Unfortunately, Desmos does not support complex numbers. Luckily, the parameterization of the complex unit circle is exactly the same as the parameterization of the unit circle in the real plane (excluding the imaginary unit of course). In the Desmos graph I took advantage of this fact to rotate each point. To rotate an entire curve (as opposed to a single point), I made a parametric function of $t$ and applied the transformation to each point $(t, f(t))$ in the domain $a\leq t \leq b$. I like this one because of the simplicity. I also like it because I had no idea how to do calculus when I created it about a year and a half ago. However, I did understand that the derivative magically yields the slope of the function at any given point. So I decided to create a little demonstration, and this is probably one of the first things that got me seriously interested in math. The black curve is the function (you can edit it from the link above), the red line is the tangent, and the blue line is the normal. This graph shows the functions that yield the $x$ and $y$ coordinates of a point as it moves around an ellipse. This is analogous to the $\sin$ and $\cos$ functions for a circle. In fact, if you set $a=c_0$ and $b=c_0$ in the graph above for some $c_0\neq 0$, the resulting functions will be the sine and cosine. The blue curve is an analog of cosine, and the red is an analog of sine. I don’t know of any applications for this, but it is pretty interesting to see how the shape of these functions change depending on the characteristics of the ellipse. For example, from the image above, you can tell the major axis is in the $x$ direction because the cosine analog has a greater range than the sine analog. The top and bottom of the blue curve are pinched together because the direction of the point moving along the circle is rapidly changing since the ellipse is wider than it is tall.
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