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lecnotes45 - 13.472J/1.128J/2.158J/16.940J COMPUTATIONAL...

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Unformatted text preview: 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 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 κ 2 ds ). 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....
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This note was uploaded on 11/08/2011 for the course AERO 16.872 taught by Professor Danielhastings during the Fall '03 term at MIT.

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lecnotes45 - 13.472J/1.128J/2.158J/16.940J COMPUTATIONAL...

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