B-spline Guide, Meaning , Facts, Information and Description
In the mathematical subfield of numerical analysis a B-spline is a special spline curve. It is a linear combination of B-splines basis curves. B-splines are a generalization of the Bézier curves and can be further generalized to NURBS, allowing the accurate modelling of more general classes of geometry.The De Boor algorithm is a numerical stable way to evalute B-splines.
The term B-spline was coined by Isaac Jacob Schoenberg and is short for basis spline.
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2 Notes 3 Examples 4 See also 5 References 6 External links |
Given m+1 knots ti in [0,1] with
Definition
a B-spline of degree n is a parametric curve
- .
The m+1 basis B-splines of degree n can be defined using the Cox-de Boor recursion formula
When the B-spline is uniform the basis B-Splines for a given degree n are just shifted copies of each other. An alternative non recursive definition for the m+1 basis B-splines is
Uniform B-spline
with
with
where
is the truncated power function
When the number of knots is the same as the degree, the B-Spline degenerates into a Bezier curve. The shape of the basis functions is determined by the position of the knots. Scaling or translating the knot vector does not alter the basis functions.
The spline is contained in the convex hull of its control points.
A basis B spline of degree n
The constant B-spline is the most simple spline. It is defined on only one knot span and is not even continuous on the knots. It is a just indicator function for the different knot spans.
The linear B-spline is defined on two consecutive knot spans and is continuous on the knots, but not differentiable.
A B-spline formulation for a single segment can be written as:
An entire set of segments, m-2 curves () defined by m+1 control points (), as one B-spline in t would be defined as:
There are two types of B-spline - uniform and non-uniform. A non-uniform B-spline is a curve where the intervals between successive control points is not, or not necessarily, equal (the knot vector of interior knot spans are not equal). A common form is where intervals are successively reduced to zero, interpolating control points.Notes
is non-zero only in the interval [ti, ti+n+1] that is
In other words if we manipulate one control point we only change the local behaviour of the curve and not the global behaviour as with Bézier curves.Examples
Constant B-spline
Linear B-spline
Cubic B-Spline
where Si is the ith B-spline segment and P is the set of control points, segment i and k is the local control point index. A set of control points would be where the is weight, pulling the curve towards control point as it increases or moving the curve away as it decreases.
where i is the control point number and t is a global parameter giving knot values. This formulation expresses a B-spline curve as a linear combination of B-spline basis functions, hence the name.