[Math] Finding a point on B-spline

algorithmsbezier-curvegeometryspline

I would like to find a point on B-spline.
For Bezier curves, only control points and degree of curve are needed find coordinates of point $u$,
$$ C(u) = \sum_{i=0}^n B_{n,i}(u) P_i $$
$$ B_{n,i}(u) = \frac {n!}{i! (n-i)!} u^i (1-u)^{n-i}$$
$$ 0 \le u \le 1 $$
For B-spline, how do I find (x,y) coordinates of a point on it? Do I need to consider knot values and fit points in calculation beside control points?

Best Answer

You need to consider knot values when evaluating a point on a b-spline curve from a given parameter as that it is part of the b-spline curve definition. You don't need to consider fit points which are merely used to construct the b-spline curve.

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[Math] Finding the mid-point of a B-spline curve.

My understanding is that a B-spline can have any number of control points. In the special case where the B-spline has exactly 4 control points, it is the same as a cubic Bezier curve with those same 4 control points.

So yes, De Casteljau's algorithm on the given control points works great:

given  midpoint  midpoint  midpoint
(0, 0)
      \
       (1/2, 0)
      /        \
(1, 0)          (3/4, 1/4)
      \        /          \
       (1, 1/2)            (3/4, 1/2)
      /        \          /
(1, 1)          (3/4, 3/4)
      \        /
       (1/2, 1)
      /
(0, 1)

So the given spline

starts at (0,0) at t=0,
goes through (3/4, 1/2) at t=1/2, and
ends up at (0,1) at t=1.

[Math] Convert a B-Spline into Bezier curves

What you need is something called "Boehm's algorithm" (after its originator, Wolfgang Boehm). It has a simple geometric interpretation, and drawing a few pictures should make it clear. There is a pretty good explanation (with pictures) in this document.

The algorithm is based on a process called "knot insertion". You keep inserting knots into the b-spline curve until each knot has multiplicity 3. Then, the b-spline control points of this refined curve give you the Bezier control points of its segments.

So, if you're writing code to do this, one approach is to write a knot insertion function first, and then call it repeatedly.

There is knot insertion code here.

Best Answer

Related Solutions

[Math] Finding the mid-point of a B-spline curve.

[Math] Convert a B-Spline into Bezier curves

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