines they intersect. This is where our observers sights intersects.
The parametrization we made above is of our two shapes combined and can be seen in the image on the above. In the image below we can see both of our shapes beside each other allowing us to see the optical illusion we made.
I wanted to choose two shapes that look different from each other. By doing this we are able to see how the math works and helps build our optical illusion. The heart was something I also used for my ruled surface, so it was easier to work with on openscad. I also like how my model turned out to be because it looks like a heart, but we can still see both shapes. When printed this should be 2.7 X 2 X 1.5 inches.
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