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De Casteljau's algorithm In the mathematical subfield of numerical analysis the de Casteljau's algorithm, named after its inventor Paul de Casteljau, is a recursive method to evaluate polynomials in Bernstein form or Bézier curves. Although the algorithm is slower for most architectures when compared with the direct approach it is numerically more stable. Definition Given a polynomial B in Bernstein form of degree n $B(t)\; =\; \backslash sum\_^\backslash beta\_b\_(t)$ the polynomial at point t0 can be evaluated with recurrence relation $\backslash beta\_i^\; :=\; \backslash beta\_i\; \backslash mbox\; i=0,\backslash ldots,n$ $\backslash beta\_i^\; :=\; \backslash beta\_i^\; (1-t\_0)\; +\; \backslash beta\_^\; t\_0\; \backslash mbox\; i\; =\; 0,\backslash ldots,n-j\; \backslash mbox\; j=\; 1,\backslash ldots\; n$ with $B(t\_0)=\backslash beta\_0^$. Notes When doing the calculation by hand it is useful to write down the coefficients in a triangle scheme as $$ \begin \beta_0 & = \beta_0^ & & & \\ & & \beta_0^ & & \\ \beta_1 & = \beta_1^ & & & \\ & & & \ddots & \\ \vdots & & \vdots & & \beta_0^ \\ & & & & \\ \beta_ & = \beta_^ & & & \\ & & \beta_^ & & \\ \beta_n & = \beta_n^ & & & \\ \end When choosing a point t0 to evaluate a Bernstein polynomial we can use the two diagonals of the triangle scheme to construct a division of the polynomial $B(t)\; =\; \backslash sum\_^n\; \backslash beta\_i^\; b\_(t)\; \backslash qquad\; \backslash mbox\; \backslash in\; [0,1]$ into $B\_1(t)\; =\; \backslash sum\_^n\; \backslash beta\_0^\; b\_(\backslash frac)\; \backslash qquad\; \backslash mbox\; \backslash in\; [0,t\_0]$ and $B\_2(t)\; =\; \backslash sum\_^n\; \backslash beta\_^\; b\_(\backslash frac)\; \backslash qquad\; \backslash mbox\; \backslash in\; [t\_0,1]$ Example We want to evaluate the Bernstein polynomial of degree 2 with the Bernstein coefficients $\backslash beta\_0^\; =\; \backslash beta\_0$ $\backslash beta\_1^\; =\; \backslash beta\_1$ $\backslash beta\_2^\; =\; \backslash beta\_2$ at the point t0. We start the recursion with $\backslash beta\_0^\; =\; \backslash beta\_0^\; (1-t\_0)\; +\; \backslash beta\_1^t\; =\; \backslash beta\_0(1-t\_0)\; +\; \backslash beta\_1\; t\_0$ $\backslash beta\_1^\; =\; \backslash beta\_1^\; (1-t\_0)\; +\; \backslash beta\_2^t\; =\; \backslash beta\_1(1-t\_0)\; +\; \backslash beta\_2\; t\_0$ and with the second iteration the recursion stops with $$ \begin \beta_0^ & = & \beta_0^ (1-t_0) + \beta_1^ t_0 \\ \ & = & \beta_0(1-t_0) (1-t_0) + \beta_1 t_0 (1-t_0) + \beta_1(1-t_0)t_0 + \beta_2 t_0 t_0 \\ \ & = & \beta_0 (1-t_0)^2 + \beta_1 2t_0(1-t_0) + \beta_2 t_0^2 \\ \end which is the expected Bernstein polynomial of degree n. Bézier curve When evaluating a Bézier curve of degree n in 3 dimensional space with n+1 control points Pi $\backslash mathbf(t)\; =\; \backslash sum\_^\; \backslash mathbf\_i\; b\_(t)\; \backslash mbox\; t\; \backslash in\; [0,1]$ with $\backslash mathbf\_i\; :=$ \begin x_i \\ y_i \\ z_i \end . we split the Bézier curve into three separate equations $B\_1(t)\; =\; \backslash sum\_^\; x\_i\; b\_(t)\; \backslash mbox\; t\; \backslash in\; [0,1]$ $B\_2(t)\; =\; \backslash sum\_^\; y\_i\; b\_(t)\; \backslash mbox\; t\; \backslash in\; [0,1]$ $B\_3(t)\; =\; \backslash sum\_^\; z\_i\; b\_(t)\; \backslash mbox\; t\; \backslash in\; [0,1]$ which we evaluate individually using de Casteljau's algorithm. Pseudocode example Here is a pseudo-code example that recursively draws a curve from point P1 to P4, tending towards points P2 and P3. The level parameter is the limiting factor of the recursion. The procedure calls itself recursively with an incremented level parameter. When level reaches the max_level global value, a line is drawn between P1 and P4 at that level. The function midpoint takes two points, and returns the middle point of a line segment between those two points. global max_level = 5 procedure draw_curve(P1, P2, P3, P4, level) if (level > max_level) draw_line(P1, P4) else L1 = P1 L2 = midpoint(P1, P2) H = midpoint(P2, P3) R3 = midpoint(P3, P4) R4 = P4 L3 = midpoint(L2, H) R2 = midpoint(R3, H) L4 = midpoint(L3, R2) R1 = L4 draw_curve(L1, L2, L3, L4, level + 1) draw_curve(R1, R2, R3, R4, level + 1) end procedure draw_curve Sample implementations Haskell Compute a point R by performing a linear interpolation of points P and Q with parameter t. Usage: linearInterp P Q t P = list representing a point Q = list representing a point t = parametric value for the linear interpolation, t<-[0..1] Returns: List representing the point (1-t)P + tQ > linearInterp :: [Float]->[Float]->Float->[Float] > linearInterp [] [] _ = [] > linearInterp (p:ps) (q:qs) t = (1-t)*p + t*q : linearInterp ps qs t Compute one level of intermediate results by performing a linear interpolation between each pair of control points. Usage: eval t b t = parametric value for the linear interpolation, t<-[0..1] b = list of control points Returns: For n control points, returns a list of n-1 interpolated points > eval :: Float->Float->Float > eval t (bi:bj:[]) = [linearInterp bi bj t] > eval t (bi:bj:bs) = (linearInterp bi bj t) : eval t (bj:bs) Compute a point on a Bezier curve using the de Casteljau algorithm. Usage: deCas t b t = parametric value for the linear interpolation, t<-[0..1] b = list of control points Returns: List representing a single point on the Bezier curve > deCas :: Float->Float->[Float] > deCas t (bi:[]) = bi > deCas t bs = deCas t (eval t bs) Compute the points along a Bezier curve using the de Casteljau algorithm. Points are returned in a list. Usage: bezierCurve n b n = number of points along the curve to compute b = list of Bezier control points Returns: A list of n+1 points on the Bezier curve Example: bezierCurve 50 [[0,0],[1,1],[0,1],[1,0]] > bezierCurve :: Int->Float->Float > bezierCurve n b = [deCas (fromIntegral x / fromIntegral n) b | x<-[0 .. n] ] References Farin, Gerald & Hansford, Dianne (2000). The Essentials of CAGD. Natic, MA: A K Peters, Ltd. ISBN 1-56881-123-3 See also Horner scheme to evaluate polynomials in Monomial form Clenshaw algorithm to evaluate polynomials in Chebyshev form Most Popular Articles: Music    Musical Instruments    Guitar    Classical Music    History    Music Definition    Country Music    Rock Music    Pop Music    Hip Hop    Rap    DJ    Video Games    Casino Games    Poker    Blackjack    Casinos    Real Estate    Loan    Auction    eBay    Marketing    Search Engine Optimization    File Sharing    MP3    Music Industry    Kazaa    Internet    Supermodel    Fashion    Fashion Designers    Erotica    Modeling    Clothing    Boots    Shoes    Diamond Engagement Rings    Diamond    Rugs    Gospel Music    Affiliate    E-Commerce    Radio This article is licensed under the GNU Free Documentation License at http://www.gnu.org/copyleft/fdl.html You may copy and modify it as long as the entire work (including additions) remains under this license. 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