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Derive: d = ½ 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(du^{2} + dv^{2}) 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(du^{2} + dv^{2}) is
small and can be neglected. Thus we get

Now

and

which becomes

Substituting 6) into 5) we get

Thus d = ½ where

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