11.3.1 Differential geometry

A parametric offset surface is a continuum of all points at a constant distance along normal to another parametric surface and defined as

where may be a positive or negative real number and is the unit normal vector of (see (3.3)). As we mentioned at the beginning of this chapter we employ the convention that the normal curvature is positive if its associated center of curvature is opposite to the direction of the surface normal (see Fig. 3.7 (b) and Table 3.2).

If is the unit normal vector of , then the relation between and is given by [444]

where and or expanding the right hand side of (11.18) and using the definitions of Gaussian curvature (3.61) and mean curvature (3.62), (11.18) can be rewritten as follows:

If we take the norm of (11.18), we obtain

and substituting into (11.18) yields

If we denote by

then (11.21) gives

From this relation we can see that and are collinear but may be directed in opposite directions, if and have opposite signs. This occurs when the offset is taken towards the concave region of the progenitor surface. Offsetting towards the concave region of a surface is equivalent to taking the offset where and where , provided the above sign convention is used. In machining, the cutter radius must not exceed the smallest concave principal radius of curvature of the surface to avoid gouging [116].

Principal curvatures of the offset surface corresponding to and of the progenitor surface can be easily obtained by adding the signed offset distance to the signed radius of principal curvature of the progenitor surface and inverting it. Taking into account the direction in which the normal vector points, we have

Gaussian and mean curvatures of the offset surface are readily computed by

Given the offset distance , the critical curvature is defined as and three categories arise [94]:

: The normal vector of the progenitor and its offset are directed in the same direction, since . Also the sign of Gaussian and principal curvatures of the offset are the same that of the progenitor.

: The normal vector of the progenitor and its offset are directed in the opposite direction, since . Also the sign of Gaussian and the principal curvature of the offset corresponding to are opposite to that of the progenitor, while the sign of the principal curvature of the offset corresponding to is the same to that of the progenitor.

: The normal vector of the progenitor and its offset are directed in the same direction ( ), while the sign of both principal curvatures of the offset are opposite to that of the progenitor and thus the sign of Gauss curvature of the offset remains the same as that of the progenitor.