awakening-soul:

This is perfect

1ucasvb:

In a previous post, I showed how to geometrically construct a sine-like function for a regular polygon.

I also pointed out how the shape of the function’s graph depends on the orientation of the polygon, since it isn’t perfectly symmetric like the circle.

This animation illustrates how the polygonal sine (dark curve) and polygonal cosines (clear curve) change as the generating polygon rotates.

Derivation

In order to find these functions for an arbitrary polygon, we first need to write the polygon in polar form. That is, we want the radius for a given angle. In a circle, this is a constant value.

A general “Polar Polygon” function is (simplest one I could find):

PP_n(x) = sec((2/n)·arcsin(sin((n/2)·x)))

Where n is the number of sides of the polygon. If n is not an integer, the curve is not closed.

Armed with this function, we can quickly find the polygonal sine and polygonal cosine:

Psin_n(x) = PP_n(x)·sin(x)
Pcos_n(x) = PP_n(x)·cos(x)

As n grows, the functions approximate the circular ones, as expected. To rotate the polygon, just add an angle offset to the x in PP_n.

So, what is it good for?

I’ve used this several times when I wanted some smooth interpolation between a circle and a polygon, in such a way that the endpoints of the interpolation are a perfect circle and a perfect, pointy polygon. It’s useful in parametric surfaces, such as in this old avatar of mine:

(via fractalfactory)

gymbooty:

I want shredded #abs. I will have shredded #abs!

(via workingmuscles)

getting-back-on-track:

Nathalia Melo <3

(Source: powerfit, via workingmuscles)