As well as for the truncated cone paraboloid datasets

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Unformatted text preview: x ⎤ ⎡ 0 ⎤⎞ ⎜⎢ ⎢ ⎥⎟ N⎜ Cψ Sψ ty ⎥ ⎢0 ⎥ ⋅ mi − ⎢ 0 ⎥⎟ . Q(t ) = ∑ ⎜ (4) i =1 ⎢ ⎥ ⎢n zi ⎥ ⎟ ⎜ ⎢− Sθ CθSψ CθCψ 0 ⎥ ⎟ ⎢ ⎥⎟ ⎜0 0 0 1⎦ ⎣ 1 ⎦⎠ ⎝⎣ The equation (4) can be simplified turning the attention to the fact that the only need is to minimize the distance of the first and the second transformed coordinate from z axis, never minding of the third component of mi. Thus, the only first two components of the vector in (4) are considered, obtaining the following objective function: Q(ψ ,θ , t x , t y ) = ∑ [Cθ ⋅ m xi + SθSψ ⋅ m yi + SθCψ ⋅ m zi + t x ] + N 2 i =1 ∑ [Cψ ⋅ m N i =1 yi + Sψ ⋅ m zi + t y ] . 2 . (5) The method of Levenberg-Marquardt is applied to the numerical optimisation of this equation. Then, the best-fit parameters for the equation (5), is t = ψ , θ , 0 , t x , t y , 0 . [ The second step of our algorithm is the minimization of the vertical distance between the measured points and the nominal surface. The res...
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