Geometric Interpretation of Hermite-Coons curves If we make the following substitutions R O R(1)=a1t(1) where t(a) is the unit tangent of the curve, the following observations can be made from Figure 4.5, relating the coefficients ao and an to the shape of the curve const= bias towards t(O) ·a1↑,ao= const→ bias towards t(1) a0,a1↑→ increase fullness ao, a1 both LARGE cusp forms(R(u)=0) or self-intersection occurs ao Increasing t(1) R(0) R(1) a0, a1 Increasing simultaneously R(1) R(0) t(1) Figure 4.5: Fullness of a Hermite-Coons curve Due to the importance of choosing ao and an appropriately, the Hermite-Coons represen- tation can be difficult for designers to use efficiently, but is much easier to understand than the Ferguson or monomial forn
Geometric Interpretation of Hermite-Coons curves. If we make the following substitutions: R˙ (0) = α0t(0) R˙ (1) = α1t(1) where t(u) is the unit tangent of the curve, the following observations can be made from Figure 4.5, relating the coefficients α0 and α1 to the shape of the curve: • α0 ↑, α1 = const ⇒ bias towards t(0) • α1 ↑, α0 = const ⇒ bias towards t(1) • α0, α1 ↑ ⇒ increase fullness • α0, α1 both LARGE ⇒ cusp forms (R˙ (u ∗ ) = 0) or self-intersection occurs R(1) t(0) R(0) t(1) simultaneously α0, α1 increasing α1 constant α0 increasing t(1) t(0) R(0) R(1) Figure 4.5: Fullness of a Hermite-Coons curve. Due to the importance of choosing α0 and α1 appropriately, the Hermite-Coons representation can be difficult for designers to use efficiently, but is much easier to understand than the Ferguson or monomial form. 6
4.3.3 Matrix forms and change of basis Monomial or Power or Ferguson form R() =U·F M 33-2-1R() 0010 R(0) 1000 R(1) Conversion between the two representations is simply a matter of matrix manipulation U·F U·MH·FH F MH·F FH Ma·F 4.3, 4 Bezier curves Bezier developed a reformulation of Ferguson curves in terms of Bernstein polynomials for the UNISURF System at Renault in France in 1970. This formulation is expressed mathematically as follows R(u)=∑RBn()0≤u≤1 Ri and Bi, n(u) represent polygon vertices and the Bernstein polynomial basis functions respec tively. The definition of a Bernstein polynomial is Bi n(u) u2(1-u)-i=0,1,2, il(n-i) The polygon joining Ro, R1,..., Rn is called the control polygon Examples of Bezier curve n=2: Quadratic Bezier Curves(Parabola), see Figures 4.6 and Figure 4.7 R(u)=Ro(1-)2+R12u(1-)+R2 220R 00R2
4.3.3 Matrix forms and change of basis • Monomial or Power or Ferguson form R(u) = h u 3 u 2 u 1 i a3 a2 a1 a0 = U · FM • Hermite-Coons R(u) = h u 3 u 2 u 1 i 2 −2 1 1 −3 3 −2 −1 0 0 1 0 1 0 0 0 R(0) R(1) R˙ (0) R˙ (1) = U · MH · FH Conversion between the two representations is simply a matter of matrix manipulation: U · FM = U · MH · FH Hence, FM = MH · FH FH = M−1 H · FM 4.3.4 B´ezier curves B´ezier developed a reformulation of Ferguson curves in terms of Bernstein polynomials for the UNISURF System at Renault in France in 1970. This formulation is expressed mathematically as follows: R(u) = Xn i=0 RiBi,n(u) 0 ≤ u ≤ 1 Ri and Bi,n(u) represent polygon vertices and the Bernstein polynomial basis functions respectively. The definition of a Bernstein polynomial is: Bi,n(u) = n i ! u i (1 − u) n−i i = 0, 1, 2, ...n where n i ! = n! i!(n − i)! The polygon joining R0, R1, ..., Rn is called the control polygon. Examples of B´ezier curves: • n=2: Quadratic B´ezier Curves (Parabola), see Figures 4.6 and Figure 4.7 R(u) = R0 (1 − u) 2 + R1 2u(1 − u) + R2 u 2 = h u 2 u 1 i 1 −2 1 −2 2 0 1 0 0 R0 R1 R2 7
B 0.0 0.0 1.0 Figure 4.6: Plot of the quadratic Bernstein basis functions. R Figure 4.7: Illustration of end conditions for a quadrati er curve n=3: Cubic Bezier Curves, see Figures 4.8 and 4.9 R(u)=R(1-x)3+R13(1-)2+R23x2(1-)+R3 3 300 R 0R oR Interpolation of Ro, Rn and tangency of curve with polygon at u=0, 1 The Bezier curve approximates and smooths control polygon Properties of Bernstein polynomials 1. Positivity B;n(u)≥0in0≤a≤1
0.0 0.0 1.0 1.0 i=0 u Bi,2(u) i=2 i=1 Figure 4.6: Plot of the quadratic Bernstein basis functions. u = 1 u = 0 R2 R1 R0 Figure 4.7: Illustration of end conditions for a quadratic B´ezier curve. • n=3: Cubic B´ezier Curves, see Figures 4.8 and 4.9 R(u) = R0 (1 − u) 3 + R1 3u(1 − u) 2 + R2 3u 2 (1 − u) + R3 u 3 = h u 3 u 2 u 1 i −1 3 −3 1 3 −6 3 0 −3 3 0 0 1 0 0 0 R0 R1 R2 R3 • Interpolation of R0, Rn and tangency of curve with polygon at u = 0, 1: • The B´ezier curve approximates and smooths control polygon. Properties of Bernstein polynomials 1. Positivity Bi,n(u) ≥ 0 in 0 ≤ u ≤ 1 8
B3() 2 0.0 0.0 1.0 Figure 4.8: Plot of the cubic bernstein basis functions 2. Partition of unity Eio Bi n(u)=[1-u+un=l(by the binomial theorem) Bi n(u) (1-u)B1,n-1()+uB2-1,n-1() ith Bi n(u)=0 for i<0. and Boo(u)=1 4. Linear Precision Property ∑=B,n(u) onversion of explicit curves to parametric Bezier curves Parametrization of straight lines y(a) 5. Degree elevation: The basis functions of degree n can be expressed in terms of those degree n+1 as +1 B1+1m+1(x),i=0,1 +1 6. Symmetry Bi, n(u)= Bn-i,n(1-u 7. Derivative B(u) =n[Bi-In-1(u)-Bin-Iu) where B-1n-1(u)=B,, n-1(u)=0 8. Basis conversion (for each of the y, a, coordinates Ferguson or monomial form f 2 u 1- a f1 f
0.0 0.0 1.0 1.0 u i=3 i=2 i=1 i=0 Bi,3(u) Figure 4.8: Plot of the cubic Bernstein basis functions. 2. Partition of unity Pn i=0 Bi,n(u) = [1 − u + u] n = 1 (by the binomial theorem) 3. Recursion Bi,n(u) = (1 − u)Bi,n−1(u) + uBi−1,n−1(u) with Bi,n(u) = 0 for i < 0, i > n and B0,0(u) = 1 4. Linear Precision Property u = Xn i=0 i n Bi,n(u) • Conversion of explicit curves to parametric B´ezier curves. • Parametrization of straight lines y = y(x) → x = u; y = y(u) 5. Degree elevation: The basis functions of degree n can be expressed in terms of those of degree n + 1 as: Bi,n(u) = 1 − i n + 1 Bi,n+1(u) + i + 1 n + 1 Bi+1,n+1(u), i = 0, 1, · · · , n 6. Symmetry Bi,n(u) = Bn−i,n(1 − u) 7. Derivative B0 i,n(u) = n[Bi−1,n−1(u) − Bi,n−1(u)] where B−1,n−1(u) = Bn,n−1(u) = 0 8. Basis conversion (for each of the x, y, z, coordinates) • Ferguson or monomial form: f(u) = X 3 i=0 fiu i = h u 3 u 2 u 1 i f3 f2 f1 f0 = U · FM 9
RI/ R R R R R Figure 4.9: Examples of cubic Bezier curves approximating their control polygons . Hermite form f(u)= U·M·F f(0) . bezier form Fo f(u)=x32u1 F1 MB:F2 =U·Mp·F F3 Conversion between form U·FM=U.MH·FH=U·MB·FB →F FH=MB·F or→FH=Mn1FM=Mn·MB·FB
R0 X R3 R2 R1 R0 R1 R3 R2 R1 R0 R3 R2 R3 R2 R1 R0 X Figure 4.9: Examples of cubic B´ezier curves approximating their control polygons. • Hermite form: f(u) = h u 3 u 2 u 1 i · MH · f(0) f(1) ˙f(0) ˙f(1) = U · MH · FH • B´ezier form: f(u) = h u 3 u 2 u 1 i · MB · F0 F1 F2 F3 = U · MB · FB • Conversion between forms: U · FM = U · MH · FH = U · MB · FB ⇒ FM = MH · FH = MB · FB or ⇒ FH = M−1 H · FM = M−1 H · MB · FB etc. 10