Derivative. Bernstein polynomials: Jacobs University Visualization and Computer Graphics Lab : ESM4A - Numerical Methods 313

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1 Derivative Bernstein polynomials: : ESM4A - Numerical Methods 313

2 Derivative Bézier curve (over [0,1]): with differences. being the first forward : ESM4A - Numerical Methods 314

3 Derivative Bézier curve (over [a,b]): Higher-order derivatives: with differences. being the rth forward : ESM4A - Numerical Methods 315

4 4.9 Piecewise Hermite Interpolation : ESM4A - Numerical Methods 316

5 Motivation & idea We observed that polynomials fulfilling many interpolation constraints tend to oscillate. The idea is to replace one high-order polynomial with several low-order polynomials, each of which covers a part of the knot sequence. The challenge is to stitch these low-order polynomials together in a way that a curve with sufficient continuity is generated. This continuity is what we want to achieve using Hermite interpolation to interpolate derivatives where curves come together : ESM4A - Numerical Methods 317

6 Design choice In the following we are targeting curves that are C 1 - continuous. In order to achieve C 1 -continuity, we have to make sure that the endpoint of the i-th curve segment and the startpoint of the (i+1)-st curve segment have the samefunctionvalueand thesamederivative. Using Hermite interpolation, we have interpolation conditions up to the k-th derivative. Here, k = 1, and therefore degree n = 2k+1 = 3. Hence, the curve segments will be cubic polynomials : ESM4A - Numerical Methods 318

7 Remark Of course, we can get C k -continuity for k > 1. However, C 1 -continuity is for many applications sufficient. Moreover, to obtain C k -continuity for k > 1, one has to use higher-order polynomials (n 5), which may insert oscillations again. Still, the method we are going to look into generalizes to k > 1 in a straight-forward manner : ESM4A - Numerical Methods 319

8 Piecewise cubic Hermite interpolation Given n+1 points p 0,,p n and the corresponding knots u 0,,u n, we generate a piecewise cubic C 1 -curve s(u) such that s([u i,u i+1 ]) is a cubic polynomial. Curve s(u) is uniquely defined by the interpolation conditions s(u i ) = p i for i=0,,n s (u i ) = d i for i=0,,n where d 0,,d n are the derivatives at knots u 0,,u n. Hence, s(u) interpolates function values p i and derivatives d i at the knots u i : ESM4A - Numerical Methods 320

9 Illustration Note that the interpolation conditions for p i and d i at knot u i apply to both cubic polynomials s([u i-1,u i ]) and s([u i,u i+1 ]), which guarantees C 1 -continuity : ESM4A - Numerical Methods 321

10 Polynomial curve s([u i,u i+1 ]) To describe the polynomial curve segment s([u i,u i+1 ]), we make use of the nice property of Bézier curves, where the derivative of a Bézier curves is again a Bézier curve of one degree less. Recall: : ESM4A - Numerical Methods 322

11 Cubic Bézier curves Notation: Let s i (u) := s([u i,u i+1 ]). Then, s i (u) with u є [u i,u i+1 ] is a cubic Bézier curve with point and derivative interpolation conditions at the endpoints. A cubic Bézier curve is uniquely determined by its 4 Bézier points. For s i (u) with u є [u i,u i+1 ] we have to find Bézier points b 3i, b 3i+1, b 3i+2, and b 3i+3. Because of the continuity criterion, we will see that Bézier points b 3i is being used as first Bézier points of s i (u) and as last Bézier points of s i-1 (u). Hence, each additional interval [u i,u i+1 ] adds 3 new Bézier points : ESM4A - Numerical Methods 323

12 Cubic Bézier curves We get with local parameter and. What is left is to determine the Bézier points b 3i, b 3i+1, b 3i+2, and b 3i : ESM4A - Numerical Methods 324

13 Computing Bézier points Computing b 3i : Point interpolation condition: As we know that a Bézier curve is interpolating the endpoints of the Bézier polygon, i.e., we directly obtain that : ESM4A - Numerical Methods 325

14 Computing Bézier points Computing b 3i+3 : Because of symmetry, we analogously obtain b 3i+3. Point interpolation condition: Endpoint interpolation: Hence: : ESM4A - Numerical Methods 326

15 Computing Bézier points Computing b 3i+1 : Derivative interpolation condition: : ESM4A - Numerical Methods 327

16 Computing Bézier points Computing b 3i+2 : Because of symmetry, we analogously obtain b 3i+2. Derivative interpolation condition: : ESM4A - Numerical Methods 328

17 Illustration of Bézier curve Bézier polygon Bézier curve : ESM4A - Numerical Methods 329

18 Illustration of piecewise Hermite interpolation overall Bézier polygon piecewise interpolating curve s(u) : ESM4A - Numerical Methods 330