  We propose a general theory of estimating interpolation error for smooth
functions in two and three dimensions. In our theory, the error of
interpolation is bound in terms of the diameter of a simplex and a geometric
parameter. In the two-dimensional case, our geometric parameter is equivalent
to the circumradius of a triangle. In the three-dimensional case, our geometric
parameter also represents the flatness of a tetrahedron. Through the
introduction of the geometric parameter, the error estimates newly obtained can
be applied to cases that violate the maximum-angle condition.
