The Cubic Interpolant
Last Updated: July 14, 1997
One technique that has resulted in higher quality surfaces is
the cubic interpolant. This is a cubic surface element that matches
second order data prescribed at the data points (second order == position,
first derivatives, second derivatives). In order to use this patch,
we must relax our continuity conditions from tangent plane continuity to
approximate continuity (ie, where the surface normals of two patches
along their common boundaries are nearly equal).
Despite having lower
order continuity, the resulting surfaces cubic interpolant appear
far smoother
than the corresponding surfaces contructed by the C1 schemes.
To construct cubic interpolant surfaces we need second order data
at the vertices. When approximating known functions, this data
is reasonable to obtain. However, when fitting surfaces to
scattered data, we are normally lacking both the first and second
derivatives at the data points. Thus, one of the research topic
are finding methods for estimating this derivative information
from data points containing only positional information.
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