Continuity algorithm:
Bézier curves can represent complex curves by increasing the degree
and thus the number of control points. Alternatively, complex curves
can be represented using composite curves, which can be formed by
joining several Bézier curves end to end. If this method is
adopted, the continuity between consecutive curves must be addressed.
One set of continuity conditions are the geometric continuity
conditions, designated by the letter
with an integer exponent. Position continuity, or
continuity, requires the endpoints of the two curves to coincide,
The superscripts denote the first and second curves.
Tangent continuity, or
continuity, requires
continuity
and in addition the tangents of the
curves to be in the same direction,
where
is the common unit tangent vector and
,
are the magnitude of
and
.
continuity is important in minimizing stress
concentrations in physical solids loaded with external forces and in
helping prevent flow separation in fluids.
Curvature continuity, or
continuity, requires
continuity
and in addition the center of
curvature to move continuously past the connection point [116],
where
is an arbitrary constant.
continuity is important
for aesthetic reasons and also for helping prevent fluid flow
separation.
More stringent continuity conditions are the parametric
continuity conditions, where
continuity requires the
th derivative (and all lower derivatives) of each curve to be equal at
the joining point. In other words,
Let us assume that the global parameter
, associated with the
-
segment of a composite degree
Bézier curve with local
parameter
(
), runs over the interval
,
. Then the
-
segment of a composite Bézier curve
is given by:
where the global parameter
and the local parameter
are related by,
If we denote
,
the
and
continuity conditions for the
-th and
+1-th segments of the composite
Bézier curve can be stated as [455,175]:
and
Figure 1.7 illustrates the connection of two
cubic Bézier curve segments at
.