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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.
Given m+1 knots ti in [0,1] with
a B-spline of degree n is a parametric curve
composed of basis B-splines of degree n
The Pi are called control points or de Boor points. A polygon can be constructed by connecting the de Boor points with lines, starting with P0 and finishing with Pn. This polygon is called the de Boor polygon.
The m+1 basis B-splines of degree n can be defined using the Cox-de Boor recursion formula
When the knots are equidistant we say the B-spline is uniform otherwise we call it non-uniform.
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
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 translatingEuclidean geometry In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. It can also be interpreted as the addition of a constant vector to every point, or as shifting the knot vector does not alter the basis functions.
The spline is contained in the convex hullIn mathematics, the convex hull for an object or a set of objects is the minimal convex set containing the given objects. It is the minimal convex set because the convex hull is a subset of any convex set which contains the given objects. The convex hull of its control points.
A basis B spline of degree n
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.
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 functionThis article is about the characteristic function in set theory. For characteristic function in probability theory see characteristic function In mathematical subfield of set theory, the indicator function or characteristic function is a function defined for the different knot spans.