You know, sine and cosine really are pretty great -
they're smooth,
they're orthogonal functions,
they're computable from Pythagoras theorem and function of a circle, then being even resp. odd functions,
there are lots of derived functions -
haversine, arcsin, hyperbolic sine, complementary sine
then one thing I've always wondered is about alternate constructions of them,
that make for ways to form constructions of more kinds of similar rich functions.

For example, imagine a (horizontal) line, and, an equilateral triangle sitting on it.
Treat the triangle as having hinges at each vertex. So, unhinge the left vertex,
then opening the other hinges at the same rate, trace the swinging end, until the
hinges are open (flat). It traces the positive part of a curve. Flip the resulting
line under, and close the hinges back, and it draw the negative part of the curve,
and results a periodic function. It's the same shape as the sine curve.

So I'm hoping someone can point me to where this is detailed, then also about
what happens for regular n-gons tracing curves these ways.

The case of just a line segment or just a 2-gon (...), it traces a semi-circle in the
positive then flips then traces the semi-circle in the negative and so on, making
the simplest sort of periodic function that is connected semi-circles.

So I'm hoping you can tell me about how it works out that these curves, basically
make it so that each n-gon does make a continuous function and a periodic function
that's symmetrical in the odd and even when centered then absolute, that
increasing that as the n-gon goes to infinitely many sides, it sort of draws a square wave,
then whether it does or not for "unrolling the circle".

"Hinged dissections"? "Swinging and Twisting"? "Linkages"? "Geared continuous hinges"?

Anyways the generality of orthogonal functions is very usual in the development of
all kinds of transforms in differential analysis.

On Sunday, May 28, 2023 at 9:34:08 AM UTC-7, Ross Finlayson wrote:
> You know, sine and cosine really are pretty great -
> they're smooth,
> they're orthogonal functions,
> they're computable from Pythagoras theorem and function of a circle, then being even resp. odd functions,
> there are lots of derived functions -
> haversine, arcsin, hyperbolic sine, complementary sine
> then one thing I've always wondered is about alternate constructions of them,
> that make for ways to form constructions of more kinds of similar rich functions.
>
> For example, imagine a (horizontal) line, and, an equilateral triangle sitting on it.
> Treat the triangle as having hinges at each vertex. So, unhinge the left vertex,
> then opening the other hinges at the same rate, trace the swinging end, until the
> hinges are open (flat). It traces the positive part of a curve. Flip the resulting
> line under, and close the hinges back, and it draw the negative part of the curve,
> and results a periodic function. It's the same shape as the sine curve.
>
> So I'm hoping someone can point me to where this is detailed, then also about
> what happens for regular n-gons tracing curves these ways.
>
> The case of just a line segment or just a 2-gon (...), it traces a semi-circle in the
> positive then flips then traces the semi-circle in the negative and so on, making
> the simplest sort of periodic function that is connected semi-circles.
>
> So I'm hoping you can tell me about how it works out that these curves, basically
> make it so that each n-gon does make a continuous function and a periodic function
> that's symmetrical in the odd and even when centered then absolute, that
> increasing that as the n-gon goes to infinitely many sides, it sort of draws a square wave,
> then whether it does or not for "unrolling the circle".
>
>
> "Hinged dissections"? "Swinging and Twisting"? "Linkages"? "Geared continuous hinges"?
>
> Anyways the generality of orthogonal functions is very usual in the development of
> all kinds of transforms in differential analysis.

It seems for a sort of axle and belt system, what results for the "geared continuous hinges",
that they are also geared together, then that the angle bisector going from the closed
equilateral triangle to the flat line, results that the center of the closed triangle results
the line bisector when it's open.

So, this sort of "un-rolling" gets into a model of "discrete curvature" that when the
triangle is closed it's curved and when it's unrolled that it's flat.

Then, there's the tracing of the interpolation of the loose or swinging end,
then also the tracing of the "center" and each of the intersections of the angle
and segment and "perimeter bisectors", about that this is a form that happens
to reproduce the trigonometric functions, their curves.