Re: bezier/B-splines with endpoint conditions


I have two pieces of curve (c0 and c2), curve c0 ends in point p0,
curve c2 ends in point p2. At these endpoints I have information about
the k first derivatives.

As stated earlier in the discussion, i am looking for a curve which
connects c0 and c2 by means of piece of curve c1 between p0 and p2 such
that it satisfies the boundary conditions (i.e., agree on the
derivatives at the endpoints).

Moreover, I want to be sure that c1 does not intersect with anything
else which might be there, except the points p0 and p2. With anything
else, I mean anything outside the circle with center p1 and going
through p0 and p2.

Reading more on this on the web (and the discussion so far) it seems
that a curve can easily be found satisfying the boundary conditions:

Using Hermite splines you can use the derivatives at the endpoints to
constrain the possible coefficients of the polynomials; using Bezier
curves you have to use the control points. There seems to be a close
connection between these two approaches in case of cubic polynomials.
The first derivative conditions in the two endpoints in the Hermite
approach give rise to two extra control points in the Bezier approach;
and vice versa. However, it is unclear how to translate the higher
derivative conditions in terms of extra control points.

Meanwhile, I also found out that the B-spline approach has the convex
hull property, which if I understand it correctly, says that the
obtained curve lies withing the convex hull of the control points. The
Hermite splines do not seem to have this property.

The convex hull property still is not sufficient to guarantee that the
curve does not intersect other parts since it might be the case that
some of the control points lie outside the circle.

Concluding, I now know how to satsify easily the boudary conditions
using Hermite splines; but it is less clear how to do this using
control points. On the other hand the control points give me a way of
computing a bounding box of the curve, while in the Hermitian approach
this is not clear.

-Sirolf

.

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