Digging deeper into curves

bezier.red

Red [
	Author: "Toomas Vooglaid"
	Date: 2018-03-19
	Inspiration: https://pomax.github.io/bezierinfo
]
lut: [[1][1 1][1 2 1][1 3 3 1][1 4 6 4 1][1 5 10 10 5 1][1 6 15 20 15 6 1]]

binomial: func [n k /local s nextRow i prev][
	while [
		n > s: length? lut
	][
		nextRow: make block! s + 1 
		prev: s - 1 
		repeat i prev [
			append nextRow lut/:s/:i + lut/:s/(i + 1)
		] 
		append/only lut append insert nextRow 1 1
	] 
	lut/:n/:k
]
bezier2a: func [t /local t2 mt mt2][
	t2: t * t
	mt: 1 - t
	mt2: mt * mt
	return mt2 + 2 * mt * t + t2
]

;probe points: collect [repeat i 11 [keep reduce [as-pair (i - 1 * 10) 100 - (100 * bezier2a i - 1 / 10.0)]]]
;[0x100 10x73 20x53 30x38 40x27 50x18 60x12 70x7 80x3 90x0 100x0]
;view compose/deep [box 150x150 draw [box 0x0 100x100 pen red spline (points)]]

;bezier: func [n t w /local sum k][sum: 0 repeat k n [sum: sum + (w/:k * (binomial n k) * ((1 - t) ** (n - k)) * (t ** k))] sum]
bezier: func [t w /local sum-x sum-y k n][
	n: length? w 
	sum-x: 0 sum-y: 0 
	repeat k n [
		sum-x: sum-x + (w/:k/x * (binomial n k) * ((1 - t) ** (n - k)) * (t ** (k - 1)))
		sum-y: sum-y + (w/:k/y * (binomial n k) * ((1 - t) ** (n - k)) * (t ** (k - 1)))
	]
	as-pair sum-x sum-y
]
bezier2: func [t w /local t2 mt mt2][
	t2: t * t 
	mt: 1 - t 
	mt2: mt * mt 
	as-pair 
		(w/1/x * mt2) + (2 * w/2/x * mt * t) + (w/3/x * t2)
		(w/1/y * mt2) + (2 * w/2/y * mt * t) + (w/3/y * t2)
]

bezier3: func [t w /local t2 t3 mt mt2 mt3][
	t2: t * t
	t3: t2 * t
	mt: 1 - t
	mt2: mt * mt
	mt3: mt2 * mt
	as-pair
		(w/1/x * mt3) + (3 * w/2/x * mt2 * t) + (3 * w/3/x * mt * t2) + (w/4/x * t3)
		(w/1/y * mt3) + (3 * w/2/y * mt2 * t) + (3 * w/3/y * mt * t2) + (w/4/y * t3)
]
;comment [
	;Example
	points: [130x50 200x50 400x700 800x150 400x-1200 150x1200 -50x150 50x50 130x50];[0x0 100x200 200x0];[120x160 35x200 220x260 220x40]
	tick: 0
	view/tight compose/deep [
		box 500x300 rate 50 draw [
			pen off 
				line (points) 
			;line-width 2 pen black 
			;	curve (points) 
			line-width 1.75 pen brick 
				c1: circle (points/1) 15
			fill-pen brick
				c2: circle (points/1) 6
				c3: rotate (points/1) 0 [circle (points/1 + 0x-15) 2]
		]
		on-time [
			tick: tick + 1 
			c1/2: c2/2: c3/3: bezier tick % 360 / 360.0 points 
			c3/4/2: c3/3 + 0x-15 
			c3/2: tick % 360 * 3
		]
	]
;]