• Bulletin of the Korean Mathematical Society
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• v.56 no.3
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• pp.675-680
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• 2019

We, in this note, decompose the invariant Laplacian of the unit complex ball of ${\mathbb{C} }^n$ by the radial part and tangential part as ${\tilde{\Delta}}={\tilde{\Delta}}_{rad}+{\tilde{\Delta}}_{tan}$. We give several properties and interpretations involved with this decomposition.

• Journal of the Korean Society for Precision Engineering
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• v.23 no.9 s.186
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• pp.200-207
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• 2006

In this paper a new approach which combines implicit surface scheme and NURBS surface interpolation method is proposed in order to generate a complete surface model from unorganized point cloud data. In the method a base surface was generated by creating smooth implicit surface from the input point cloud data through which the actual surface would pass. The implicit surface was defined by a combination of shape functions including quadratic polynomial function, cubic polynomial functions and radial basis function using adaptive domain decomposition method. In this paper voxel data which can be extracted easily from the base implicit surface were used in order to generate rectangular net with good quality using the normal projection and smoothing scheme. After generating the interior points and tangential vectors in each rectangular region considering the required accuracy, the NURBS surface were constructed by interpolating the rectangular array of points using boundary tangential vectors which assure C$^1$ continuity between rectangular patches. The validity and effectiveness of this new approach was demonstrated by performing numerical experiments for the various types of point cloud data.