& Lesson 12: c_ngon
& Clayson walks you through this step by step
& It's just high school trigonometry
& But that probably strikes terror into the hearts of PhD students in the humanities
& So I've adapted his solution.
& He walks you carefully through it on pp. 24-31.

fn ngon! (sides, size) -> repeat sides {
	fd! (size)
	rt! (inv (sides))
}

fn c_ngon! (sides, radius) -> {
	& first, some fancy trigonometry
	& don't read this if you don't want to
	& but: there are two hard problems here
	& how far to turn before drawing the ngon
	& and how long each side should be to fit inside the radius
	let entry_turn = sub (
		0.5
		mult (
			0.25 
			sub (sides, 2)
			inv (sides)))

	let side_length = mult (
		2
		radius
		sin (div (0.5, sides)))

	& now that we have the math: what do we need the turtle to do
	pu! ()
	fd! (radius)
	pd! ()
 	rt! (entry_turn)
	ngon! (sides, side_length)
	lt! (entry_turn)
	pu! ()
	bk! (radius)
	pd! ()
}