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 beamLecture 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 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 ∝ 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