Beta-Splines: A Flexible Model - Bezier and Splines in Image Processing and Machine Vision - page 134

Double Vertices

Double vertices mean a vertex is considered twice to generate a piece of curve.
So, when V 0 and V m are considered twice, we get two more pieces of the
complete curve, one at the beginning and the other at the terminal end. This
means, instead of P 2 ( t ), P 3 ( t ),

,
P m− 1 ( t ), P m ( t ). Additional pieces of the curve are then P 1 ( t )and P m ( t ).
These two pieces of the curve are given by

··· , P m− 1 ( t ), we get P 1 ( t ), P 2 ( t ), P 3 ( t ), ···

P 1 ( t )=( w − 2 ( t )+ w − 1 ( t )) V 0 + w 0 ( t ) V 1 + w 1 ( t ) V 2 ,
P m ( t )= w − 2 ( t ) V m− 2 + w − 1 ( t ) V m− 1 +( w 0 ( t )+ w 1 ( t )) V m .

With these two additional pieces of curves, β -spline curve starts at

2
γ 0 ) V 0 +

2
γ 0 V 1 ,

P 1 (0) = (1 −

and ends at

P m (1) = 2 α 1 ,m
γ m

2 α 1 ,m
γ m

.V m− 1 +(1 − ) V m .

2
γ 0

along the vector from V 0 to V 1 and the The initial point of the curve is

2 α 1 ,m terminal point is (1

γ m ) along the vector from V m− 1 to V m . At both the
end points, the curve is tangent to the control polygon.
The first derivative vector at the end points can be easily shown to be

−

P 1 (0) = 6 α 1 , 0 ( V 1 − V 0 /γ 0 ) ,

P m (1) = 6 α 1 ,m ( V m −

V m− 1 /γ m ) ,

while the second derivative at each end point of the curve can be derived to
be

P 1 (0) = 6(2 α 1 , 0 + α 2 , 0 )( V 1 − V 0 ) /γ 0 ) ,

P m (1) = 6(2 α 1 ,m + α 2 ,m )( V m− 1 −

V m ) /γ m ) .
From the above expressions for the first and second derivative vectors at each
end point of the curve, we get after some algebraic manipulations

P 1 (0) = P 1 (0) , {

(2 α 1 , 0 + α 2 , 0 ) /α 1 , 0 }

P m (1) = P m (1) .

(2 α 1 ,m + α 2 ,m ) /α 1 ,m }

{−

Triple Vertices

For the use of double vertices, we get two extra pieces of curves at each end
of the complete curve and these extra pieces of curves are P 1 ( t )and P m ( t ).
When we use triple vertices, we get one more piece of curve at each end, i.e.,
we get P 0 ( t )and P m +1 ( t ). These pieces of curves are given by