![]() Recently in my math class we went over “contour plots” which show the curves of intersection between a function $f(x,y)$ and a number of constant planes of the form $z=c$. You can make the blob more interesting by composing and summing additional sinusoids, but it slows down the blob due to greater computation time, which ruins the effect. I added in $\cos(x)$ to the $x$ coordinate to create the jellyfish effect and it works pretty well. Luckily, math gives us the perfect tool for this: sinusoidal functions. I wanted to give the blob an expanding and contracting motion, sort of like a jellyfish. So began the first blob type (“blobby blob”). Otherwise, the blob is a sort of ellipsoid shape that points in the direction of the function. Note that $f(x)$ is a function of $x$, so chances are it isn’t a constant. The blob has to follow the function, though, so I set $b=f(x)$. I wanted the blob to move along the $x$ axis as $c$ varies so I set $a=c$. The general form of a circle centered at $(a, b)$ with radius $r$ is The second blob (“not so blobby blob”) is easier to explain, so I’ll start with that. The difference between the two blobs is the nature of their shapes. Press the play button on the $c$ slider or simply move it back and forth manually to see the blob move. Both blobs follow the given function $f(x)$ as $c$ changes. Basically, you can select 1 of 2 blob types. Try it yourself: The blob moves along the function (orange curve) and always stays between the red curves.Īlthough it has no purpose, this is still one of my favorites. 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. 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. 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. The blue curve is an analog of cosine, and the red is an analog of sine. 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. This is analogous to the $\sin$ and $\cos$ functions for a circle. This graph shows the functions that yield the $x$ and $y$ coordinates of a point as it moves around an ellipse. 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. So I decided to create a little demonstration, and this is probably one of the first things that got me seriously interested in math. However, I did understand that the derivative magically yields the slope of the function at any given point. I also like it because I had no idea how to do calculus when I created it about a year and a half ago. ![]() I like this one because of the simplicity. 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 $a\leq t \leq b$. In the Desmos graph I took advantage of this fact to rotate each point. 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). Unfortunately, Desmos does not support complex numbers. This means a $z$ can be rotated about the origin $a$ radians by multiplying by $i^$) and $\theta$ is the angle of the same vector with the positive x axis. Note that this means each “point” is really just a single number. The blue curve is the original function and the orange curves are the rotated versions.Ī complex number $z=x yi$ is plotted in the complex plane by putting the real part $x$ on the horizontal axis and the imaginary part $y$ on the vertical axis. I accomplished this by first looking to rotations in the complex plane, then translating the method to the real plane. The goal of this graph is to take a function $f(x)$ and rotate it around the origin by an arbitrary radian amount in order to demonstrate the concept of rotation in the complex plane. ![]() You can click on the links after each header to edit and interact with the graphs in the browser. There are some additional interesting graphs at the bottom that are cool as well. Recently I went through my old Desmos ( ) graphs and I’d like to share 5 of my favorites here.
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