Have you ever had your mind blown by some sort of obstacle illusion? Where, when you look at some object one way, you see one shape and by slightly changing your point of view you see something completely different? This idea of constructing what feels to be an impossible shape was our task for this week! The goal is that by having two observers looking down at the object at different spots, they will see two completely different shapes. Below you will find an example of an ambiguous object so you can begin to get some sort of familiarity of what all is going on throughout the rest of this post!
As you can see, when looking directly at the object from one direction you see one shape, but when changing your point of view, you see something else. While the concept of funny shapes that appear to be different may seem simple, since we are able to see many examples of them on the internet, they are fairly difficult to construct. I will begin by showing you the shape I created, and then walk through the math behind it, as it may be easier to see the model while reading the behind the scenes. Here, I constructed an object where if you stare at it one way then you see a square, while someone on the opposite side will see a triangle instead. I thought it was nice to see what the change between the basic four sided shape to the three sided shape. Here is the shape, and the two different views.
So now that you can see what the goal is to obtained, lets talk about how you generally do it and the math for this example. You first by finding the curves you want the observer to see, or some way to create it. In particular, the curves need to be hidden, more like blended in, on one part so that we see a flat side. In order to do this, you take the curves you want to show and create some vectors such that the viewers will see their respective shapes. How do you do this? Well, lets say you want Viewer 1 to see the curve $f(x)$ and Viewer 2 to see the curve $g(x)$. Then, we want to parameterize some vector $\overline{r}$ so that Viewer 1 looks along the curve to see $(x, f(x), 0)$ and Viewer 2 looks down and see $(x, g(x), 0)$. By parametizing our functions for $\overline{r}(t)$, we can arrive at this vector.
Lets continue by fixing $t$ and then solving for the intersection of the line passing through the point $(t,f(t),0)$, where the line is parallel to $<0,1,1>$, and the line passing through the point $(t,g(t),0)$ which is parallel to $<0,-1,1>$ (this is to make sure the observers are on different sides of the shape). By solving through the intersection, you see you arrive at $\overline{r}= < t, .5*(f(t) + g(t), .5(g(t) - f(t)))>$. From there, you plug back in you original equations and you end with two vectors, where you split it between positive and negative y's if having two curves.
For my example, I split the shape into the four quadrants and built the shape from there. The big thing to note is that I just used $x$ as the parameter for each function throughout. Moreover, each function was defined for a total interval length of $1$ for $x$, with steps of $.5$, as this is enough to make the curvature appear with the functions but keep certain sides straight. For my first two equations wanting to be shown, I chose $f(x) = \sin{x} * (.5-|x+.5|)$ and $g(x) = (1-x^2)^2$. I chose these two because from different angles, they straighten out ($g(x)$), and curve slightly enough so that we a more of a bending on a side ($f(x)$). The $\sin{x}$ part really came in handy in order to actually make the side curve enough so that it disappears from one direction but look straight from another. The second equation was identical, besides $f(x) = \sin{x} + (.5-|x-.5|)$, just to get the same curve but going a bit opposite from the first.
Following from this discussion, the third and fourth parts were similar to each other. The third quadrant was fined by the functions $f(x) = -.5*(1-|x|)$ and $g(x) = -(1-x^2)^2$. Notice, no sine function was added here since we wanted these sides to be a bit more straight. Finally, the fourth part of the overall shape was created with the functions from the third quadrant, exactly the same. With the even power of functions we chose as well as making the function go between the positive values of $x$, we replicate the exact side made in the third quadrant. Putting these four together made the overall shape from above!
Now, to discuss a little bit more about why I chose these speific shapes. I wanted to have something that is very familar to anyone looking at the object, leaving little up to the imagnation of the mind to see a shape. It also shows off how one can use some simple equations in order to get this curvature of one side and the sharpness on the other, thus changing the perspective of one observer and another.
As you can see, when looking directly at the object from one direction you see one shape, but when changing your point of view, you see something else. While the concept of funny shapes that appear to be different may seem simple, since we are able to see many examples of them on the internet, they are fairly difficult to construct. I will begin by showing you the shape I created, and then walk through the math behind it, as it may be easier to see the model while reading the behind the scenes. Here, I constructed an object where if you stare at it one way then you see a square, while someone on the opposite side will see a triangle instead. I thought it was nice to see what the change between the basic four sided shape to the three sided shape. Here is the shape, and the two different views.
So now that you can see what the goal is to obtained, lets talk about how you generally do it and the math for this example. You first by finding the curves you want the observer to see, or some way to create it. In particular, the curves need to be hidden, more like blended in, on one part so that we see a flat side. In order to do this, you take the curves you want to show and create some vectors such that the viewers will see their respective shapes. How do you do this? Well, lets say you want Viewer 1 to see the curve $f(x)$ and Viewer 2 to see the curve $g(x)$. Then, we want to parameterize some vector $\overline{r}$ so that Viewer 1 looks along the curve to see $(x, f(x), 0)$ and Viewer 2 looks down and see $(x, g(x), 0)$. By parametizing our functions for $\overline{r}(t)$, we can arrive at this vector.
Lets continue by fixing $t$ and then solving for the intersection of the line passing through the point $(t,f(t),0)$, where the line is parallel to $<0,1,1>$, and the line passing through the point $(t,g(t),0)$ which is parallel to $<0,-1,1>$ (this is to make sure the observers are on different sides of the shape). By solving through the intersection, you see you arrive at $\overline{r}= < t, .5*(f(t) + g(t), .5(g(t) - f(t)))>$. From there, you plug back in you original equations and you end with two vectors, where you split it between positive and negative y's if having two curves.
For my example, I split the shape into the four quadrants and built the shape from there. The big thing to note is that I just used $x$ as the parameter for each function throughout. Moreover, each function was defined for a total interval length of $1$ for $x$, with steps of $.5$, as this is enough to make the curvature appear with the functions but keep certain sides straight. For my first two equations wanting to be shown, I chose $f(x) = \sin{x} * (.5-|x+.5|)$ and $g(x) = (1-x^2)^2$. I chose these two because from different angles, they straighten out ($g(x)$), and curve slightly enough so that we a more of a bending on a side ($f(x)$). The $\sin{x}$ part really came in handy in order to actually make the side curve enough so that it disappears from one direction but look straight from another. The second equation was identical, besides $f(x) = \sin{x} + (.5-|x-.5|)$, just to get the same curve but going a bit opposite from the first.
Following from this discussion, the third and fourth parts were similar to each other. The third quadrant was fined by the functions $f(x) = -.5*(1-|x|)$ and $g(x) = -(1-x^2)^2$. Notice, no sine function was added here since we wanted these sides to be a bit more straight. Finally, the fourth part of the overall shape was created with the functions from the third quadrant, exactly the same. With the even power of functions we chose as well as making the function go between the positive values of $x$, we replicate the exact side made in the third quadrant. Putting these four together made the overall shape from above!
Now, to discuss a little bit more about why I chose these speific shapes. I wanted to have something that is very familar to anyone looking at the object, leaving little up to the imagnation of the mind to see a shape. It also shows off how one can use some simple equations in order to get this curvature of one side and the sharpness on the other, thus changing the perspective of one observer and another.
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