BUILT-IN MATHEMATICA SYMBOL

BSplineCurve

is a graphics primitive that represents a non-uniform rational B-spline curve with control points

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MORE INFORMATION

BSplineCurve can be used in both Graphics and Graphics3D (two- and three-dimensional graphics).

The positions of control points can be specified either in ordinary coordinates as or , or in scaled coordinates as Scaled or Scaled.

In two dimensions, Offset and ImageScaled can be used to specify coordinates.

The following options can be given:

SplineDegree	Automatic	degree of polynomial basis
SplineKnots	Automatic	knot sequence for spline
SplineWeights	Automatic	control point weights
SplineClosed	False	whether to make the spline closed

By default, BSplineCurve uses cubic splines.

The option setting SplineDegree->d specifies that the underlying polynomial basis should have maximal degree d.

By default, knots are chosen uniformly in parameter space, with additional knots added so that the curve starts at the first control point and ends at the last one.

With an explicit setting for SplineKnots, the degree of the polynomial basis is determined from the number of knots specified and the number of control points.

With the default setting SplineWeights->Automatic, all control points are chosen to have equal weights, corresponding to a polynomial B-spline curve.

Curve thickness can be specified using Thickness or AbsoluteThickness, as well as Thick and Thin.

Curve dashing can be specified using Dashing or AbsoluteDashing, as well as Dashed, Dotted, etc.

Curve shading or coloring can be specified using CMYKColor, GrayLevel, Hue, Opacity, or RGBColor.

Individual coordinates and lists of coordinates in BSplineCurve can be Dynamic objects.

EXAMPLES

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Basic Examples (1)

A B-spline curve and its control points in 2D:

A B-spline curve and its control points in 3D:

A B-spline curve and its control points in 2D:

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A B-spline curve and its control points in 3D:

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Scope (12)

A cubic B-spline curve:

B-spline curves with the same control points and different degrees:

By default, a B-spline curve is open:

A closed B-spline curve automatically adds the first control point at the end:

Knots can be explicitly specified to control the smoothness of a curve:

Weights can be specified to each point:

B-spline curves with different thicknesses:

Thickness in scaled size:

Thickness in printer's points:

Dashed curves:

Colored curves:

Use Scaled coordinates:

Use ImageScaled coordinates in 2D:

Use Offset coordinates in 2D:

Generalizations & Extensions (4)

By default, knots are generated in such a way that the curve is smooth overall:

By repeating knots, you can decrease the smoothness of the curve:

generates uniform knots, and the curve will not go through the endpoints:

Unclamped knots combined with SplineClosed will make a uniform periodic B-spline curve:

By default, all the control points have equal weights:

By giving more weight to a control point, the curve will be attracted to that point:

Applications (5)

By using weights, you can make a rational B-spline, such as a circle:

Choose six points to be interpolated:

Compute distances between control points:

Compute normalized parameters wrt the distances (chord length parametrization):

A cubic B-spline curve with clamped knots will be used:

Set up the square basis matrix to solve:

Solve the linear system to get control points:

Show the interpolating curve with the original data:

Choose 3D points to be interpolated:

Compute distances between control points:

Compute normalized parameters wrt the distances (chord length parametrization):

A cubic B-spline curve with clamped knots will be used:

Set up the square basis matrix to solve:

Solve the linear system to get control points:

Show the interpolating curve with the original data:

Sample a list of points to be approximated with random noise:

Use uniform parametrization:

Define a function to generate clamped knots for a given number of control points and degrees:

Define the basis matrix for least squares:

A cubic B-spline curve with 12 control points will be used for fitting:

Show the data with the curve:

The results with various numbers of control points:

The results with 12 control points and different degrees:

Linear transition from one B-spline curve to another:

Properties & Relations (6)

A B-spline curve with degree 1 is equivalent to a line:

A Bézier curve is affine invariant:

A B-spline curve lies in the union of convex hulls of subsets of control points:

In 3D, a B-spline curve with planar control points lies in the plane:

BSplineBasis can be used to build up B-spline curve objects:

The individual basis functions have bounded support:

Changing the knots affects the basis functions just as it does the BSplineCurve:

A B-spline curve generated from the average of two sets of control points:

The new curve is the average of two B-spline curves: