13472J/1.128J/2158J/16940J COMPUTATIONAL GEOMETRY Lecture 13 N. M. Patrikalakis Massachusetts Institute of Technology Cambridge MA 02139-4307. USA Copyright @2003 Massachusetts Institute of Technology Contents 1 3 Offsets of Parametric curves and surfaces 13.1 Motivation 13.2 Parametric offset curves 13.2.1 Differential geometry of parametric offset curves 13.2.2 Singularities of parametric offset curves 13.2.3 Approximations 13.3 Parametric offset surfaces 3.1 Differential geometry of parametric offset surfaces 13.3.2 Singularities of parametric offse et surfaces 13.3.3 Tracing algorithm 13 13.3.4 Self-intersections of offsets of explicit quadratic surfaces 13.3.5 Approximations Bibliography Reading in the Textbook Chapter 11, pp. 293-353
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 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) of Bal Tool Driving Plane Figure 13. 1: NC machining
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
• 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. e feature recognition through construction of skeleto 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 3D curve [18] geodesic offsets when the progenitor is curve on a surface[20[25][11]
• 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 3D curve [18]. • geodesic offsets when the progenitor is curve on a surface [20] [25] [11]. 4
13.2 Parametric offset curves 13.2.1 Differential geometry of parametric offset curves lanar parametric curve r(t)is given by r(t)={x(t,y(切),t∈[0.,1 where a and y are differentiable functions of a parameter t The unit normal vector of a plane curve, which is orthogonal to t, is given by n=t×ez ((t),-i(t) √x(t)+y2(t) where e2=(0, 0, 1) is a unit vector perpendicular to the plane of the curve, see Figure For a plane curve, the Frenet formulae reduce to dt ds ds where K is the signed curvature of the curve given by 2)2 (13.4) where v=r(t)I is the parametric speed. The curvature k of a curve at point P is positive when the direction of n and PC are opposite where C is the center of the curvature of the curve at point P, see Figure 13.6 y (t) h Figure 13.6: Definitions of unit tangent and normal vectors
13.2 Parametric offset curves 13.2.1 Differential geometry of parametric offset curves • A planar parametric curve r(t) is given by r(t) = [x(t), y(t)] , t ∈ [0, 1] (13.1) where x and y are differentiable functions of a parameter t. • The unit normal vector of a plane curve, which is orthogonal to t, is given by n = t × ez = (y˙(t), −x˙(t)) p x˙ 2 (t) + y˙ 2 (t) (13.2) where ez = (0, 0, 1) is a unit vector perpendicular to the plane of the curve, see Figure 13.6. • For a plane curve, the Frenet formulae reduce to dt ds = −κn, dn ds = κt (13.3) where κ is the signed curvature of the curve given by κ = (r˙ × ¨r) · ez v 3 = x˙y¨ − y˙x¨ (x˙ 2 + y˙ 2) 3 2 (13.4) where v = |r˙(t)| is the parametric speed. The curvature κ of a curve at point P is positive when the direction of n and P~C are opposite where C is the center of the curvature of the curve at point P, see Figure 13.6. C P r(t) n t x y ez Figure 13.6: Definitions of unit tangent and normal vectors. 5