Solving the above equations yields expression for the coefficients, ais, in terms of the geometric end conditions of the curve. a0= R(O R(0 3R(1)-R(O)]-2R(0)-R(1) a3=2R(0)-R(1)+R(0)+R(1) 4.3.2 Hermite-Coons curves By substituting the coefficients into the Ferguson representation, we can rewrite Equation 4.3 R(u)=R(0)(t)+R(1)mn(u)+R(0)m2(u)+R(1)(u) (4.4) 0( n(a)=3x2-23 m(u)=u-2n2+u3 The new basis functions, ni(u), are known as Hermite polynomials or blending functions see Fi igure 4 ere first used for 3D curve representation in a computer environ the 60,s by the late Steven Coons, an MIT professor, participant in the famous ARPa project MAC 1.0 7 0.0 72 0.0 Figure 4. 4 Plot of hermite basis functions Note that these basis functions, ni(u), satisfy the following boundary conditions, see Fig- 7()=1,m(1)=m0(0)=m6(1)=0; mn(1)=1,m(0)=mh(0)=m1(1)=0; n2(0)=1,m2(0)=n(1)=m2(1)=0; n3(1)=1,m3(O)=n(0)=3(1)=0. These boundary conditions also allow the computation of the cubic Hermite polynomials, ni(u), by setting up and solving a system of 16 linear equations in the coefficients of theseSolving the above equations yields expression for the coefficients, ai ’s, in terms of the geometric end conditions of the curve. a0 = R(0) a1 = R˙ (0) a2 = 3[R(1) − R(0)] − 2R˙ (0) − R˙ (1) a3 = 2[R(0) − R(1)] + R˙ (0) + R˙ (1) 4.3.2 Hermite-Coons curves By substituting the coefficients into the Ferguson representation, we can rewrite Equation 4.3 as: R(u) = R(0)η0(u) + R(1)η1(u) + R˙ (0)η2(u) + R˙ (1)η3(u) (4.4) where η0(u) = 1 − 3u 2 + 2u 3 η1(u) = 3u 2 − 2u 3 η2(u) = u − 2u 2 + u 3 η3(u) = u 3 − u 2 The new basis functions, ηi(u), are known as Hermite polynomials or blending functions, see Figure 4.4. They were first used for 3D curve representation in a computer environment in the 60’s by the late Steven Coons, an MIT professor, participant in the famous ARPA project MAC. 0.0 1.0 0.0 1.0 0 1 2 3 Figure 4.4: Plot of Hermite basis functions. Note that these basis functions, ηi(u), satisfy the following boundary conditions, see Fig￾ure 4.4: η0(0) = 1, η0(1) = η 0 0 (0) = η 0 0 (1) = 0; η1(1) = 1, η1(0) = η 0 1 (0) = η 0 1 (1) = 0; η 0 2 (0) = 1, η2(0) = η 0 2 (1) = η 0 2 (1) = 0; η 0 3 (1) = 1, η3(0) = η 0 3 (0) = η 0 3 (1) = 0. These boundary conditions also allow the computation of the cubic Hermite polynomials, ηi(u), by setting up and solving a system of 16 linear equations in the coefficients of these polynomials. 5