EN
GR

Control points as the Center of the Support

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

Control Point – Basis Function Correspondence

Each basis function corresponds to a certain control point. There are n basis functions and n control points in a B-SPLine curve.
 (a)
 (b)
 Figure 27. (a) Physical Space and (b) basis functions with the corresponding Control Points.
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. 

Interpolation to the Curve

The first and last control points are interpolatory to the curve. Any internal control point corresponding to C0 continuous basis function is also interpolatory to the curve.
 (a)
 (b)
 Figure 28. Control Point interpolation. (a) B-SPLine curve and (b) the reciprocal basis functions.
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 C0 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 C0 for internal.
 (a)
 (b)
 (c)
 Figure 29. (a) B-SPLine Surface and (b), (c) the corresponding basis functions in axes ξ,η.
 (a)
 (b)
 (c)
 (d)
 Figure 30. (a) Solid in the Physical Space
with (b), (c), (d) the associated basis functions in axes ξ,η,ζ.

​Convex Hull

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.
 Figure 31. Step-by-step convex hull creation for a B-SPLine curve.
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.

​Control Point Local Support

 (a)​​
 (b)​​
 Figure 32. Control Point Local support (a) in Physical and (b) in Parameter Space.
Local support of control points is also expanded by tensor product properties. The local support of a multi-directional control point is the tensor product of the respective supports.
 (a)​​
 (b)​​
 Figure 33. Local support of a 2D Control Point.
Surface (a) in Physical and (b) in Parameter Space.

 (a)​​
 (b)​​
 Figure 34. Local support of a 3D Control Point.
Solid (a) in Physical and (b) in Parameter Space.

​Control polygon approximation

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.
 Figure 35. Control Polygon approximation through Refinement.
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.
 Figure 36. Control Net approximation through Surface h-Refinement.
For example, in figure 2.36, the refinements, that were made, brought the control net closer to the surface.

Multiple control points

It is possible to use multiple control points with the same coordinates. This can prove to be very useful.
 (a)​​
 (b)​​
 Figure 37. (a) Physical Space and (b) Convex Hull creation for a curve
with two coincident Control Points (drawn in deep red).

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.