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