cstypes

bezier

Type bezier specifies a sequence of Bezier curves of similar degree, each
beginning where the prior terminated.  If there are multiple curves in the
sequence, then the initial control point of a latter curve is not explicitly
specified since it is equal to the last control point of the prior curve.

Consider the following example.

    curv2 -6 -5 -4 -3 -2 -1 -6

When the degree is 3, the above specifies two Bézier curves: C₁=(-6,-5,-4,-3) and
C₂=(-3,-2,-1,-6).

The Bézier curve functions are taken to be branches of a peicewise function C
obtianed by dividing the parameter space into regions and mapping each region
to a different curve.  For example,

    parm u 0.00 1.00 2.00

specifies two intervals, [0,1] and [1,2].  If the parameter is between 0 and 1,
it will be mapped via the first curve C₁ and if it is between 1 and 2, it will
be mapped via the second curve C₂.  In order to maintain continuity, an affine
transformation will be applied first to obtain a value between 0 and 1 so that
C(1.5) = C₂(0.5).