Control Point

which means that a control point of even degree can either be on a knot, or in the middle of a knot span.

Each basis function corresponds to a certain control point. There are n basis functions and n control points in a B-SPLine curve.

In figure 2.27, control points are represented both in parameter and physical Space. Each point controls a specific basis function. This property also applies for multiple directions. Every control point of the surface or the solid is tensor product of a control point in directions ξ, η and ζ. By extension, the corresponding B-SPLine is tensor product of the basis functions.

The first and last control points are interpolatory to the curve. Any internal control point corresponding to C^{0} continuous basis function is also interpolatory to the curve.

In figure 2.28, the first and the last control point, which have C^{-1} continuity, are interpolatory to the curve. This can be explained with the help of the equation of the curve:

For ξ = 0, it applies that:

where,

so,

And for ξ = 3:

so,

Likewise, the internal control point, with C^{0} continuity across ξ = 2 is interpolatory to the curve because:

so,

Observe that both the form of the curve and the form of the basis functions indicate that this geometry could be represented by two different sets of knot vectors and control points, with absolutely no deflections from the current representation. This will be examined thoroughly later.
Interpolation also applies for surfaces and solids, when appropriately reduced continuity is used for all directions at a knot. C^{-1} continuity is required for external knots and C^{0} for internal.

B-SPLine curves possess strong convex hull property. The convex hull of the curve is defined as the sum of the convex hulls of p+1 consecutive control points. The curve is always contained in the convex hull.

The curve in figure 2.31 has a degree of p=2. The convex hull is formed by connecting each control point with the p=2 successive ones. As we can easily see in the figure, the union of the convex hulls contains the curve. The convex hull is a way to assume the general form of a B-SPLine curve.

The control polygon represents a piecewise linear approximation to the curve. Due to convex hull properties, refinement by knot insertion or order elevation brings the control polygon closer to the curve.

In figure 2.35 a curve of degree p=3 is designed. The control polygon already represents a linear approximation to the curve. When consecutive h- or p- refinements are applied, the control polygon is brought even closer to the curve. Refined control polygons provide a general idea of the form of the curve. This property also applies for multiple directions.

For example, in figure 2.36, the refinements, that were made, brought the control net closer to the surface.

It is possible to use multiple control points with the same coordinates. This can prove to be very useful.

In figure 2.37.a, a quadratic curve with a double control point is designed. The curve is interpolatory at these points and a sharp edge is formed. This is explained in figure 2.37.b, where the convex hull of the curve is designed. The curve is always contained in the convex hull, therefore a sharp edge has to be formed exactly at the double point coordinates. Inductively, this applies when p coincident control points are used in a curve of polynomial degree p.