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lecnotes13 - 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 Lecture 13 N. M. Patrikalakis Massachusetts Institute of Technology Cambridge, MA 02139-4307, USA Copyright c 2003 Massachusetts Institute of Technology Contents 13 Offsets of Parametric Curves and Surfaces 2 13.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 13.2 Parametric offset curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 13.2.1 Differential geometry of parametric offset curves . . . . . . . . . . . . . 5 13.2.2 Singularities of parametric offset curves . . . . . . . . . . . . . . . . . . 6 13.2.3 Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 13.3 Parametric offset surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 13.3.1 Differential geometry of parametric offset surfaces . . . . . . . . . . . . 10 13.3.2 Singularities of parametric offset surfaces . . . . . . . . . . . . . . . . . 11 13.3.3 Tracing algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 13.3.4 Self-intersections of offsets of explicit quadratic surfaces . . . . . . . . . 14 13.3.5 Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Bibliography 22 Reading in the Textbook Chapter 11, pp. 293 - 353 1 Lecture 13 Offsets of Parametric Curves and Surfaces 13.1 Motivation Offsets are defined as the locus of points at a signed distance d along the normal of a planar curve or surface. A literature survey on offset curves and surfaces up to 1992 was carried out by Pham [24], while the overview of the literature after 1992 and those which were not cited in [24] is given by Maekawa [14]. Offset curves and surfaces are widely used in various engineering applications, such as Tool path generation for pocket(2.5D), 3D and 5D NC machining [9, 1]. (See Figure 13.1). Generator Surface Tool Driving Plane Center of Ball Endmill Ball Endmill Tool Path Offset Surface Figure 13.1: NC machining. 2 Definition of tolerance regions [4, 26, 21]. (See Figure 13.2). Figure 13.2: Definition of tolerance regions. Access space representations in robotics [12]. (See Figure 13.3) Figure 13.3: Access space representations in robotics. 3 Curved plate (shell) representation in solid modeling [23]. (See Figure 13.4) Figure 13.4: Plate representation. Feature recognition through construction of skeletons or medial axes of geometric models [22, 29]. (See Figure 13.5). The medial axis is made up of boundary offset intersections. Figure 13.5: Medial Axis. The concept of offset curves generalizes to pipe surfaces when the progenitor is a general 3 D curve [18]. geodesic offsets when the progenitor is curve on a surface [20] [25] [11]....
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lecnotes13 - 13.472J/1.128J/2.158J/16.940J COMPUTATIONAL...

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