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8.5.4 Examples
To illustrate constant curvature contouring, we
used a wave-like bicubic integral Bézier surface patch (see Fig.
8.11). The boundary twelve control points are coplanar so that
the boundary curves form a square. The remaining four interior
control points are not on the same plane. The control points are given
as follows:
Therefore the surface is anti-symmetric with respect to
.
The wave-like surface can also be expressed in a graph form
as
with
(8.132)
where
and
,
. Although the wireframe of
the surface looks simple (see Fig. 8.11), the surface is
rich in its variety of differential geometry properties. The wave-like
surface has four spherical umbilics at
,
,
,
with principal curvature values
,
,
,
, and one flat umbilic at
. None of them are local extrema according to the
criterion 9.5.1.
To display the curvature of the subdivided surface clearly, we
assigned discrete color to each closed region based on curvature
level. The level was determined by taking the average value of the
curvature values of the contour curves excluding the boundary lines
which form the closed region. We assigned
,
and
to
the minimum, zero and maximum curvature values of the whole domain.
The color of the curvature values in between is linearly
interpolated.