Construction of Bezier curves, step by step.

At each step the number of points decreases by one. The last step leaves us only one point. What happens when we increase the order of curves? And how can the computation cost be reduced, the curves easier to use? The motivation and blinding algebra are worked out here: [url]http://poppersdreamland.blogspot.com/2013/01/splines-and-bezier-curves-part-i.html[/url] ....a gentle warning that mathematical induction can be laborious and hard to follow. These are personal projects. Please let me know how I can make these materials more useful to you. ____________________ Bézier Curves [b]→1. Construction[/b] 2. Higher Order Curves: [url]http://www.geogebratube.org/material/show/id/27627[/url] 3. Basic Implementation: [url]http://www.geogebratube.org/material/show/id/27631[/url] 4. Weights: [url]http://www.geogebratube.org/material/show/id/28313[/url] 5. Continuous path: [url]http://www.geogebratube.org/material/show/id/81193[/url] 6. ...

[b]Tool (.ggt):[/b] [url]http://www.geogebratube.org/material/show/id/83831[/url] [math] \;\;\;\;[/math]Command: BezierCf[n] [math] \;\;\;\;[/math]Output: (n+1)×(n+1) matrix [math] C [/math]

Given a list of Points [math]\;\;\; {\rm Pts} = [A_0, A_1, .... A_n], [/math] and a variable of motion, [i]t[/i] [math] \;\;\;\tau(t) = [1, t, t^2 , ..., t^n]^T,[/math] In GGB,[i]t[/i] can be used as a scalar parameter, with the Locus[] command. For a Curve[], trade t for x: [math] \;\;\;[/math]τ = Sequence[x^k, k, 0, n] The x- and y- components of the curve are then [math] \;\;\;[/math]βx = Sum[Zip[a b, a, Join[{x(Pts)} C], b, τ]] [math] \;\;\;[/math]βy = Sum[Zip[a b, a, Join[{y(Pts)} C], b, τ]] From which the Bézier curve is [math] \;\;\;[/math]β = Curve[βx(a), βy(a), a, 0, 1] [i]What?[/i] GGBScript is not math. In math, the product [math]\;\;\;\;[/math]β = Pts C τ(t), 0≤t≤1 is immediate and unambiguous. It's cute that math people wonder why students have a hard time with math at the computer. ? When did we make it possible for them to do math at the computer? ________ Bézier Curves: 2. High Order Curves [list] [*] Standard Curve: [math] \;\;\;[/math][b]→ Coefficient Matrix[color=#1551b5](+TOOL)[/color][/b] [math] \;\;\;[/math] Bézier Curve of Order n[color=#1551b5] (+TOOL)[/color]: [url]http://www.geogebratube.org/material/show/id/83844[/url] [/list] [list] [*]Curve from Path: [url]http://www.geogebratube.org/material/show/id/84017[/url] [math] \;\;\;[/math]Coefficient Matrix [color=#1551b5](+TOOL)[/color]: [url]http://www.geogebratube.org/material/show/id/84218[/url] [math] \;\;\;[/math]Bézier Path of Order n [color=#1551b5](+TOOL)[/color]: [url]http://www.geogebratube.org/material/show/id/84231[/url] [/list]

The 3rd order Bezier curve is pretty compact. It may still prove useful to have a zeroed prototype of unit size, so here's one. Let's see if I can improve the implementation...

1. Given three arbitrary points in space, A, B, C, the second order Bézier curve has equation: [math]A + 2(B-A)t + (C-2B +A)t^2, \;\;\; 0 ≤ t ≤1[/math] 2. Let ΔABC be an Archimedes triangle with base (chord) AC, and tangents AB, BC. ΔABC encloses a parabola section. [i]The two curves are the same.[/i]

__________________ Archimedes Triangle: [url]http://www.geogebratube.org/material/show/id/35781[/url] Vector Equation of a Parabola: [url]http://www.geogebratube.org/material/show/id/32226[/url] Tool: Parabola from Archimedes Triangle (two points and the intersection of their tangents): [url]http://www.geogebratube.org/material/show/id/38279[/url]