Bxi y j cy2 ) = 0 ji =1 j =1 i =1 j =1 M N i =1 j =1 M NM N(16)The terms x, y, and z are defined as: xi m ; y=Mx=i =j =yj n ; z=Ni =1 j =z ( xi , y j ) mn (17)M NSubstituting PX-478 web Equation (17) in to the initial equation of Equation (16), the following equation might be obtained: z ( xi , y j ) mnM Na = z – bx – cy =i =1 j =- b i =1 mxiM-cj =yj n (18)NSubstituting Equation (18) in to the second and third equations of Equation (16), the following equation is often obtained: M N M N M N M N z( xi , y j ) xi – axi – bx2 – cxi y j = 0 i i =1 j =1 i =1 j =1 i =1 j =1 i =1 j =1 (19) M N M N M N M N z( x , y )y – ay – bx y – cy2 = 0 i j j j i j ji =1 j =1 i =1 j =1 i =1 j =1 i =1 j =Equation (20) is obtained by means of mathematical transformation: M N M x =N x i i =1 j =1 i i =1 M N N yj = M yj i =1 j =1 j =1 M N M N xi y j = xi y ji =1 j =1 i =1 j =(20)Substituting Equation (20) into Equation (19), the following equation is usually obtained: M N N M yj z( xi ,y j ) xi i =1 j =1 j =1 1 a= – b i=M – c N MN M N z( xi ,y j ) xi – MNxz b = i =1 j =1 M (21) two N xi – MNx2 i =1 M N z( xi ,y j )y j – MNyz c = i =1 j =1 N M y2 – MNy2 jj =Substituting Equation (21) into Equation (17), the least-squares datum plane can be determined, there’s a exceptional least-squares fitting datum within the sampling location, andMicromachines 2021, 12,eight ofthe corresponding least-squares datum plane equation might be obtained by giving the coordinate values of arbitrary points. 3.two. The Arithmetic Square Root Deviation Sa with the Machined Surface The arithmetic square root deviation Sa in the machined surface will be the arithmetic imply distance between the measured contour surface and also the datum plane along the z-axis within the sampling area. It may be expressed mathematically as [16]: Sa = 1 N M z a xi , y j MN j i =1 =1 (22)where, M and N would be the variety of sampling points inside the x-axis and y-axis
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