Figure 4.3: Section of simply supported beam between pins. The shear force and bending moment at the ends of the section are illustrated Upon substituting Equation 4.2 into Equation 4.1 and integrating, we get dy AoI+A Y()Co+C1C+C2:c-+C3c(cubic) Therefore, in order to replicate the shape of physical splines, the CAd community developed shape representation methods based on cubic polynomials Generically, polynomials have the following additional advantages easy to store as sequences of coefficients efficient to compute and trace efficiently easy to differentiate, integrate, and adapt to matrix and vector algebra; and easy to piece together to construct composite curves with a certain order of continuity, a feature important in increasing complexity of a curve or surface 4.3 Parametric polynomial curves 4.3.1 Ferguson representation In 1963, Ferguson at boeing developed a polynomial representation of space curves R(u)=a+a1+a22+a3n3 where< us l by convention. Note there are 12 coefficients, ai, defining the curve r(u) This representation is also known as the power basis or monomial forn The coefficients, ai, are difficult to interpret geometrically, so we can express ai in terms of R(O, R(1), R(O, and R(I)(where R denotes derivative with respect to u R(1) a0+a1+a2+a3 R(O) R(1x A0 A1 A1 M(x) Figure 4.3: Section of simply supported beam between pins. The shear force and bending moment at the ends of the section are illustrated. • Upon substituting Equation 4.2 into Equation 4.1 and integrating, we get: φ ∼= dY dx ∼= − 1 EI [A0x + A1 x 2 2 ] + A2 Y (x) ∼= C0 + C1x + C2x 2 + C3x 3 (cubic) Therefore, in order to replicate the shape of physical splines, the CAD community developed shape representation methods based on cubic polynomials. Generically, polynomials have the following additional advantages: • easy to store as sequences of coefficients; • efficient to compute and trace efficiently; • easy to differentiate, integrate, and adapt to matrix and vector algebra; and • easy to piece together to construct composite curves with a certain order of continuity, a feature important in increasing complexity of a curve or surface. 4.3 Parametric polynomial curves 4.3.1 Ferguson representation In 1963, Ferguson at Boeing developed a polynomial representation of space curves: R(u) = a0 + a1u + a2u 2 + a3u 3 (4.3) where 0 ≤ u ≤ 1 by convention. Note there are 12 coefficients, ai , defining the curve R(u). This representation is also known as the power basis or monomial form. The coefficients, ai , are difficult to interpret geometrically, so we can express ai in terms of R(0), R(1), R˙ (0), and R˙ (1) (where R˙ denotes derivative with respect to u): R(0) = a0 R(1) = a0 + a1 + a2 + a3 R˙ (0) = a1 R˙ (1) = a1 + 2a2 + 3a3 4