Complejo de Edipo: Mujer

For the parametrization of curves, surfaces and solids a special basis of the Bernstein polynomials defined on a parameter interval [a, b] is used. Since we can transform each interval [a, b] :={t ∈ R|a ≤ t ≤ b} into the unit interval I := [0, 1] := {λ ∈ R|0 ≤ λ ≤ 1} by an affine transformation λ = (t− a)/(b − a) we consider the definition and properties of these special polynomials on this unit interval.

Definition 1.2.1 (Bernstein Polynomials). The i’th Bernstein polynomial of degree n on the interval I is defined as (cf. Fig. 1.1)

B_in(λ) = n!

(n− i)!i!(1− λ)

n−i_λi_{, i = 0, 1, . . . , n. (1.1)}

Figure 1.1: Bernstein Polynomials of degree one (left), two (middle), and three (right), where the colors (from blue to red) indicate the ith polynomial.

There are some properties of the Bernstein polynomials which turned out to be useful for application and proofs. However, the most important ones are symmetry, boundedness and the maximum principle on the unit interval. Finally the recursion formula is stated in

Theorem 1.2.1 (Bernstein Polynomial Recursion Formula). The Bernstein polynomials satisfy the recursion

B_in(λ) = (1− λ)B_in−1(λ) + λB_i−1n−1(λ), λ∈ R, i = 1, 2, . . . , n. (1.2)

CHAPTER 1. INTRODUCTION

and the explicit derivatives are specified in

Theorem 1.2.2 (Bernstein Polynomial Derivatives). The derivatives of the Bernstein polynomials are given by

∂ ∂λB n i (λ) = ⎧ ⎨ ⎩ −nBn−1 0 (λ) if i = 0, +n[B_i−1n−1(λ)− B_in−1(λ)] if i = 1, 2, . . . , n− 1, +nBn−1_n−1(λ) if i = n. (1.3)

These Bernstein polynomials define the basis functions for B´ezier curves. The geometric properties of B´ezier curves were developed independently by de Casteljau and B´ezier and later it was found by Forrest that there is a connection between B´ezier curves and the original Bernstein polynomials (cf. notes below and [Far02]). However, we arrive at the so called Bernstein-B´ezier form

s|_I(λ) =

n



i=0

b_iB_in(λ), λ∈ I (1.4)

of a polynomial curve, where b_i ∈ Rd_{are called B´ezier points and λ the local coordinate.}

Together with the Bernstein polynomials the Bernstein-B´ezier curve s|_I(λ) is defined. The B´ezier points as well as the curve have an illustrative and geometric meaning. That means, if the B´ezier points become b_i := (i/n) ∈ I and are connected according to the natural order of their indices, then we obtain the associated B´ezier grid (or net), which defines the convex hull of the polynomial piece (cf. Fig. 1.2). However, if the B´ezier points become b_i := (i/n, a_i) ∈ I × R the resulting spline can be considered as a function-valued formula and the coefficients a_i ∈ R as ordinates associated with abscissae (i/n) ∈ I, i.e (λ, s|_I) defines a surface in I × R. Such B´ezier splines can be used to approximate a one-dimensional function f (t) as shown in Fig. 1.2. B´ezier points of the form b_i := (i/n, c_i) ∈ I × Rl can be used to describe a vector-valued function

f(t) := (f_x1(t), f_x2(t), . . . , f_xl(t))∈ Rl.

Figure 1.2: Bernstein-B´ezier curve (left) using Bernstein polynomials of degree two (right). The red dots represent some data (e.g. obtained from an explicit polynomial), whereas the blue circles represent the B´ezier points usually computed from the data points and the blue curve was reconstructed using Equ. (1.4).

These curves in Bernstein-B´ezier form have some interesting and useful properties as well, which result often directly from the Bernstein polynomials. One should have in mind the most important properties, when developing algorithms using this kind of piecewise polynomials. First, the 1’st derivative of Equ. 1.4 can be computed by

∂

∂λs|I(λ) = n n−1_

i=0

(b_i+1− b_i)B_in−1(λ), (1.5)

where the 2’nd derivative is written as

∂2

∂2λs|I(λ) = n(n− 1) n−2_

i=0

(b_i+2− 2b_i+1+ b_i)Bn−2_i (λ), (1.6)

and both equations directly follow from Equ. 1.3. Then, the convex hull property, which is one of the most important properties, speeding up intersection computations between diffenent curves.

Theorem 1.2.3 (Convex Hull Property). The set of all B´ezier points

M :=  s|_I(λ) = n  i=0 b_iB_in(λ)| λ ∈ I  (1.7)

is contained in the convex hull of the n + 1 B´ezier points b_i ∈ Rd, i = 0, 1, . . . , n.

Smoothness properties are very important and can be used to connect piecewise poly- nomials in Bernstein-B´ezier form, i.e. to set neighboring B´ezier points of two different pieces in a way that for example one obtains a total C1-continuous curve (see Fig. 1.2). In special we have – without loss of generality – two Bernstein-B´ezier polynomials s|_Iand

s|_J defined on two intervals I := [a, b] and J := [c, d], where h_b := b− a and h_d= d− c. Then, if b_n,b = b_0,d and b_n,b = hd

hb+hdbn−1,b +

hb

hb+hdb1,d we obtain a C

1_-continuous patch which is built up by these two piecewise polynomials. The final property which is useful to evaluate piecewise polynomials in Bernstein-B´ezier form is the de Casteljau algorithm. This algorithm is numerically stable and directly results from the recursion formula (1.2).

Algorithm 1.2.1 (Univariate de Casteljau). The input to this algorithm is an array of n + 1 B´ezier points b_i ∈ Rd, i = 0, 1, . . . , n (i.e. the polynomial piece is of total degree n) and the local coordinate λ∈ I. Here b0_i := b_i are the B´ezier points from Equ. (1.4) defining the convex hull of the polynomial piece and the algorithm to compute the value s|_I(λ) is

1: for j = 1 <= n do

2: for i = 0 <= n− j do

3: {Compute second derivative.} 4: if j==n-1 then 5: ∂2 ∂2λs|I(λ) = n(n− 1)(b j−1 i+2− 2bj−1i+1+ bj−1i ) 6: end if {Compute first derivative.} 7: if j==n then

CHAPTER 1. INTRODUCTION 8: ∂ ∂λs|I(λ) = n(b j−1 i+1 − bj−1i ) 9: end if {Compute value.} 10: if j==n then 11: s|_I(λ) = (1− λ)bj−1_i + λbj−1_i+1 12: end if

{Recursion to compute new values on layer j.}

13: bj

i = (1− λ)bj−1i + λbj−1i+1

14: end for

15: end for

As output we obtain the result s|_I(λ) = bn₀, i.e. the value on the one-dimensional curve, and the first and second partial derivatives (see also Equ. 1.5 and 1.6). Note that in step j the old values bj−1_i and bj−1_i+1 in the array can be replaced by the newly computed values bj_i. There is no need to allocate space for another temporary array.

However, the Bernstein polynomials are used to construct one-dimensional curves in Bernstein-B´ezier form. Similarly, but with little variation of the notation, the same poly- nomials are utilized for the parametrization of surfaces and solids defined over triangular and tetrahedral partitions. For the one-dimensional case the Bernstein polynomials can be rewritten as follows.

Definition 1.2.2 (Bernstein Polynomials). The Bernstein polynomials of degree n on the interval I are defined as

B_τn_0,τ1(λ₀, λ₁) = n! τ₀!τ₁!λ τ0 0 λτ11, |τ0+ τ1| = n, (1.8) where Bn τ0,τ1(λ0, λ1) = 0 if τν ∈ {0, 1, . . . , n} for some ν ∈ {0, 1}, λ/ ν ≥ 0, ₁ ν=0λν = 1, and λ_ν are called the barycentric coordinates on the interval I. Finally, if τ₀:= (n− i),

τ₁ := i, λ₀:= (1− λ), and λ₁:= λ we arrive at Def. 1.2.1.

Therefore, given an interval I with {λ_ν ∈ R, ν ∈ {0, 1} | 0 ≤ λ_ν ≤ 1, 1_ν=0λ_ν = 1} and two points p₀, p₁ at the interval boundaries any point on a line can be expressed in terms of barycentric coordinates λ_ν with respect to this non degenerate interval I as

p =_νλ_νp_ν. Now, Equ. (1.4) can be rewritten as

s|_I(λ₀, λ₁) =  |τ0+τ1|=n b_τ0,τ1B_τn_0,τ1(λ₀, λ₁) (1.9) =  |τ0+τ1|=n b_τ0,τ1 n! τ₀!τ₁!λ τ0 0 λτ11, (1.10)

where b_τ0,τ1 ∈ R2 are called the B´ezier points defined as b_τ0,τ1 := ((τ₀, τ₁)/n, a_τ0,τ1), where a_τ0,τ1 are the B´ezier ordinates associated with the abscissas ((τ₀, τ₁)/n).

Note, the classical Horner scheme – also numerically stable – is an analogous method to evaluate a polynomial at a given point location λ = λ₀ which is written in terms of the monomial basis. Whereas, the de Boor algorithm is a generalization of de Casteljau’s algorithm for evaluating B-spline curves, which is fast and numerically stable as well.

Remark (Bernstein-B´ezier Curves). Sergei Natanovich Bernstein an Ukrainian math- ematician (1880-1968) has introduced new polynomials in a constructive proof of the ”Stone-Weierstrass” approximation theorem. These ”Bernstein” polynomials have been used later by Paul de Casteljau (1910 - 1999) – an engineer at Citro¨an – for the ap- proximation of curves. At the same time Pierre B´ezier (1910-1999) came to the same curves working for the competitor Renault. Robin Forrest realized that de Casteljau and B´ezier did the same job. Then, the noun ”Bernstein-B´ezier” curve appears, and due to the progress of computer graphics these curves become more and more popular (cf. [Far02]).

Figure 1.3: Biographical profiles of Sergei Natanovich Bernstein (left), and Pierre B´ezier (right). By courtesy of Wikipedia

2 Tensor Product B´ezier Splines

Tensor product splines in Bernstein-B´ezier form are usually defined on finite rectilinear, volumetric domains Ω⊂ R3.