Corollary. (f є C n+1 [a,b]). Proof: This follows directly from the preceding theorem using the inequality

Transcription

1 Corollary For equidistant knots, i.e., u i = a + i (b-a)/n, we obtain with (f є C n+1 [a,b]). Proof: This follows directly from the preceding theorem using the inequality : ESM4A - Numerical Methods 263

2 4.6 Bézier Curves : ESM4A - Numerical Methods 264

3 Motivation We have seen that polynomial interpolation does not necessarily converge to the to-be-interpolated function when increasing the number of samples. It would be desirable to have such a convergence. As polynomials are simple, it is also desirable to use polynomial bases. We are now looking into a polynomial scheme that does not interpolate the points, but that converges to the to-be-interpolated function when increasing the number of samples : ESM4A - Numerical Methods 265

4 Observation Computation of binomial expansion: polynomial of degree n : ESM4A - Numerical Methods 266

5 Definition The polynomials are the Bernstein polynomials of degree n (for i=0, n). They go back to Сергей Натанович Бернштейн ( ) : ESM4A - Numerical Methods 267

6 Examples Degree 0: Degree 1: Degree 2: : ESM4A - Numerical Methods 268

7 Properties The Bernstein polynomials are linearly independent, i.e., they form a basis. (Proof: Every monomial can be written as a linear combination of Bernstein polynomials.) Symmetry: Roots: Partition of unity: Positiveness: : ESM4A - Numerical Methods 269

8 Recursion Recursive definition: with. (proof by induction) : ESM4A - Numerical Methods 270

9 Definition The polynomial curve is the Bézier representation of degree n. The nodes b i are called Bézier points. The Bézier points b 0,,b n form the Bézier polygon. Each polynomial curve of degree n (or lower) has a unique Bézier representation of degree n. This goes back to Pierre Bézier ( ) : ESM4A - Numerical Methods 271

10 Remark The Bézier representation is defined over the interval [0,1]. It can be easily generalized to any interval [a,b] by using the substitution : ESM4A - Numerical Methods 272

11 Endpoint interpolation Any curve b(u) in Bézier representation over interval [a,b] interpolates the endpoints of the Bézier polygon, i.e., b(a) = b 0 and b(b) = b n. The other Bézier points do, in general, not get interpolated : ESM4A - Numerical Methods 273

12 Example Cubic Bézier curve: Bézier points Bézier polygon Bézier curve convex hull : ESM4A - Numerical Methods 274

13 Convergence theorem Let f(u) be a continuous function. Moreover, let b(u) be the Bézier curve that has nodes b i = f(i/n), i.e., an equidistant sampling of f over interval [0,1]. We have Then, lim n-> sup uє[0,1] f(u) b(u) = : ESM4A - Numerical Methods 275

14 Proof Let K be a random variable with binomial distribution with parameters n and x, i.e., in n independent Bernoulli trials we have probability x of success. Hence, the expected value is E(K/n) = x. The weak law of large numbers of probability theory delivers for any δ > 0 that As f is continuous, we can deduce over a closed interval that : ESM4A - Numerical Methods 276

15 Proof Consequently, and In particular, we obtain that the second probability must go to zero. However, as K is the only probabilistic variable and appears in the scope of the expectation operator, the second probability is either 0 or 1, i.e., it is 0. Furthermore, E(f(K/n)) is exactly the Bézier curve b with nodes b i = f(i/n) : ESM4A - Numerical Methods 277

16 De Casteljau algorithm The De Casteljau algorithm evaluates a Bézier curve at a given position u. It is of similar nature as the Aitken algorithm for Lagrange interpolation curves. It also relies on a recursion. It goes back to Paul de Faget de Casteljau (1930- ) : ESM4A - Numerical Methods 278

17 De Casteljau algorithm : ESM4A - Numerical Methods 279

18 De Casteljau algorithm Scheme: : ESM4A - Numerical Methods 280