Trigonometric Ambiguity

In the second dimension, two different curves could never be portrayed as one curve yet still be observed separately. If one has a different \(y\) value at a certain \(x\), then there's no way they can be portrayed as one curve. This changes when you add another dimension, though. In the second dimension, the only way perspective can change is by rotating the plane, translating the plane, or making the plane smaller. These perspective changes don't make the curve any different though. A perspective change in the third dimension, however, can make a curve look drastically different. A circle in the \(xy\) plane looks like a circle when looking in the \(z\) direction. When looking in the \(x\) or \(y\) direction, though, a circle would look like a straight line. This kind of property enables us to make one curve look like two different curves depending on the perspective. The two perspectives we will be observing our curve from are orthogonal to the \(yz\) plane and the \(-yz\) plane. Now, we know that it's possible to make one curve look like two curves, but how do we do it? Well, we first take two curves in the \(xy\) plane and parameterize them. We need to make sure that their \(x\) values are the same. If your curve can be portrayed as a function of one variable, then just make the \(x\) value for both parameterizations \(t\). Because both curves are in the \(xy\) plane and their \(x\) values are the same, the only difference is their \(y\) values. Now, we make the parameterization for an entirely new curve. This new curve will still have the same \(x\) value as the other two curves. Its \(y\) and \(z\) values are new, though. If the \(y\) value in one of the curves is \(f(t)\) and the \(y\) value in the other curve is \(g(t)\), then the \(y\) value in the new curve is \(\frac{f(t)+g(t)}{2}\) and the \(z\) value is \(\frac{f(t)-g(t)}{2}\). Now, when we look at this curve at a 45 degree angle, we see either \(f(x)\) or \(g(x)\). The two curves become one in the third dimension.
  The two curves I chose to make one curve from are \(\sin(x)\) and \(\sin(x)-\cos(x)\). In order to make the two curves into a closed shape, the curves were rotated 180 degrees. The parameterization of the two curves are \([t, \sin(t), 0]\) and \([t, \sin(t)-\cos(t), 0]\). The parameterization of the new curve is \([t, \sin(t) - \frac{\cos(t)}{2}, \frac{\cos(t)}{2}]\). I chose these two curves to demonstrate how a concave shape and a convex shape can both be observed from one curve.

The two pictures above demonstrate how the curve looks different depending on the perspective. It could be considered an illusion made from math. The illusion is based on the fact that if you flattened out the curve in the plane orthogonal to your perspective, the resulting curve is one of the original curves we made the object with. For the print, in order to know where to look at the curve from, a shell was built underneath the curve to keep it oriented properly. I also made plates besides the object to show what the curve is supposed to look like from that perspective. The curve observed from above looks like a combination of the two original curves.