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Derive: d = ½ ole.gif where

ole1.gif


Derivation. Let d = PQ ole2.gif be the projection of PQ onto the unit normal ole3.gif at P as shown in Fig. 1. Point P corresponds to ole4.gif (u, v) and point Q corresponds to ole5.gif (u + du, v + dv). So


ole6.gif


By Taylor’s formula

 

ole7.gif


where e(du2 + dv2) is an infinitesimal of higher order. Substituting 2) into 1) we get


ole8.gif  


or

 

   ole9.gif


Now ole10.gif since the vector ole11.gif lies in the tangent plane and the term e(du2 + dv2) is small and can be neglected. Thus we get

ole12.gif

                                                            

ole13.gif


Now                                                     


             ole14.gif


and


             ole15.gif


which becomes


ole16.gif


Substituting 6) into 5) we get


ole17.gif


Thus d = ½ ole18.gif where


ole19.gif


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