Website owner: James Miller
Derive: d = ½ω where
Derivation. Let d = PQ
be the projection of PQ onto the unit normal
at P as shown in Fig.
1. Point P corresponds to
(u, v) and point Q corresponds to
(u + du, v + dv). So
By Taylor’s formula
where e(du2 + dv2) is an infinitesimal of higher order. Substituting 2) into 1) we get
or
Now
since the vector
lies in
the tangent plane and the term e(du2 + dv2) is
small and can be neglected. Thus we get
Now
and
which becomes
Substituting 6) into 5) we get
Thus d = ½ω where
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