13.3 Parametric offset surfaces 13.3.1 Differential geometry of parametric offset surfaces ● Definition A parametric offset surface r(u, v) is a continuum of all points at a constant distance d along normal to another parametric surface r(u, v) and defined as r(u, v)=r(u, u)+dn(u, u) (13.15) where d may be a positive or negative real number and n is the unit normal vector of r(u,) Sign convention for normal curvature The normal curvature is typically considered positive if its associated center of curvature is opposite to the direction of the surface normal Relation between n and n] If n(u, u) is the unit normal vector of r(u, u), then the relation between n and n is given Sn=(1+dmaa)(1+domin)s where S=furu and S=ru xrul or expanding the right hand side of equation(13 16) and using the definitions of Gaussian curvature K and mean curvature h min,H (13.17) quation(13 16)can be rewritten as follows Sn=S(1+2Hd+Kd If we take the norm of equation(13.16), we obtain S=SI(1 +dkmaz)(1+domin ) 13.19) and substituting S into equation(13. 16) yields n≈(1+dma)(1+ demit l(1+dkmaz)(1+drmir ainn From this relation n and f are collinear but may be directed in opposite directions, if domar <-1 or dOmin < -1. This occurs when the offset is taken towards the concave region of the progenitor Offsetting towards concave region of a surface is equivalent to taking the offset d >0 where kmin <0 and d<o where kmax >0, provided the above sign convention is used13.3 Parametric offset surfaces 13.3.1 Differential geometry of parametric offset surfaces • Definition A parametric offset surface ˆr(u, v) is a continuum of all points at a constant distance d along normal to another parametric surface r(u, v) and defined as ˆr(u, v) = r(u, v) + dn(u, v) (13.15) where d may be a positive or negative real number and n is the unit normal vector of r(u, v). • Sign convention for normal curvature The normal curvature is typically considered positive if its associated center of curvature is opposite to the direction of the surface normal. • Relation between n and nˆ [28] If nˆ(u, v) is the unit normal vector of ˆr(u, v), then the relation between n and nˆ is given by Sˆnˆ = (1 + dκmax)(1 + dκmin)Sn (13.16) where Sˆ = |ˆru׈rv| and S = |ru×rv| or expanding the right hand side of equation (13.16) and using the definitions of Gaussian curvature K and mean curvature H K = κmaxκmin, H = κmax + κmin 2 (13.17) equation (13.16) can be rewritten as follows: Sˆnˆ = S(1 + 2Hd + Kd 2 )n (13.18) If we take the norm of equation (13.16), we obtain Sˆ = S|(1 + dκmax)(1 + dκmin)| (13.19) and substituting Sˆ into equation (13.16) yields nˆ = (1 + dκmax)(1 + dκmin) |(1 + dκmax)(1 + dκmin)| n (13.20) From this relation n and nˆ are collinear but may be directed in opposite directions, if dκmax < −1 or dκmin < −1. This occurs when the offset is taken towards the concave region of the progenitor. • Offsetting towards concave region of a surface is equivalent to taking the offset d > 0 where κmin < 0 and d < 0 where κmax > 0, provided the above sign convention is used. 10