200 COMPUTER AIDED ENGINEERING DESIGN
Hint: One may express nu and nv as respective linear combinations of ru and rv, that is,
nu = c 1 ru + c 2 rv
nv = d 1 ru + d 2 rv
wherec 1 ,c 2 ,d 1 and d 2 are scalars. Taking dot product of the above with ru and rv would yield the values of
c 1 ,c 2 ,d 1 and d 2 in terms of L,M and N. Elimination of the scalars leads to the Weingarten equations. To get
the third relation, consider the vector product of Weingarten relations and simplify.- Show that (nu×nv) = Kru×rv where K is the Gaussian curvature.
- Show that the following surfaces are not developable.
r(u,v) = u cos vi + u sin vj + sin nvk
r(u,v) = ui + vj + uvk