Introduction
If you want to make somebody’s brain hurt, a good bet is to show them an optical illusion. These little tricks are fun to show off and a great way to challenge your mind. The only way to increase the fun would be to use mathematics to create your own optical illusion! This can be done through structures called ambiguous objects, which take on a different shape depending on the angle from which you look at them.
Math of Ambiguous Objects
The objective of an ambiguous object is that an individual standing on one side of the object will see a certain shape, while the person standing on the other side of the object will see a completely different shape. How do we make this work? First, let’s decide which shapes we want the observers to see, and then we’ll graph them in the same coordinate plane:
The first shape will be a star, which is parametrized by the following equations: \[ x(t) = 4 \sin(t) + \cos(4t) \] \[ y(t) = 4 \sin(t) - \sin(4t) \] You can also see that our second shape will be the curve that resembles a bat (you may need to use your imagination a little to fully appreciate the bat). It is parameterized by the following equations: \[ x(t) = 4 \sin(t) + \cos(4t) \] \[ y(t) = 10 \sin(t) - \sin(3t) \]
Now, imagine our observers are standing on opposite sides of these curves. We assume that they are looking down on the object at a 45-degree angle:
Let’s say that we want our red friend (let’s call her Sue) to see the bat and for our blue friend (let’s call him Bob) to see the star. If we were to draw a line connecting Sue’s eyes to any point on the bat, we know that every point on that line would appear to be in the same spot from Sue’s perspective. If we do the same for Bob with the corresponding point on the star, we can graph Bob and Sue’s line of sight:
Notice that these two lines intersect at a certain point. This is the point that Sue will perceive as being a point on the bat and that Bob will perceive as being a point on the star. Let’s calculate this mathematically. Say that the vector representing Bob’s eyes is \( \langle 0, 1, 1 \rangle \) and the vector representing Sue’s eyes is \( \langle 0 -1, 1 \rangle \).
Notice that the parametrization for the \( x \) coordinate is the same for both the bat and the star (this will make our math much easier). We will denote this by \( x(t) \). Let’s represent the \( y \) parameterization of our star by \( f(t) \) and the \( y \) parametrization of the bat by \( g(t) \). Then, a coordinate on the star and bat respectively can be represented by: \[ (x(t), f(t), 0) \] \[ (x(t), g(t), 0) \]
In order to generate the lines we saw in the previous image that represented Bob and Sue’s line of sight, we want to find the line that passes through the coordinates above and is also parallel to our observer’s line of sight. These lines are given below, where \( S \) represents Sue and \( B \) represents Bob: \[ r_{B}(q) = (x(t), f(t)+q, q) \] \[ r_{S}(s) = (x(t), g(t)+s, s)\]
Next, in order to find the intersection, we set these equations equal: \[ r_{B}(q) = r_{S}(s) \] \[ (x(t), f(t)+q, q) = (x(t), g(t)+s, s)\] Notice that the \( x \) coordinates are already the same. By setting the \( z \) coordinates equal we can substitute \( q \) for \( s \), and simplify the equation for the \( y \) coordinate: \[ f(t)+q = g(t)+s \] \[ f(t)+s = g(t)+s \] \[ s = \frac{1}{2}\ (g(t)-f(t))\] Plugging this in gives us the resulting coordinate of intersection: \[ (x(t), \frac{1}{2}\ (g(t)+f(t)), \frac{1}{2}\ (g(t)-f(t))\]
When we do this calculation for each point on the original curves, we end up with the following composite:
If we give it some depth, we have a structure that can be 3D printed and placed on the table. It is shown from the side below:
From this angle our object just looks like a weird star-bat hybrid. But when we look at it from Bob’s perspective, he sees the following (he can also see Sue waving from the other end of the table!):
What Sue sees when she looks at the object is the following (she can also see Bob waving from the other end of the table!):
So, with that bit of math we have created an object that takes on two very different shapes when looked at from opposite angles.
Why this example?
I chose these shapes because, first of all, I think the star parametrization is very cool looking and I’ve used it in a lot of previous assignments, so I wanted to keep the trend going. Secondly, I thought these two shapes were different enough for the illusion to be mind-boggling, since the bat shape is much wider and less pointy than the star. Both shapes have a lot of curves, so it’s hard to imagine at first how one object could appear to be both. I also wanted the shapes I used to be easily recognizable. Thirdly, these shapes worked out well logistically since they have the same parametrization for the \( x \) coordinate.
Author: Sarah Bombrys
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