13472J/1.128J/2158J/16940J COMPUTATIONAL GEOMETRY Lectures 4 and 5 N. M. Patrikalakis Massachusetts Institute of Technology Cambridge MA 02139-4307. USA Copyright 2003 Massachusetts Institute of Technology Contents 4 Introduction to Spline Curves 4.1 Introduction to parametric spline curves 4.2 Elastic deformation of a beam in bending 4.3 Parametric polynomial curves 4.3.1 Ferguson representation 4.3.2 Hermite- Coons curves 4.3.3 Matrix forms and change of basis 13. 4 bezier curves 4.3.5 Bezier surfaces 4.4 Composite curves 4.4.1 Motivation 2224457778889 4.4.3 Continuity conditions 4.4.4 Bezier composite curves(splines) 4.4.5 Uniform Cubic B-splines Bibliography Reading in the Textbook Chapter 1, pp
13.472J/1.128J/2.158J/16.940J COMPUTATIONAL GEOMETRY Lectures 4 and 5 N. M. Patrikalakis Massachusetts Institute of Technology Cambridge, MA 02139-4307, USA Copyright c 2003 Massachusetts Institute of Technology Contents 4 Introduction to Spline Curves 2 4.1 Introduction to parametric spline curves . . . . . . . . . . . . . . . . . . . . . . 2 4.2 Elastic deformation of a beam in bending . . . . . . . . . . . . . . . . . . . . . 2 4.3 Parametric polynomial curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4.3.1 Ferguson representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4.3.2 Hermite-Coons curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4.3.3 Matrix forms and change of basis . . . . . . . . . . . . . . . . . . . . . . 7 4.3.4 B´ezier curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4.3.5 B´ezier surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.4 Composite curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.4.2 Lagrange basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.4.3 Continuity conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.4.4 B´ezier composite curves (splines) . . . . . . . . . . . . . . . . . . . . . . 20 4.4.5 Uniform Cubic B-splines . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Bibliography 28 Reading in the Textbook • Chapter 1, pp.6 - pp.33 1
Lecture 4 Introduction to Spline Curves 4.1 Introduction to parametric spline curves Parametric formulation =r(u),y=y(u), z=2(u) or R=R(u)(vector notation) Usually applications need a finite range for u(e.g. 0<u<1) For free-form shape creation, representation, and manipulation the parametric representa tion is preferrable, see Table 1.2 in textbook. Furthermore, we will use polynomials for the following reasons Cubic polynomials are good approximations of physical splines. (Historical note: Shape of a long flexible beam constrained to pass through a set of points- Draftsman's Splines) Parametric polynomial cubic spline curves are the" smoothest"curves passing through a set of points; (i.e. they minimize the bending strain energy of the beam x Jo xds) 4.2 Elastic deformation of a beam in bending Within the Euler beam theory Center of curvature dA Length ds a中 Fiber igure 4.1: Differential segment of an Euler beam
Lecture 4 Introduction to Spline Curves 4.1 Introduction to parametric spline curves Parametric formulation x = x(u), y = y(u), z = z(u) or R = R(u) (vector notation) Usually applications need a finite range for u (e.g. 0 ≤ u ≤ 1). For free-form shape creation, representation, and manipulation the parametric representation is preferrable, see Table 1.2 in textbook. Furthermore, we will use polynomials for the following reasons: • Cubic polynomials are good approximations of physical splines. (Historical note: Shape of a long flexible beam constrained to pass through a set of points → Draftsman’s Splines). • Parametric polynomial cubic spline curves are the “smoothest” curves passing through a set of points; (i.e. they minimize the bending strain energy of the beam ∝ R L 0 κ 2ds). 4.2 Elastic deformation of a beam in bending Within the Euler beam theory: R y dA dA N.A. y Fiber Length ds d Center of curvature ✁✁✁ ✁✁✁ ✂✁✂✁✂✁✂ ✂✁✂✁✂✁✂ Figure 4.1: Differential segment of an Euler beam. 2
Plane sections of the beam normal to the neutral axis(N A remain plane and normal to the neutral axis after deformation Elongation strain of a fiber at distance y above the NA (y+ r)do-rdo y The radius of curvature can be related to the differential angle and arc length Bdb=ds→k=1=2 The stress experienced by a fiber at distance y from the N. A can be found from Hooke's law relating stress and strain g= E y The bending moment about the N.A. can be calculated by integrating the stress due to bending times the moment arm over the surface of the cross-section Loya=R/dA=-EL ds =-EI For small deflections, the displacement Y= Y() of the N.A., the following approxima tions can be made d 2 ds ds dr2 e Therefore M (4.1) Consider a beam with simple supports and no distributed loads, as shown in Figure 4.2. Figure 4.2: Simply supported beam Looking at a section of the beam between two supports(Figure 4.3), the bending moment can be expressed as a linear equation M=Ao+Ala where Ao and Al are the bending moment and shearing force at a =0 in Figure 4.3
• Plane sections of the beam normal to the neutral axis (N.A.) remain plane and normal to the neutral axis after deformation. • Elongation strain of a fiber at distance y above the N.A.: = (−y + R)dφ − Rdφ Rdφ = − y R • The radius of curvature can be related to the differential angle and arc length: Rdφ = ds ⇒ κ = 1 R = dφ ds • The stress experienced by a fiber at distance y from the N.A. can be found from Hooke’s law relating stress and strain. σ = E = −E y R • The bending moment about the N.A. can be calculated by integrating the stress due to bending times the moment arm over the surface of the cross-section: M = Z A σydA = − E R Z A y 2 dA = −EI 1 R = −EI dφ ds = −EIκ. • For small deflections, the displacement Y = Y (x) of the N.A., the following approximations can be made: φ ≈ dY dx , ds ≈ dx, dφ ds ≈ d 2Y dx2 • Therefore, − M EI ≈ d 2Y dx2 (4.1) • Consider a beam with simple supports and no distributed loads, as shown in Figure 4.2. Figure 4.2: Simply supported beam. Looking at a section of the beam between two supports (Figure 4.3), the bending moment can be expressed as a linear equation: M = A0 + A1x (4.2) where A0 and A1 are the bending moment and shearing force at x = 0 in Figure 4.3. 3
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(1
x 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
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 these
Solving 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 Figure 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