If we make the spacing between the first (K+1) knots to be the same
as the spacing between the last (K+1) knots, where K is the order,
and
let the first (K-1) vertices overlap with the last (K-1) vertices, we
have a closed curve. This is called closed end condition. The curve
remains closed as long as the knot spacings are kept the same, and
the
vertices are overlapped. Furthermore, if the knots are wrapped around
and made into a periodic sequence, then the curve is a periodic
curve.
Note that the basis functions are connected in a smooth way. This
is why the curve ends are merged with C^{K-2} continuity,
where K is
the order.

Some designers prefer the curve interpolate the ends of the control
polygon by enforcing multiple knots with multiplicity equals to the
order, and call it open end condition. Furthermore, they also make
all
the interior knots to be evenly spaced. Other people, however, still
prefer uniform knot spacing, mostly for simplicity. Let us return our
attention to the curve space and start playing with vertices.

Suppose we line up K consecutive vertices, where K is the order. The
curve will contain a line segment. This is easy to see because the
convex
hull formed by these K vertices is squeezed into a line segment.
Next,
let us demonstrate the effect of multiple vertices. If we move these
vertices to make a triple vertex, the curve passes through it. In
general,
stacking up (K-1) vertices will effectively force a positional
interpolation.
But there is more. In addition to forcing an interpolation, it also
makes the curve contain linear segments. This is because the convex
hull formed by the (K-1) identical vertices plus the K-th one is also
squeezed into a line segment. Therefore when one stacks up vertices
to force an interpolation, be aware of this extra side effect that
may
or may not be desirable.

Consider a cubic curve with uniform knots. The same set of control
vertices can be used for a quadratic curve with uniform knots. Over
each interval we have exactly three basis functions, and the convex
hulls are formed by successive three control vertices. Notice that
these
curve segments are also known as conic sections. When the same set of
control vertices are used for a linear curve with uniform knots, the
curve coincides with the control polygon. This is a quartic curve
with
uniform knots. Over each interval we have exactly five basis
functions,
and the convex hulls are formed by five successive control vertices.
In all the cases, the affine invariance, convex hull property,
locality
properties, continuity and variation diminishing property, they all
hold.

Going downward with the degrees, let us see the cases of open end
condition, with uniform interior knots. You probably have noticed by
now that as we lower the
degrees,
the curve gets closer and closer to the control polygon and finally,
when it is linear case, the curve is identical to its control
polygon.