1ucasvb

Quick update: I'm alive! More stuff soon.

Hey, everyone! Sorry for the lack of updates, but I’m very much alive.

I recently arrived in the United States and will be studying at Georgetown University in Washington-DC for the next year. Expect new stuff from me soon. I just need to settle down on the new class schedule first. :)

Cheers!

I thought these GIFs by 1ucasvb were nifty & wanted to have them all in a single post. 

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The familiar trigonometric functions can be geometrically derived from a circle.

But what if, instead of the circle, we used a regular polygon?

In this animation, we see what the “polygonal sine” looks like for the square and the hexagon. The polygon is such that the inscribed circle has radius 1.

We’ll keep using the angle from the x-axis as the function’s input, instead of the distance along the shape’s boundary. (These are only the same value in the case of a unit circle!) This is why the square does not trace a straight diagonal line, as you might expect, but a segment of the tangent function. In other words, the speed of the dot around the polygon is not constant anymore, but the angle the dot makes changes at a constant rate.

Since these polygons are not perfectly symmetrical like the circle, the function will depend on the orientation of the polygon.

More on this subject and derivations of the functions can be found in this other post

Now you can also listen to what these waves sound like.

This technique is general for any polar curve. Here’s a heart’s sine function, for instance

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Pi’s parametric coordinate functions

The polygonal trigonometric functions I described earlier were based on the interior angle, instead of the length along the polygon’s border.

This simplified things a lot, and created some interesting uses for the functions. However, since I could only have one value of radius for each angle (they were based on polar equations), I could not draw arbitrary shapes with a continuous line based on the [0,2π] interval.

The solution is to extend the idea to general closed curves, by using the position along the curve to define the sine and cosine analogues. In other words, we want “path trigonometric functions” for which the input parameter is the position along the path, and whose periods are the curve’s total arc-length.

But the concept of “sine” and “cosine”, as well as “trigonometric”, completely lose their meaning at this point. It has nothing to do with triangles or angles.

We’re now dealing with the functions x(s) (in blue) and y(s) (in red) that together describe the curve, by being used in the parametric equation r(s) = ( x(s) , y(s) ), where r(s) is a vector function and s is the arc-length. This is very standard stuff, so it isn’t incredibly exciting anymore.

Notice that if the green curve was a unit circle, the functions would become the usual sine and cosine.

But we do get to see what these functions look like and what they are doing. So here’s the coordinate functions for the arc-length parametrization of a pi curve!

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A 5-pointed star’s “sine” and “cosine” functions

Based on the same principle as the polygonal trigonometric functions.

This was requested a few times, but I had to figure how to draw polar stars first. Finally got around to it.

I'm now open for donations

I’ve been making these educational animations for Wikipedia for years now. It has always been a completely volunteer effort, and I never really got anything from it other than the joy of learning new things about math and physics, and being able to show them to others in a memorable, intuitive and visually pleasing way.

I just wanted people to understand how cool math and physics are, and how simple some complicated looking things can be if you look at them the right way. And to my great pleasure, it has worked really well so far!

Over the years, the reactions have been very positive, especially since I created this tumblr blog. It’s been very exciting to see so many people getting enthusiastic about math and physics because of my work.

But lately, things have been a bit rough, and free time has been scarce. Being a college student takes its toll on anyone, and living off a tiny student grant and my personal savings (one of the reasons why I started college so late) is pretty tough.

So I decided to open up for donations. If you like my work and feel that it is worth something to the world, then consider making a donation. Anything would help me worry less about money and survival, and more about learning stuff and getting me motivated to spend more time on making more cool animations to help others.

You can find the donate button at the top right, on the blog header, or you can follow this link. Any help would be immensely appreciated.

Thank you all for the love and support so far! It means the world to me.