Pi’s parametric coordinate functions
This is the second of three animations I’ll be posting today (here’s the first). Be sure to check them out later if you miss them!
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!
Happy Pi day!