Graphics Reference
In-Depth Information
11.15
Summary
It is worthwhile to summarize the main properties of the various types of parametric
curves we discussed in this chapter. First of all, recall that Bézier curves are really
special cases of B-splines, so that when one refers to “Bézier curves” and “B-spline
curves” one is really referring to their
representation
and not to the underlying curve.
They are simply different ways of controlling a curve and based on different basis
functions. One can convert between the two representations.
Differences between the Bézier and B-spline curves:
(1) For splines one needs to specify knots t
i
. Bézier curves do not need knots.
(2) One can force an order k B-spline to pass through a control point by giving
that point a multiplicity k - 1, although a cusp may result. Similarly, a knot
of order k - 1 will cause the B-spline to pass through the corresponding control
point.
(3) B-splines use fewer control points than the corresponding Bézier curve.
(4) The Bézier basis functions are easier to compute.
Similarities between Bézier and B-spline curves:
(1) The shape of the curves roughly follow the outlines of the control polygon,
although B-spline curves offer more control over the shape.
(2) B-splines can be made to start and end at control points like the Bézier curves.
(3) The tangents at the ends of the curve is determined by the slope of the secants at
the ends.
(4) They are symmetric.
(5) They are affinely invariant, but this is true for B-splines only if we use uniformly
spaced knots.
(6) They are invariant under affine parameter transformations.
(7) They satisfy the variation diminishing property.
Other properties of Bézier curves:
(1) They satisfy the convex hull property, that is, they lie in the convex hull of their
characteristic polygon.
(2) They interpolate the first and last control points.
(3) Their tangent vector at the beginning and end is parallel to the line between the
first and last two control points.
(4) They only have pseudo local control: Although changing any control point changes
the
entire
curve, the change in the curve drops off rapidly as one moves away
from the point that was changed.
Other properties of B-spline curves:
(1) They satisfy a strengthened version of the convex hull property, namely, they lie
inside the union of triangles defined from consecutive triples of control points.
See Figure 11.20.
