B-spline Curves: Moving Control Points

Moving control points is the most obvious way of changing the shape of a B-spline curve. The local modification scheme discussed in an earlier page states that changing the position of control point p_i only affects the curve p(u) on interval [u_i, u_i+p+1), where p is the degree of the B-spline curve. In fact, the shape change is translational in the direction of the control point being moved. More precisely, if control point p_i is moved in certain direction to a new position q_i, then point p(u), where u is in [u_i, u_i+p+1), will be moved in the same direction from p_i to q_i. However, the distance moved is different from point to point.

In the following figures, control point p₄ is moved from the position in the left figure to a new position in the middle figure and finally to the its final position in right figure. As you can see those points corresponding to knots (marked with little triangles) are moved in the same direction.

Some Useful Consequences from the Strong Convex Hull Property

1. Force a curve segment to become a line segment: let p+1 adjacent control points be collinear
   If u lies in knot span [u_i,u_i+1), then p(u) lies in the convex hull defined by p+1 control points p_i, p_i-1, ..., p_i-p+1, p_i-p. Since this is true for all u in that span, the curve segment on this knot span lies entirely in this convex hull. If all of these p+1 control points are collinear (i.e., on a straight line), the convex hull collapses to a line segment and so does the curve segment it contains. As a result, the curve segment on knot span [u_i,u_i+1) becomes a line segment. Note that in this case only this curve segment becomes a line segment. Other curve segments are still non-linear.
   

   Let us take a look at an example. Figures above are defined by n = 15 (i.e., 16 control points), p = 3 (degree 3) and m = 19 (i.e., 20 knots). Note that the first four and last four knots are clamped. The left figure on the top row is a given B-spline curve. Let us make p₉, p₈, p₇ and p₆ collinear. Therefore, the curve segment on [u₉,u₁₀) lies in the convex hull defined by p₉, p₈, p₇ and p₆. Since this convex hull is a line segment, the curve segment must also be a line segment. Keep in mind that the first four knots are clamped and hence the first three knot spans do not exists. Since [u₉,u₁₀) is the seventh knot span, the seventh segment collapses to the line segment p₇p₈. This is illustrated by the second and third figures of the top row and the left figure of the bottom row.

   But, why is the curve segment on [u₉,u₁₀) the only collapsed curve segment? Look at the second figure on the top row. The shaded area is the convex hull just before u enters [u₉,u₁₀). This convex hull is defined by control points p₈, p₇, p₆ and p₅, which is still not a line segment yet. Once u enters [u₉,u₁₀), the curve segment collapses (the third figure on the top row). Immediately after u leaving [u₉,u₁₀), a new convex hull appears (first figure on the bottom row).

   The second figure on the bottom row has p₅ collinear with its four successors. You see the curve contains one more line segment. The third figure on the bottom row has p₁₀ collinear with its five predecessors; however, it is moved to a position between p₈ and p₉. This would make part of the corresponding curve segment a straight line (why?)

2. Force a B-spline curve to pass a control point: let p adjacent control points be identical
   Consider control p_i. Since the curve segment on knot span [u_i, u_i+1) lies entirely in the convex hull defined by p_i, ..., p_i-p+1,p_i-p, if we make the first p control point identical (i.e., p_i = p_i-1 = ... = p_i-p+1), the convex hull collapse to a line segment p_i-pp_i and the curve must pass p_i.
   

   The curve in the left figure above is of degree 3. If p₅ is moved and made identical with p₆, the curve is moved closer to p₆ but not passing though it yet. This is shown in the middle figure. Note that the number of curve segments is not changed due to this move; however, the little triangle marker near to p₅ is move closer to p₆.

   If p₄ is moved and made identical to p₆ = p₅, the curve passes through p₆ and the point that corresponds to a knot becomes identical to control point p₄ due to this move.

3. Force a B-spline curve tangent to a leg of control polyline: let p_i-p, p_i-p+1 = p_i-p+2 = .... = p_i-1 = p_i and p_i+1 be collinear
   In the above, p adjacent control points are made identical. At that control point, the continuity is C⁰ because the curve has a cusp (see the right figure above). However, since with all simple knots a B-spline curve is C^p-1 at the knots and infinitely differentiable at other places, the curve is C^p-1 continuous at control point p_i (the collapsed control point renumbered to i) on leg p_i-pp_i and is also C^p-1 continuous at control point p_i on leg p_ip_i+1. Therefore, if we make control points p_i-p, p_i and p_i+1 collinear, as long as the two adjacent curve segments do not have a cusp at the knot, they are C^p-1 continuous at p_i.
   

   In the above figure figures, the curve is of degree 2. If we make control points 2, 3, 4 and 5 collinear and 3 and 4 identical, we have the right figure. The collinearity makes sure the curve segment lies on the line while the identical control points enforce the C^3-1 = C² continuous.